Classical Layer-Resolving Scheme for a System of Two Singularly Perturbed Time-Dependent Problems with Discontinuous Source Terms and Spatial Delay

Abstract

In this paper, a classical layer-resolving finite difference scheme is formulated to solve a system of two singularly perturbed time-dependent initial value problems with discontinuity occurring at $(\mathfrak{y},\mathfrak{t})$ in the source terms and Robin initial conditions. The delay term occurs in the spatial variable, and the leading term of the spatial derivative of each equation is multiplied by a distinct small positive perturbation parameter, inducing layer behaviors in the solution domain. Due to the presence of perturbation parameters, discontinuous source terms, and delay terms, initial and interior layers occur in the solution domain. In order to capture the abrupt change that occurs due to the behavior of these layers, the solution is further decomposed into smooth and singular components. Layer functions are also formulated in accordance with layer behavior. Analytical results and bounds of the solution and its components are derived. The formulation of a finite difference scheme involves discretization of temporal and spatial axes by uniform and piecewise uniform meshes, respectively. The formulated scheme achieves first-order convergence in both time and space. At last, to bolster the numerical scheme, example problems are computed to prove the efficacy and accuracy of our scheme.

1. Introduction

Solving singular perturbation problems (SPPs) is very tedious due to the presence of layer phenomena in the solution domain. They have a wide range of applications in the field of applied mathematics and engineering due to their complex boundary layer phenomena. Their application is prevalent in areas such as aerodynamics to examine boundary layer behavior; in fluid mechanics to model flow transitions and turbulence; in elasticity to study material deformation under stress; and also in the fields of optimal control theory, chemical processes, biological modeling, population dynamics, and ecology [1,2,3,4,5,6,7,8,9], demonstrating their wide-ranging relevance. Singularly perturbed delay differential equations (SPDDEs) have a wide range of applications in diverse fields [10,11,12,13,14].

The presence of discontinuity in the source term further complicates solving SPPs as the solution tends to behave unevenly or non-smoothly at the discontinuity point. This leads to the formation of an interior layer at the region in the domain where the discontinuity occurs. To tackle such difficulties in estimating the solution of SPPs, classical numerical schemes, such as fitted mesh methods [15], are modified by employing highly refined meshes (i.e., Shishkin mesh or Bakhvalov mesh) to provide numerical approximations by preserving the monotonicity of the original problem. Many specialized techniques have been introduced by researchers across the globe in order to overcome difficulties arising in solving SPPs.

Abagero et al. [16] employed a fitted nonstandard numerical method to solve SPPs with Robin-type boundary conditions and discontinuous source terms. Sahoo & Gupta [17] employed a first-order upwind scheme on a Shishkin mesh to solve convection–diffusion SPPs with discontinuous convective and source terms. They further employed the Richardson extrapolation scheme to enhance the order of convergence. Singh et al. [18] devised spline-based numerical techniques, such as the Crank–Nicolson scheme and trigonometric B-spline basis function, to solve two-parameter SPPs with discontinuity in the convection coefficient and source term. The Richardson extrapolation scheme was implemented by the authors to enhance the accuracy in the spatial direction. Chawla et al. [19] employed a backward difference scheme on Shishkin and Bakhvalov meshes to solve first-order singularly perturbed differential equations (SPDEs) with discontinuous source terms. Cen et al. [20] developed a quadratic B-spline collocation method on a Shishkin-type mesh to solve a semilinear reaction–diffusion SPP with a discontinuous source term. Soundararajan et al. [21] employed the backward Euler Method for time discretization and the streamline-diffusion Finite Element Method (FEM) on a Shishkin mesh to solve 1D parabolic SPPs with a discontinuous source term. Ajay Singh Rathore & Vembu Shanthi [22] proposed an exponentially fitted mesh method to solve a singularly perturbed Fredholm integro-differential equation with a discontinuous source term.

In [23], Ismail & Elmekkawy utilized the Restrictive Padé Approximation to solve a singularly perturbed first-order hyperbolic Partial Differential Equation, where a small perturbation is multiplied by the term containing the time derivative. The present paper differs from [23] in the treatment of the numerical analysis as well as in the problem formulation. The problems considered in this article involve solving a system of two singularly perturbed time-dependent initial value problems. These problems contain distinct perturbation parameters that multiply the spatial term involving the highest-order derivative. Additionally, they contain a spatial delay, and a discontinuity occurs at $(\mathfrak{y},\mathfrak{t})$ in the source terms. Initial and interior layers occur in the solution domain due to the presence of perturbation parameters, discontinuous source terms, and delay terms. This divergent problem structure and the complex behavior of the solution necessitate developing a classical layer-resolving finite difference scheme (FDS) [24] using a highly refined Shishkin mesh. This method is able to precisely capture the layer behavior of solutions and achieves first-order convergence in both time and space. Numerical illustrations confirm the effectiveness of the proposed method and clearly exhibit its first-order convergence. The subsequent section outlines the organizing framework of the paper. The notation and terminology are discussed in Section 2. Section 3 outlines the problem statement. The solution components and layer functions are discussed in Section 4. In Section 5, the bounds on the solution and its components are derived. Discrete solution components and error estimates are discussed in Section 6. A numerical scheme is constructed for example problems in Section 7.

2. Notation and Terminology

This section defines the key notation and terminology used throughout the paper to ensure clarity and consistency.

• $\epsilon$     singular perturbation parameter
• $\varkappa ,\mathfrak{t}$     independent variables
• $\overrightarrow{\vartheta }$    solution of the continuous problem (1)
• $\overrightarrow{\gamma }$    smooth component of the solution
• $\overrightarrow{\varsigma }$    singular component of the solution
• $\overrightarrow{G}$    solution of the discrete problem (29)
• $\overrightarrow{{\rm Y}}$    discrete smooth component of the solution
• $\overrightarrow{\zeta }$    discrete singular component of the solution
• ${\overrightarrow{\aleph }}^{\pm }$     continuous barrier function
• ${\overrightarrow{\theta }}^{\pm }$     discrete barrier function
• $D_{\varkappa }^{-},D_{\mathfrak{t}}^{-}$  backward difference operators
• $D_{\varkappa }^{+},D_{\mathfrak{t}}^{+}$  forward difference operators
• $(\mathfrak{y},\mathfrak{t})$   is the point at which discontinuity occurs in the domain $\tilde{\mathrm{\Lambda }}$

The jump at $(\mathfrak{y},\mathfrak{t})$ is denoted by a function $\overrightarrow{\mathfrak{g}}$ where $\left[\overrightarrow{\mathfrak{g}}\right]=\overrightarrow{\mathfrak{g}}(\mathfrak{y}+,\mathfrak{t})-\overrightarrow{\mathfrak{g}}(\mathfrak{y}-,\mathfrak{t})$

For any n-vector $\overrightarrow{U}$, the norm

$$||\overrightarrow{U}||=\underset{1\le \kappa \le n}{max}|U_{\kappa }|.$$

For any scalar-valued function $y\in \mathbb{C}\left(V\right),V$ is a closed set in $\overline{\tilde{\mathrm{\Lambda }}}$,

$$||y||=\underset{(\varkappa ,\mathfrak{t})\in V}{sup}|y(\varkappa ,\mathfrak{t})|.$$

For any vector-valued function y,

$$||\overrightarrow{y}||=\underset{1\le \kappa \le n}{max}|y_{\kappa }|.$$

For any function $\overrightarrow{W}$, the vector discrete maximum norm defined on the Shishkin mesh

$$||\overrightarrow{W}{||}_{{\overline{\tilde{\mathrm{\Lambda }}}}^{M,N}}=\underset{\iota ,j,\kappa }{max}{|W_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|}_{{\overline{\tilde{\mathrm{\Lambda }}}}^{M,N}}.$$

Domain defined for continuous case

$$\begin{array}{cc}\hfill \tilde{\mathrm{\Lambda }}& =(0,2]\times (0,\mathbb{T}],\mathrm{\Lambda }=[-1,0]\times (0,\mathbb{T}],\hfill \\ \hfill \overline{\tilde{\mathrm{\Lambda }}}& =\tilde{\mathrm{\Lambda }}\cup \Omega ,\Omega ={\Omega }_{\mathfrak{t}}\cup {\Omega }_{\varkappa }\mathrm{where}{\Omega }_{\mathfrak{t}}=0\times [0,\mathbb{T}],{\Omega }_{\varkappa }=[0,2]\times 0,\hfill \\ \hfill {\widehat{\mathrm{\Lambda }}}^{*}& ={\tilde{\mathrm{\Lambda }}}_1\cup {\tilde{\mathrm{\Lambda }}}_2\cup {\tilde{\mathrm{\Lambda }}}_3\cup {\tilde{\mathrm{\Lambda }}}_4,{\widehat{\mathrm{\Lambda }}}^{★}={\tilde{\mathrm{\Lambda }}}_5\cup {\tilde{\mathrm{\Lambda }}}_6\cup {\tilde{\mathrm{\Lambda }}}_7,\hfill \\ \hfill \overline{{\widehat{\mathrm{\Lambda }}}^{*}}& ={\overline{\tilde{\mathrm{\Lambda }}}}_1\cup {\overline{\tilde{\mathrm{\Lambda }}}}_2\cup {\overline{\tilde{\mathrm{\Lambda }}}}_3\cup {\overline{\tilde{\mathrm{\Lambda }}}}_4,\overline{{\widehat{\mathrm{\Lambda }}}^{★}}={\overline{\tilde{\mathrm{\Lambda }}}}_5\cup {\overline{\tilde{\mathrm{\Lambda }}}}_6\cup {\overline{\tilde{\mathrm{\Lambda }}}}_7,\hfill \end{array}$$

$$\begin{array}{cccccccc}\hfill {\tilde{\mathrm{\Lambda }}}_1& =(0,\mathfrak{y})\times (0,\mathbb{T}],\hfill & \hfill {\tilde{\mathrm{\Lambda }}}_2& =(\mathfrak{y},1]\times (0,\mathbb{T}],\hfill & \hfill {\tilde{\mathrm{\Lambda }}}_3& =(1,1+\mathfrak{y})\times (0,\mathbb{T}],\hfill & \hfill {\tilde{\mathrm{\Lambda }}}_4& =(1+\mathfrak{y},2]\times (0,\mathbb{T}],\hfill \\ \hfill {\tilde{\mathrm{\Lambda }}}_5& =(0,1]\times (0,\mathbb{T}],\hfill & \hfill {\tilde{\mathrm{\Lambda }}}_6& =(1,\mathfrak{y})\times (0,\mathbb{T}],\hfill & \hfill {\tilde{\mathrm{\Lambda }}}_7& =(d,2]\times (0,\mathbb{T}],\hfill & \hfill {\tilde{\mathrm{\Lambda }}}_8& =(1,2]\times (0,\mathbb{T}],\hfill \\ \hfill {\overline{\tilde{\mathrm{\Lambda }}}}_1& =[0,\mathfrak{y}]\times [0,\mathbb{T}],\hfill & \hfill {\overline{\tilde{\mathrm{\Lambda }}}}_2& =[\mathfrak{y},1]\times [0,\mathbb{T}],\hfill & \hfill {\overline{\tilde{\mathrm{\Lambda }}}}_3& =[1,1+\mathfrak{y}]\times [0,\mathbb{T}],\hfill & \hfill {\overline{\tilde{\mathrm{\Lambda }}}}_4& =[1+\mathfrak{y},2]\times [0,\mathbb{T}],\hfill \\ \hfill {\overline{\tilde{\mathrm{\Lambda }}}}_5& =[0,1]\times [0,\mathbb{T}],\hfill & \hfill {\overline{\tilde{\mathrm{\Lambda }}}}_6& =[1,\mathfrak{y}]\times [0,\mathbb{T}],\hfill & \hfill {\overline{\tilde{\mathrm{\Lambda }}}}_7& =[\mathfrak{y},2]\times [0,\mathbb{T}],\hfill & \hfill {\overline{\tilde{\mathrm{\Lambda }}}}_8& =[1,2]\times [0,\mathbb{T}].\hfill \end{array}$$

Domain defined for discrete case

Temporal domain

$(0,\mathbb{T}]$: ${\tilde{\mathrm{\Lambda }}}_{\mathfrak{t}}^M={\left\{{\mathfrak{t}}_{\kappa }\right\}}_{\kappa =1}^M$,  $[0,\mathbb{T}]$: ${\overline{\tilde{\mathrm{\Lambda }}}}_{\mathfrak{t}}^M={\left\{{\mathfrak{t}}_{\kappa }\right\}}_{\kappa =0}^M.$

Spatial domain

$(0,2]$: ${\tilde{\mathrm{\Lambda }}}_{\varkappa }^N={\left\{{\varkappa }_j\right\}}_{j=1}^N$,  $[0,2]$: ${\overline{\tilde{\mathrm{\Lambda }}}}_{\varkappa }^N={\left\{{\varkappa }_j\right\}}_{j=0}^N$.

The presence of discontinuity $(\mathfrak{y},\mathfrak{t})$ in domain $\tilde{\mathrm{\Lambda }}$ necessitates the consideration of two different cases. In Case 1, the discontinuity occurs in the interval ${\tilde{\mathrm{\Lambda }}}_5$, and, in Case 2, the discontinuity occurs in the interval ${\tilde{\mathrm{\Lambda }}}_8$.

Case 1: Here, ${\varkappa }_{\frac{N}{4}}=\mathfrak{y}$, ${\varkappa }_{\frac{N}{2}}=1$, and ${\varkappa }_{\frac{3N}{4}}=1+\mathfrak{y},$

$$\begin{array}{cc}\hfill {\tilde{\mathrm{\Lambda }}}_{{\varkappa }_1}^N& ={\left\{{\varkappa }_j\right\}}_{j=1}^{\frac{N}{4}-1},{\tilde{\mathrm{\Lambda }}}_{{\varkappa }_2}^N={\left\{{\varkappa }_j\right\}}_{j=\frac{N}{4}+1}^{\frac{N}{2}},{\tilde{\mathrm{\Lambda }}}_{{\varkappa }_3}^N={\left\{{\varkappa }_j\right\}}_{j=\frac{N}{2}+1}^{\frac{3N}{4}-1},{\tilde{\mathrm{\Lambda }}}_{{\varkappa }_4}^N={\left\{{\varkappa }_j\right\}}_{j=\frac{3N}{4}+1}^N,\hfill \\ \hfill {\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_1}^N& ={\left\{{\varkappa }_j\right\}}_{j=0}^{\frac{N}{4}},{\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_2}^N={\left\{{\varkappa }_j\right\}}_{j=\frac{N}{4}}^{\frac{N}{2}},{\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_3}^N={\left\{{\varkappa }_j\right\}}_{j=\frac{N}{2}}^{\frac{3N}{4}},{\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_4}^N={\left\{{\varkappa }_j\right\}}_{j=\frac{3N}{4}}^N,\hfill \\ \hfill {\tilde{\mathrm{\Lambda }}}_{\varkappa }^N& ={\tilde{\mathrm{\Lambda }}}_{{\varkappa }_1}^N\cup {\tilde{\mathrm{\Lambda }}}_{{\varkappa }_2}^N\cup {\tilde{\mathrm{\Lambda }}}_{{\varkappa }_3}^N\cup {\tilde{\mathrm{\Lambda }}}_{{\varkappa }_4}^N,{\overline{\tilde{\mathrm{\Lambda }}}}_{\varkappa }^N={\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_1}^N\cup {\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_2}^N\cup {\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_3}^N\cup {\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_4}^N,\hfill \\ \hfill {\tilde{\mathrm{\Lambda }}}^{M,N}& ={\tilde{\mathrm{\Lambda }}}_{\mathfrak{t}}^M\times {\tilde{\mathrm{\Lambda }}}_{\varkappa }^N,{\overline{\tilde{\mathrm{\Lambda }}}}^{M,N}={\overline{\tilde{\mathrm{\Lambda }}}}_{\mathfrak{t}}^M\times {\overline{\tilde{\mathrm{\Lambda }}}}_{\varkappa }^N,\mathrm{and}{\Omega }^{M,N}=\Omega \cap {\overline{\tilde{\mathrm{\Lambda }}}}^{M,N},\hfill \\ \hfill {\tilde{\mathrm{\Lambda }}}_1^{M,N}& ={\tilde{\mathrm{\Lambda }}}_{\mathfrak{t}}^M\times {\tilde{\mathrm{\Lambda }}}_{{\varkappa }_1}^N,{\overline{\tilde{\mathrm{\Lambda }}}}_1^{M,N}={\overline{\tilde{\mathrm{\Lambda }}}}_{\mathfrak{t}}^M\times {\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_1}^N,\hfill \\ \hfill {\tilde{\mathrm{\Lambda }}}_2^{M,N}& ={\tilde{\mathrm{\Lambda }}}_{\mathfrak{t}}^M\times {\tilde{\mathrm{\Lambda }}}_{{\varkappa }_2}^N,{\overline{\tilde{\mathrm{\Lambda }}}}_2^{M,N}={\overline{\tilde{\mathrm{\Lambda }}}}_{\mathfrak{t}}^M\times {\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_2}^N,\hfill \\ \hfill {\tilde{\mathrm{\Lambda }}}_3^{M,N}& ={\tilde{\mathrm{\Lambda }}}_{\mathfrak{t}}^M\times {\tilde{\mathrm{\Lambda }}}_{{\varkappa }_3}^N,{\overline{\tilde{\mathrm{\Lambda }}}}_3^{M,N}={\overline{\tilde{\mathrm{\Lambda }}}}_{\mathfrak{t}}^M\times {\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_3}^N,\hfill \\ \hfill {\tilde{\mathrm{\Lambda }}}_4^{M,N}& ={\tilde{\mathrm{\Lambda }}}_{\mathfrak{t}}^{\mathbb{M}}\times {\tilde{\mathrm{\Lambda }}}_{{\varkappa }_4}^N,{\overline{\tilde{\mathrm{\Lambda }}}}_4^{M,N}={\overline{\tilde{\mathrm{\Lambda }}}}_{\mathfrak{t}}^M\times {\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_4}^N.\hfill \end{array}$$

Case 2: Here, ${\varkappa }_{\frac{N}{3}}=1$ and ${\varkappa }_{\frac{2N}{3}}=\mathfrak{y}$,

$$\begin{array}{cc}\hfill {\tilde{\mathrm{\Lambda }}}_{\varkappa }^N& ={\tilde{\mathrm{\Lambda }}}_{{\varkappa }_5}^N\cup {\tilde{\mathrm{\Lambda }}}_{{\varkappa }_6}^N\cup {\tilde{\mathrm{\Lambda }}}_{{\varkappa }_7}^N\mathrm{where}{\tilde{\mathrm{\Lambda }}}_{{\varkappa }_5}^N={\left\{{\varkappa }_j\right\}}_{j=1}^{\frac{N}{3}},{\tilde{\mathrm{\Lambda }}}_{{\varkappa }_6}^N={\left\{{\varkappa }_j\right\}}_{j=\frac{N}{3}+1}^{\frac{2N}{3}-1},{\tilde{\mathrm{\Lambda }}}_{{\varkappa }_7}^N={\left\{{\varkappa }_j\right\}}_{j=\frac{2N}{3}+1}^N,\hfill \\ \hfill {\overline{\tilde{\mathrm{\Lambda }}}}_{\varkappa }^N& ={\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_5}^N\cup {\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_6}^N\cup {\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_7}^N\mathrm{where}{\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_5}^N={\left\{{\varkappa }_j\right\}}_{j=0}^{\frac{N}{3}},{\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_6}^N={\left\{{\varkappa }_j\right\}}_{j=\frac{N}{3}}^{\frac{2N}{3}},{\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_7}^N={\left\{{\varkappa }_j\right\}}_{j=\frac{2N}{3}}^N,\hfill \\ \hfill {\overline{\tilde{\mathrm{\Lambda }}}}^{M,N}& ={\overline{\tilde{\mathrm{\Lambda }}}}_{\mathfrak{t}}^M\times {\overline{\tilde{\mathrm{\Lambda }}}}_{\varkappa }^N,{\tilde{\mathrm{\Lambda }}}^{M,N}={\tilde{\mathrm{\Lambda }}}_{\mathfrak{t}}^M\times {\tilde{\mathrm{\Lambda }}}_{\varkappa }^N,\mathrm{and}{\Omega }^{M,N}=\Omega \cap {\overline{\tilde{\mathrm{\Lambda }}}}^{M,N},\hfill \\ \hfill {\tilde{\mathrm{\Lambda }}}_5^{M,N}& ={\tilde{\mathrm{\Lambda }}}_{\mathfrak{t}}^M\times {\tilde{\mathrm{\Lambda }}}_{{\varkappa }_5}^N,{\overline{\tilde{\mathrm{\Lambda }}}}_5^{M,N}={\overline{\tilde{\mathrm{\Lambda }}}}_{\mathfrak{t}}^M\times {\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_5}^N,\hfill \\ \hfill {\tilde{\mathrm{\Lambda }}}_6^{M,N}& ={\tilde{\mathrm{\Lambda }}}_{\mathfrak{t}}^M\times {\tilde{\mathrm{\Lambda }}}_{{\varkappa }_6}^N,{\overline{\tilde{\mathrm{\Lambda }}}}_6^{M,N}={\overline{\tilde{\mathrm{\Lambda }}}}_{\mathfrak{t}}^M\times {\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_6}^N,\hfill \\ \hfill {\tilde{\mathrm{\Lambda }}}_7^{M,N}& ={\tilde{\mathrm{\Lambda }}}_{\mathfrak{t}}^M\times {\tilde{\mathrm{\Lambda }}}_{{\varkappa }_7}^N,{\overline{\tilde{\mathrm{\Lambda }}}}_7^{M,N}={\overline{\tilde{\mathrm{\Lambda }}}}_{\mathfrak{t}}^M\times {\overline{\tilde{\mathrm{\Lambda }}}}_{{\varkappa }_7}^N.\hfill \end{array}$$

3. The Problem Statement

In this section, we consider the following system of two singularly perturbed time-dependent initial value problems with spatial delay and discontinuity source terms.

$$\overrightarrow{\mathfrak{L}}\overrightarrow{\vartheta }(\varkappa ,\mathfrak{t})=(\frac{\partial \overrightarrow{\vartheta }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\vartheta }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\vartheta })(\varkappa ,\mathfrak{t})+\mathbb{S}(\varkappa ,\mathfrak{t})\overrightarrow{\vartheta }(\varkappa -1,\mathfrak{t})=\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_1\cup {\tilde{\mathrm{\Lambda }}}_7$$

(1)

with Robin initial conditions in space variable and Dirichlet initial condition in time variable

$$\overrightarrow{\xi }\overrightarrow{\vartheta }(\varkappa ,\mathfrak{t})=\overrightarrow{\vartheta }(\varkappa ,\mathfrak{t})-\mathbb{E}\frac{\partial \overrightarrow{\vartheta }}{\partial \varkappa }(\varkappa ,\mathfrak{t})={\overrightarrow{\varpi }}_L(\varkappa ,\mathfrak{t})on\mathrm{\Lambda }$$

(2)

$$\overrightarrow{\vartheta }(\varkappa ,0)={\overrightarrow{\varpi }}_B\left(\varkappa \right)on{\Omega }_{\varkappa }.$$

(3)

For all $(\varkappa ,\mathfrak{t})\in \overline{\tilde{\mathrm{\Lambda }}}$,$\overrightarrow{\vartheta }(\varkappa ,\mathfrak{t})={({\vartheta }_1(\varkappa ,\mathfrak{t}),{\vartheta }_2(\varkappa ,\mathfrak{t}))}^T$,$\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})={({\mathfrak{f}}_1(\varkappa ,\mathfrak{t}),{\mathfrak{f}}_2(\varkappa ,\mathfrak{t}))}^T$,$\mathbb{E}=diag\left(\overrightarrow{\epsilon }\right),\overrightarrow{\epsilon }=({\epsilon }_1,{\epsilon }_2)$ with $0<{\epsilon }_1<{\epsilon }_2<<1$, $\mathbb{R}(\varkappa ,\mathfrak{t})\mathrm{is}2\times 2\mathrm{matrix}$, and $\mathbb{S}=diag\left(\overrightarrow{\mathfrak{s}}\right),\overrightarrow{\mathfrak{s}}=({\mathfrak{s}}_1(\varkappa ,\mathfrak{t}),{\mathfrak{s}}_2(\varkappa ,\mathfrak{t}))$. The functions ${\mathfrak{r}}_{\iota j}(\varkappa ,\mathfrak{t}),{\mathfrak{s}}_{\iota }(\varkappa ,\mathfrak{t}),1\le \iota ,j\le 2$ are assumed to be in ${\mathbb{C}}^{\left(2\right)}([0,2]\times [0,\mathbb{T}])$, and $\overrightarrow{\mathfrak{L}}$ can operate on functions in the domain ${\mathbb{C}}^{\left(0\right)}([0,2]\times [0,\mathbb{T}])\cap {\mathbb{C}}^{\left(1\right)}((0,\mathfrak{y})\times (0,\mathbb{T}]\cup (\mathfrak{y},2]\times (0,\mathbb{T}])$.

Furthermore, for all $(\varkappa ,\mathfrak{t})\in [0,2]\times (0,\mathbb{T}]$, the components ${\mathfrak{r}}_{\iota j}(\varkappa ,\mathfrak{t}),{\mathfrak{s}}_{\iota }(\varkappa ,\mathfrak{t})$ of $\mathbb{R}(\varkappa ,\mathfrak{t})$ and $\mathbb{S}(\varkappa ,\mathfrak{t})$, respectively, satisfy the following conditions:

$$\left.\begin{array}{cc}\hfill {\mathfrak{r}}_{\iota \iota }(\varkappa ,\mathfrak{t})& >\sum _{\begin{array}{c}j\ne \iota \\ j=1\end{array}}^2|{\mathfrak{r}}_{\iota j}(\varkappa ,\mathfrak{t})+{\mathfrak{s}}_{\iota }(\varkappa ,\mathfrak{t})|,\iota =1,2\hfill \\ \hfill {\mathfrak{s}}_{\iota }(\varkappa ,\mathfrak{t}),{\mathfrak{r}}_{\iota j}(\varkappa ,\mathfrak{t})& \le 0\mathrm{for}0\le \iota \ne j\le 2,\hfill \end{array}\right\}$$

(4)

and, for any positive number $\delta$,

$$0<\delta <\underset{\begin{array}{c}\varkappa \in \overline{\tilde{\mathrm{\Lambda }}}\\ 0\le \iota \le 2\end{array}}{min}\left\{\sum _{j=1}^2{\mathfrak{r}}_{\iota j}(\varkappa ,\mathfrak{t})+{\mathfrak{s}}_{\iota }(\varkappa ,\mathfrak{t})\right\}.$$

(5)

The function $\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})$ is discontinuous at $(\mathfrak{y},\mathfrak{t})$ due to a finite jump. So, the solution $\overrightarrow{\vartheta }$ does not possess a continuous first-order derivative at $(\mathfrak{y},\mathfrak{t})$.

Case 1:  $(\mathfrak{y},\mathfrak{t})\in (0,1]\times (0,\mathbb{T}]$

Problem (1) can be reformulated as

$$\begin{array}{cc}\hfill \overrightarrow{\mathfrak{L}}\overrightarrow{\vartheta }(\varkappa ,\mathfrak{t})& =\left\{\begin{array}{}\mathrm{(6)}& {\overrightarrow{\mathfrak{L}}}_1\overrightarrow{\vartheta }(\varkappa ,\mathfrak{t})\hfill & =\left(\frac{\partial \overrightarrow{\vartheta }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\vartheta }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\vartheta }\right)(\varkappa ,\mathfrak{t})=\overrightarrow{\chi }(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\Lambda }}_1\cup {\tilde{\Lambda }}_2,\hfill \mathrm{(7)}& {\overrightarrow{\mathfrak{L}}}_2\overrightarrow{\vartheta }(\varkappa ,\mathfrak{t})\hfill & =\left(\frac{\partial \overrightarrow{\vartheta }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\vartheta }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\vartheta }\right)(\varkappa ,\mathfrak{t})+\mathbb{S}(\varkappa ,\mathfrak{t})\overrightarrow{\vartheta }(\varkappa -1,\mathfrak{t})=\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\Lambda }}_3\cup {\tilde{\Lambda }}_4,\hfill \end{array}\right.\hfill \end{array}$$

where $\overrightarrow{\chi }(\varkappa ,\mathfrak{t})=\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})-\mathbb{S}(\varkappa ,\mathfrak{t}){\overrightarrow{\varpi }}_L(\varkappa -1,\mathfrak{t})$.

For problems (6) and (7), the solution components ${\vartheta }_1$ and ${\vartheta }_2$ exhibit an initial layer at $\varkappa =0$ and interior layers at $\varkappa =1$, $\varkappa =\mathfrak{y}$, and $\varkappa =1+\mathfrak{y}$, each of width $\mathcal{O}\left({\epsilon }_2\right)$. Furthermore, the solution component ${\vartheta }_1$ exhibits additional layers of width $\mathcal{O}\left({\epsilon }_1\right)$ at $\varkappa =0$, $\varkappa =1$, $\varkappa =\mathfrak{y}$, and $\varkappa =1+\mathfrak{y}$.

Case 2:  $(\mathfrak{y},\mathfrak{t})\in (1,2]\times (0,\mathbb{T}]$

Problem (1) can be reformulated as

$$\begin{array}{cc}\hfill \overrightarrow{\mathfrak{L}}\overrightarrow{\vartheta }(\varkappa ,\mathfrak{t})& =\left\{\begin{array}{}\mathrm{(8)}& {\overrightarrow{\mathfrak{L}}}_1\overrightarrow{\vartheta }(\varkappa ,\mathfrak{t})\hfill & =\left(\frac{\partial \overrightarrow{\vartheta }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\vartheta }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\vartheta }\right)(\varkappa ,\mathfrak{t})=\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})-\mathbb{S}(\varkappa ,\mathfrak{t}){\overrightarrow{\varpi }}_L(\varkappa -1,\mathfrak{t})\mathrm{on}{\tilde{\Lambda }}_5\hfill \mathrm{(9)}& {\overrightarrow{\mathfrak{L}}}_2\overrightarrow{\vartheta }(\varkappa ,\mathfrak{t})\hfill & =\left(\frac{\partial \overrightarrow{\vartheta }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\vartheta }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\vartheta }\right)(\varkappa ,\mathfrak{t})+\mathbb{S}(\varkappa ,\mathfrak{t})\overrightarrow{\vartheta }(\varkappa -1,\mathfrak{t})=\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\Lambda }}_6\cup {\tilde{\Lambda }}_7.\hfill \end{array}\right.\hfill \end{array}$$

For problems (8) and (9), the solution components ${\vartheta }_1$ and ${\vartheta }_2$ exhibit an initial layer at $\varkappa =0$ and interior layers at $\varkappa =1$ and $\varkappa =\mathfrak{y}$, each of width $\mathcal{O}\left({\epsilon }_2\right)$. Furthermore, the solution component ${\vartheta }_1$ exhibits additional layers of width $\mathcal{O}\left({\epsilon }_1\right)$ at $\varkappa =0$, $\varkappa =1$, and $\varkappa =\mathfrak{y}$.

The existence of the solution of problem (1) is discussed in the following theorem by adopting similar procedure to that in [24].

Theorem 1. 

The system of Equations (1)–(3) has a solution $\overrightarrow{\vartheta }\in \mathbb{C}$, where $\mathbb{C}={\mathbb{C}}^{\left(0\right)}([0,2]\times [0,\mathbb{T}])\cap {\mathbb{C}}^{\left(1\right)}((0,2]\times [0,\mathbb{T}]\setminus \left\{\mathfrak{y}\right\}).$

Proof. 

The proof is by construction.

Case 1: Let ${\overrightarrow{\mathfrak{u}}}_1,{\overrightarrow{\mathfrak{v}}}_1,{\overrightarrow{\mathfrak{u}}}_2,{\overrightarrow{\mathfrak{v}}}_2$ be the particular solutions of

$$\begin{array}{cc}\hfill \left(\frac{\partial {\overrightarrow{\mathfrak{u}}}_1}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\mathfrak{u}}}_1}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\mathfrak{u}}}_1\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})-\mathbb{S}(\varkappa ,\mathfrak{t}){\overrightarrow{\phi }}_{\mathfrak{t}}(\varkappa -1,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_1,\hfill \\ \hfill \left(\frac{\partial {\overrightarrow{\mathfrak{v}}}_1}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\mathfrak{v}}}_1}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\mathfrak{v}}}_1\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})-\mathbb{S}(\varkappa ,\mathfrak{t}){\overrightarrow{\phi }}_{\mathfrak{t}}(\varkappa -1,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_2,\hfill \\ \hfill \left(\frac{\partial {\overrightarrow{\mathfrak{u}}}_2}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\mathfrak{u}}}_2}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\mathfrak{u}}}_2\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})-\mathbb{S}(\varkappa ,\mathfrak{t}){\overrightarrow{\mathfrak{u}}}_2(\varkappa -1,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_3,\hfill \\ \hfill \left(\frac{\partial {\overrightarrow{\mathfrak{v}}}_2}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\mathfrak{v}}}_2}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\mathfrak{v}}}_2\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})-\mathbb{S}(\varkappa ,\mathfrak{t}){\overrightarrow{\mathfrak{v}}}_2(\varkappa -1,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_4.\hfill \end{array}$$

Consider the function

$$\overrightarrow{\vartheta }(\varkappa ,\mathfrak{t})=\left\{\begin{array}{c}{\overrightarrow{\mathfrak{u}}}_1(\varkappa ,\mathfrak{t})+\overrightarrow{\xi }(\overrightarrow{\vartheta }(0,\mathfrak{t})-{\overrightarrow{\mathfrak{u}}}_1(0,\mathfrak{t})){\overrightarrow{\Xi }}_1(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_1\hfill \\ {\overrightarrow{\mathfrak{v}}}_1(\varkappa ,\mathfrak{t})+\overrightarrow{\mathfrak{P}}{\overrightarrow{\Xi }}_2(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_2\hfill \\ {\overrightarrow{\mathfrak{u}}}_2(\varkappa ,\mathfrak{t})+\overrightarrow{\mathfrak{Q}}{\overrightarrow{\Xi }}_3(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_3\hfill \\ {\overrightarrow{\mathfrak{v}}}_2(\varkappa ,\mathfrak{t})+\overrightarrow{\mathfrak{U}}{\overrightarrow{\Xi }}_4(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_4\hfill \end{array}\right.$$

where ${\overrightarrow{\Xi }}_1,{\overrightarrow{\Xi }}_2,{\overrightarrow{\Xi }}_3$, and ${\overrightarrow{\Xi }}_4$ are solutions of

$$\begin{array}{cc}\hfill \left(\frac{\partial {\overrightarrow{\Xi }}_1}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\Xi }}_1}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\Xi }}_1\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{0}\mathrm{on}{\tilde{\mathrm{\Lambda }}}_1,\overrightarrow{\xi }{\overrightarrow{\Xi }}_1(0,\mathfrak{t})=\overrightarrow{1},{\overrightarrow{\Xi }}_1(\varkappa ,0)=\overrightarrow{0}\hfill \\ \hfill \left(\frac{\partial {\overrightarrow{\Xi }}_2}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\Xi }}_2}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\Xi }}_2\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{0}\mathrm{on}{\tilde{\mathrm{\Lambda }}}_2,\overrightarrow{\xi }{\overrightarrow{\Xi }}_2(\mathfrak{y},\mathfrak{t})=\overrightarrow{1},{\overrightarrow{\Xi }}_2(\varkappa ,0)=\overrightarrow{0}\hfill \\ \hfill \left(\frac{\partial {\overrightarrow{\Xi }}_3}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\Xi }}_3}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\Xi }}_3\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{0}\mathrm{on}{\tilde{\mathrm{\Lambda }}}_3,\overrightarrow{\xi }{\overrightarrow{\Xi }}_3(1,\mathfrak{t})=\overrightarrow{1},{\overrightarrow{\Xi }}_3(\varkappa ,0)=\overrightarrow{0}\hfill \\ \hfill \left(\frac{\partial {\overrightarrow{\Xi }}_4}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\Xi }}_4}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\Xi }}_4\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{0}\mathrm{on}{\tilde{\mathrm{\Lambda }}}_4,\overrightarrow{\xi }{\overrightarrow{\Xi }}_4(1+\mathfrak{y},\mathfrak{t})=\overrightarrow{1},{\overrightarrow{\Xi }}_4(\varkappa ,0)=\overrightarrow{0}\hfill \end{array}$$

and ${\overrightarrow{\mathfrak{u}}}_1(0,\mathfrak{t})={\overrightarrow{\alpha }}_1,{\overrightarrow{\mathfrak{v}}}_1(1,\mathfrak{t})={\overrightarrow{\alpha }}_2,{\overrightarrow{\mathfrak{u}}}_2(\mathfrak{y},\mathfrak{t})={\overrightarrow{\alpha }}_3,{\overrightarrow{\mathfrak{v}}}_2(1+\mathfrak{y},\mathfrak{t})={\overrightarrow{\alpha }}_4$, where ${\overrightarrow{\alpha }}_{\iota },\iota =1,2,3,4$ are any particular vector constants. $\overrightarrow{\mathfrak{P}},\overrightarrow{\mathfrak{Q}},\overrightarrow{\mathfrak{U}}$ can be derived in the following way so as to have $\overrightarrow{\vartheta }\in \mathbb{C}$.

$$\begin{array}{cc}\hfill \overrightarrow{\mathfrak{P}}& ={\overrightarrow{\mathfrak{u}}}_1(\mathfrak{y}-,\mathfrak{t})+\overrightarrow{\xi }(\overrightarrow{\vartheta }(0,\mathfrak{t})-{\overrightarrow{\alpha }}_1){\overrightarrow{\Xi }}_1(\mathfrak{y},\mathfrak{t})-{\overrightarrow{\mathfrak{v}}}_1(\mathfrak{y}+,\mathfrak{t})\hfill \\ \hfill \overrightarrow{\mathfrak{Q}}& ={\overrightarrow{\mathfrak{v}}}_1(\mathfrak{y}-,\mathfrak{t})+\overrightarrow{\mathfrak{P}}-{\overrightarrow{\mathfrak{u}}}_2(1+,\mathfrak{t})\hfill \\ \hfill \overrightarrow{\mathfrak{U}}& ={\overrightarrow{\mathfrak{u}}}_2((1+\mathfrak{y})-,\mathfrak{t})+\overrightarrow{\mathfrak{Q}}{\overrightarrow{\Xi }}_3((1+\mathfrak{y})-,\mathfrak{t})-{\overrightarrow{\alpha }}_4.\hfill \end{array}$$

The product between vectors is the Schur product of vectors.

Case 2: Let ${\overrightarrow{\mathfrak{u}}}_1,{\overrightarrow{\mathfrak{v}}}_1,{\overrightarrow{\mathfrak{v}}}_2$ be the particular solutions of

$$\begin{array}{cc}\hfill \left(\frac{\partial {\overrightarrow{\mathfrak{u}}}_1}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\mathfrak{u}}}_1}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\mathfrak{u}}}_1\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})-\mathbb{S}(\varkappa ,\mathfrak{t}){\overrightarrow{\phi }}_{\mathfrak{t}}(\varkappa -1,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_5,\hfill \\ \hfill \left(\frac{\partial {\overrightarrow{\mathfrak{v}}}_1}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\mathfrak{v}}}_1}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\mathfrak{v}}}_1\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})-\mathbb{S}(\varkappa ,\mathfrak{t}){\overrightarrow{\mathfrak{u}}}_1(\varkappa -1,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_6,\hfill \\ \hfill \left(\frac{\partial {\overrightarrow{\mathfrak{v}}}_2}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\mathfrak{v}}}_2}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\mathfrak{v}}}_2\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})-\mathbb{S}(\varkappa ,\mathfrak{t}){\overrightarrow{\mathfrak{u}}}_1(\varkappa -1,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_7.\hfill \end{array}$$

Consider the function

$$\overrightarrow{\vartheta }(\varkappa ,\mathfrak{t})=\left\{\begin{array}{c}{\overrightarrow{\mathfrak{u}}}_1(\varkappa ,\mathfrak{t})+\overrightarrow{\xi }(\overrightarrow{\vartheta }(0,\mathfrak{t})-{\overrightarrow{\mathfrak{u}}}_1(0,\mathfrak{t})){\overrightarrow{\Xi }}_1(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_5\hfill \\ {\overrightarrow{\mathfrak{v}}}_1(\varkappa ,\mathfrak{t})+\overrightarrow{\mathfrak{P}}{\overrightarrow{\Xi }}_2(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_6\hfill \\ {\overrightarrow{\mathfrak{v}}}_2(\varkappa ,\mathfrak{t})+\overrightarrow{\mathfrak{Q}}{\overrightarrow{\Xi }}_3(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_7\hfill \end{array}\right.$$

where ${\overrightarrow{\Xi }}_1,{\overrightarrow{\Xi }}_2$, and ${\overrightarrow{\Xi }}_3$ are solutions of

$$\begin{array}{cc}\hfill \left(\frac{\partial {\overrightarrow{\Xi }}_1}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\Xi }}_1}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\Xi }}_1\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{0}\mathrm{on}{\tilde{\mathrm{\Lambda }}}_5,\overrightarrow{\xi }{\overrightarrow{\Xi }}_1(0,\mathfrak{t})=\overrightarrow{1},{\overrightarrow{\Xi }}_1(\varkappa ,0)=\overrightarrow{0}\hfill \\ \hfill \left(\frac{\partial {\overrightarrow{\Xi }}_2}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\Xi }}_2}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\Xi }}_2\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{0}\mathrm{on}{\tilde{\mathrm{\Lambda }}}_6,\overrightarrow{\xi }{\overrightarrow{\Xi }}_2(1,\mathfrak{t})=\overrightarrow{1},{\overrightarrow{\Xi }}_2(\varkappa ,0)=\overrightarrow{0}\hfill \\ \hfill \left(\frac{\partial {\overrightarrow{\Xi }}_3}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial {\overrightarrow{\Xi }}_3}{\partial \varkappa }+\mathbb{R}{\overrightarrow{\Xi }}_3\right)(\varkappa ,\mathfrak{t})& =\overrightarrow{0}\mathrm{on}{\tilde{\mathrm{\Lambda }}}_7,\overrightarrow{\xi }{\overrightarrow{\Xi }}_3(\mathfrak{y},\mathfrak{t})=\overrightarrow{1},{\overrightarrow{\Xi }}_3(\varkappa ,0)=\overrightarrow{0}\hfill \end{array}$$

and ${\overrightarrow{\mathfrak{u}}}_1(0,\mathfrak{t})={\overrightarrow{\widehat{\alpha }}}_1,{\overrightarrow{\mathfrak{v}}}_1(1,\mathfrak{t})={\overrightarrow{\widehat{\alpha }}}_2,{\overrightarrow{\mathfrak{v}}}_2(\mathfrak{y},\mathfrak{t})={\overrightarrow{\widehat{\alpha }}}_3$, where ${\overrightarrow{\widehat{\alpha }}}_{\iota },\iota =1,2,3$ are any particular vector constants. $\overrightarrow{\mathfrak{P}},\overrightarrow{\mathfrak{Q}}$ can be determined as follows to ensure that $\overrightarrow{\vartheta }\in \mathbb{C}$.

$$\begin{array}{cc}\hfill \overrightarrow{\mathfrak{P}}& ={\overrightarrow{\mathfrak{u}}}_1(1-,\mathfrak{t})+\overrightarrow{\xi }(\overrightarrow{\vartheta }(0,\mathfrak{t})-{\overrightarrow{\widehat{\alpha }}}_1){\overrightarrow{\Xi }}_1(1,\mathfrak{t})-{\overrightarrow{\mathfrak{v}}}_1(1+,\mathfrak{t})\hfill \\ \hfill \overrightarrow{\mathfrak{Q}}& ={\overrightarrow{\mathfrak{v}}}_1(\mathfrak{y}-,\mathfrak{t})+\overrightarrow{\mathfrak{P}}{\overrightarrow{\Xi }}_2(\mathfrak{y}-,\mathfrak{t})-{\overrightarrow{\widehat{\alpha }}}_3.\hfill \end{array}$$

A similar approach confirms the existence of a solution for $(\mathfrak{y},\mathfrak{t})=(1,\mathfrak{t})$. When $(\mathfrak{y},\mathfrak{t})\ne (1,\mathfrak{t}),\overrightarrow{\vartheta }(1,\mathfrak{t})$ exists and is continuous at $(1,\mathfrak{t})$ as $\overrightarrow{\mathfrak{f}}(1,\mathfrak{t})$ is both well defined and continuous at this point. The proof of the theorem is complete. □

Uniqueness and Stability Analysis

• Continuous case

Lemma 1. 

Let $\mathbb{R}(\varkappa ,\mathfrak{t})$ and $\mathbb{S}(\varkappa ,\mathfrak{t})$ satisfy (4) and (5). Consider $\overrightarrow{\phi }(\varkappa ,\mathfrak{t})={({\phi }_1(\varkappa ,\mathfrak{t}),{\phi }_2(\varkappa ,\mathfrak{t}))}^T$ as a function defined in the domain of $\overrightarrow{\mathfrak{L}}$ such that it satisfies $\overrightarrow{\xi }\overrightarrow{\phi }(\varkappa ,\mathfrak{t})\ge \overrightarrow{0}\varkappa \mathit{on} \Lambda \mathit{and} \overrightarrow{\phi }(\varkappa ,0)\ge \overrightarrow{0}$ on ${\Omega }_{\varkappa }$. Then, $\overrightarrow{\mathfrak{L}}\overrightarrow{\phi }(\varkappa ,\mathfrak{t})\ge \overrightarrow{0}$ on ${\tilde{\mathrm{\Lambda }}}_1\cup {\tilde{\mathrm{\Lambda }}}_7,\left[\overrightarrow{\phi }\right](\mathfrak{y},\mathfrak{t})=\overrightarrow{0},\left[\frac{\partial \overrightarrow{\phi }}{\partial \varkappa }\right](\mathfrak{y},\mathfrak{t})\le \overrightarrow{0}$ implies that $\overrightarrow{\phi }(\varkappa ,\mathfrak{t})\ge \overrightarrow{0}$ on $\overline{\mathrm{\Lambda }}.$

Proof. 

For $\acute{\iota },\acute{\varkappa },\acute{\mathfrak{t}}$, let us consider ${\displaystyle {\phi }_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}})=\underset{\iota =1,2}{min}\underset{\overline{\mathrm{\Lambda }}}{min}{\phi }_{\iota }(\varkappa ,\mathfrak{t})}$. If ${\phi }_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}})\ge 0$, there is nothing to prove. Let us assume that ${\phi }_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}})<0$. Suppose $\acute{\varkappa }=0,$ and then

$$\begin{array}{cc}\hfill {\left(\overrightarrow{\xi }\overrightarrow{\phi }\right)}_{\acute{\iota }}(0,\acute{\mathfrak{t}})& ={\phi }_{\acute{\iota }}(0,\acute{\mathfrak{t}})-{\epsilon }_{\acute{\iota }}\frac{\partial {\phi }_{\acute{\iota }}}{\partial \varkappa }(0,\acute{\mathfrak{t}})<0,\hfill \end{array}$$

which contradicts our assumption, and, for $\acute{\mathfrak{t}}=0,{\phi }_{\acute{\iota }}(\acute{\varkappa },0)<0,$ which also contradicts our assumption. Therefore, $(\acute{\varkappa },\acute{\mathfrak{t}})\notin \Omega$. Also, $\frac{\partial {\phi }_{\acute{\iota }}}{\partial \varkappa }(\acute{\varkappa },\acute{\mathfrak{t}})\le 0$ and $\frac{\partial {\phi }_{\acute{\iota }}}{\partial \mathfrak{t}}(\acute{\varkappa },\acute{\mathfrak{t}})\le 0$.

Thus, for $(\acute{\varkappa },\acute{\mathfrak{t}})\in {\tilde{\mathrm{\Lambda }}}_5\setminus \left\{\mathfrak{y}\right\}$, it follows that

$$\begin{array}{cc}\hfill {\left(\overrightarrow{\mathfrak{L}}\overrightarrow{\phi }\right)}_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}})& ={\left({\overrightarrow{\mathfrak{L}}}_1\overrightarrow{\phi }\right)}_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}})\hfill \\ \hfill & =\frac{\partial {\phi }_{\acute{\iota }}}{\partial \mathfrak{t}}(\acute{\varkappa },\acute{\mathfrak{t}})+{\epsilon }_{\acute{\iota }}\frac{\partial {\phi }_{\acute{\iota }}}{\partial \varkappa }(\acute{\varkappa },\acute{\mathfrak{t}})+{\mathfrak{r}}_{\acute{\iota }\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}}){\phi }_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}})+\sum _{\genfrac{}{}{0pt}{}{j=1}{j\ne \acute{\iota }}}^2{\mathfrak{r}}_{\acute{\iota }j}(\acute{\varkappa },\acute{\mathfrak{t}}){\phi }_j(\acute{\varkappa },\acute{\mathfrak{t}})\hfill \\ \hfill & \le \sum _{j=1}^2{\mathfrak{r}}_{\acute{\iota }j}(\acute{\varkappa },\acute{\mathfrak{t}}){\phi }_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}})<0,\hfill \end{array}$$

which contradicts our assumption.

For $(\acute{\varkappa },\acute{\mathfrak{t}})\in {\tilde{\mathrm{\Lambda }}}_8\setminus \left\{\mathfrak{y}\right\}$, it follows that

$$\begin{array}{cc}\hfill {\left(\overrightarrow{\mathfrak{L}}\overrightarrow{\phi }\right)}_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}})& ={\left({\overrightarrow{\mathfrak{L}}}_2\overrightarrow{\phi }\right)}_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}})\hfill \\ \hfill & =\frac{\partial {\phi }_{\acute{\iota }}}{\partial \mathfrak{t}}(\acute{\varkappa },\acute{\mathfrak{t}})+{\epsilon }_{\acute{\iota }}\frac{\partial {\phi }_{\acute{\iota }}}{\partial \varkappa }(\acute{\varkappa },\acute{\mathfrak{t}})+{\mathfrak{r}}_{\acute{\iota }\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}}){\phi }_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}})+\sum _{\genfrac{}{}{0pt}{}{j=1}{j\ne \acute{\iota }}}^2{\mathfrak{r}}_{\acute{\iota }j}(\acute{\varkappa },\acute{\mathfrak{t}}){\phi }_j(\acute{\varkappa },\acute{\mathfrak{t}})\hfill \\ \hfill & +{\mathfrak{s}}_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}}){\phi }_{\acute{\iota }}(\acute{\varkappa }-1,\acute{\mathfrak{t}})\hfill \\ \hfill & \le \sum _{j=1}^2{\mathfrak{r}}_{\acute{\iota }j}(\acute{\varkappa },\acute{\mathfrak{t}}){\phi }_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}})+{\mathfrak{s}}_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}}){\phi }_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}})<0,\hfill \end{array}$$

which contradicts our assumption. If $(\acute{\varkappa },\acute{\mathfrak{t}})=(\mathfrak{y},\mathfrak{t})$, then  $\left({\displaystyle \sum _{j=1}^2}{\mathfrak{r}}_{\iota j}(\mathfrak{y},\mathfrak{t})+{\mathfrak{s}}_{\iota }(\mathfrak{y},\mathfrak{t})\right){\phi }_{\iota }(\mathfrak{y},\mathfrak{t})<0,$  and there exists a neighborhood $N_{\mathfrak{h}}=(\mathfrak{y}-\mathfrak{h},\mathfrak{y})\times (0,\mathbb{T})$ such that $\left({\displaystyle \sum _{j=1}^2}{\mathfrak{r}}_{\iota j}(\varkappa ,\mathfrak{t})+{\mathfrak{s}}_{\iota }(\varkappa ,\mathfrak{t})\right){\phi }_{\iota }(\varkappa ,\mathfrak{t})<0$ for all $(\varkappa ,\mathfrak{t})\in N_{\mathfrak{h}}$. If $\frac{\partial {\phi }_{\acute{\iota }}}{\partial \mathfrak{t}}({\varkappa }^{*},\mathfrak{t})<0$ for any point $({\varkappa }^{*},\mathfrak{t})\in N_{\mathfrak{h}}$, and then ${\left(\overrightarrow{\mathfrak{L}}\overrightarrow{\phi }\right)}_{\acute{\iota }}({\varkappa }^{*},\mathfrak{t})<0$. If $\frac{\partial {\phi }_{\acute{\iota }}}{\partial \mathfrak{t}}(\varkappa ,\mathfrak{t})>0$ for all $(\varkappa ,\mathfrak{t})\in N_{\mathfrak{h}}$, it can be noted that ${\phi }_{\acute{\iota }}$ is an increasing function in $N_{\mathfrak{h}}$; as a consequence, ${\phi }_{\acute{\iota }}(\varkappa ,\mathfrak{t})$ cannot attain its minimum at $(\varkappa ,\mathfrak{t})=(\mathfrak{y},\mathfrak{t})$, which contradicts our assumption. This implies that ${\phi }_{\acute{\iota }}(\acute{\varkappa },\acute{\mathfrak{t}})\ge 0$ and for all $(\varkappa ,\mathfrak{t})\in \overline{\mathrm{\Lambda }}$, and then $\overrightarrow{\phi }(\varkappa ,\mathfrak{t})\ge \overrightarrow{0}$. The proof of the lemma is complete. □

Lemma 2. 

Let conditions (4) $and$ (5) hold for $\mathbb{R}(\varkappa ,\mathfrak{t})$ and $\mathbb{S}(\varkappa ,\mathfrak{t})$. Consider a function  $\overrightarrow{\phi }(\varkappa ,\mathfrak{t})={({\phi }_1(\varkappa ,t),{\phi }_2(\varkappa ,\mathfrak{t}))}^T$ in the domain of $\overrightarrow{\mathfrak{L}},$ and then

$${\displaystyle |\overrightarrow{\phi }(\varkappa ,\mathfrak{t})|\le max\left\{\Vert \overrightarrow{\xi }\overrightarrow{\phi }(0,\mathfrak{t})\Vert ,\Vert \overrightarrow{\phi }(\varkappa ,0)\Vert ,{\displaystyle \frac{1}{\alpha }}{\Vert \overrightarrow{\mathfrak{L}}\overrightarrow{\phi }\Vert }_{{\tilde{\mathrm{\Lambda }}}_1\cup {\tilde{\mathrm{\Lambda }}}_7}\right\}.}$$

Proof. 

Consider the barrier functions

$$\begin{array}{cc}\hfill {\overrightarrow{\aleph }}^{\pm }(\varkappa ,t)& {\displaystyle =max\left\{\Vert \overrightarrow{\xi }\overrightarrow{\phi }(0,\mathfrak{t})\Vert ,\Vert \overrightarrow{\phi }(\varkappa ,0)\Vert ,{\displaystyle \frac{1}{\alpha }}{\Vert \overrightarrow{\mathfrak{L}}\overrightarrow{\phi }\Vert }_{{\tilde{\mathrm{\Lambda }}}_1\cup {\tilde{\mathrm{\Lambda }}}_7}\right\}\pm \overrightarrow{\phi }(\varkappa ,\mathfrak{t})}\hfill \\ \hfill {\overrightarrow{\aleph }}^{\pm }(\varkappa ,\mathfrak{t})& =\mathbb{M}\pm \overrightarrow{\phi }(\varkappa ,\mathfrak{t}),\hfill \end{array}$$

where ${\displaystyle \mathbb{M}=max\left\{\Vert \overrightarrow{\xi }\overrightarrow{\phi }(0,\mathfrak{t})\Vert ,\Vert \overrightarrow{\phi }(\varkappa ,0)\Vert ,{\displaystyle \frac{1}{\alpha }}{\Vert \overrightarrow{\mathfrak{L}}\overrightarrow{\phi }\Vert }_{{\tilde{\mathrm{\Lambda }}}_1\cup {\tilde{\mathrm{\Lambda }}}_7}\right\}.}$

$$\begin{array}{cc}\hfill \overrightarrow{\xi }{\overrightarrow{\aleph }}^{\pm }(0,\mathfrak{t})& ={\overrightarrow{\aleph }}^{\pm }(0,\mathfrak{t})-\mathbb{E}\frac{\partial {\overrightarrow{\aleph }}^{\pm }}{\partial \varkappa }(0,\mathfrak{t})\pm \overrightarrow{\xi }\overrightarrow{\phi }(0,\mathfrak{t})\hfill \\ \hfill & ={\overrightarrow{\aleph }}^{\pm }(0,\mathfrak{t})-\mathbb{E}\left(0\right)\pm {\overrightarrow{\varpi }}_L(0,\mathfrak{t})\ge \overrightarrow{0}.\hfill \\ \hfill {\overrightarrow{\aleph }}^{\pm }(\varkappa ,0)& =\mathbb{M}\pm \overrightarrow{\phi }(\varkappa ,0)\hfill \\ \hfill & =\mathbb{M}\pm {\overrightarrow{\varpi }}_B(\varkappa ,0)\ge \overrightarrow{0}.\hfill \end{array}$$

If $(\varkappa ,\mathfrak{t})\in {\tilde{\mathrm{\Lambda }}}_1\cup {\tilde{\mathrm{\Lambda }}}_7$, then

$$\begin{array}{cc}\hfill \overrightarrow{\mathfrak{L}}{\overrightarrow{\aleph }}^{\pm }(\varkappa ,\mathfrak{t})& =\frac{\partial {\overrightarrow{\aleph }}^{\pm }}{\partial \mathfrak{t}}(\varkappa ,\mathfrak{t})+\mathbb{E}\frac{\partial {\overrightarrow{\aleph }}^{\pm }}{\partial \varkappa }(\varkappa ,\mathfrak{t})+\mathbb{R}(\varkappa ,\mathfrak{t}){\overrightarrow{\aleph }}^{\pm }(\varkappa ,\mathfrak{t})+\mathbb{S}(\varkappa ,\mathfrak{t}){\overrightarrow{\aleph }}^{\pm }(\varkappa -1,\mathfrak{t})\pm \overrightarrow{\mathfrak{L}}\overrightarrow{\phi }(\varkappa ,\mathfrak{t})\ge \overrightarrow{0}\hfill \end{array}$$

and

$$\left[{\overrightarrow{\aleph }}^{\pm }\right](\mathfrak{y},\mathfrak{t})=\pm \left[{\overrightarrow{\phi }}^{\pm }\right](\mathfrak{y},\mathfrak{t})=\overrightarrow{0}\mathrm{and}\left[\frac{\partial {\overrightarrow{\aleph }}^{\pm }}{\partial \varkappa }\right](\mathfrak{y},\mathfrak{t})=\pm \left[\frac{\partial {\overrightarrow{\phi }}^{\pm }}{\partial \varkappa }\right](\mathfrak{y},\mathfrak{t})\le \overrightarrow{0}.$$

Lemma 1 implies that ${\overrightarrow{\aleph }}^{\pm }(\varkappa ,\mathfrak{t})\ge \overrightarrow{0}$ on $\overline{\mathrm{\Lambda }}$. Then,

$${\displaystyle |\overrightarrow{\phi }(\varkappa ,\mathfrak{t})|\le max\left\{\Vert \overrightarrow{\xi }\overrightarrow{\phi }(0,\mathfrak{t})\Vert ,\Vert \overrightarrow{\phi }(\varkappa ,0)\Vert ,{\displaystyle \frac{1}{\alpha }}{\Vert \overrightarrow{\mathfrak{L}}\overrightarrow{\phi }\Vert }_{{\tilde{\mathrm{\Lambda }}}_1\cup {\tilde{\mathrm{\Lambda }}}_7}\right\}.}$$

The proof of the lemma is complete. □

4. Solution Components and Layer Functions

The continuous solution $\overrightarrow{\vartheta }(\varkappa ,\mathfrak{t})$ of (1) is decomposed into $\overrightarrow{\gamma }(\varkappa ,\mathfrak{t})+\overrightarrow{\varsigma }(\varkappa ,\mathfrak{t})$.

$\overrightarrow{\gamma }={({\gamma }_1(\varkappa ,\mathfrak{t}),{\gamma }_2(\varkappa ,\mathfrak{t}))}^T$ represents the solution of the following equation.

$$\overrightarrow{\mathfrak{L}}\overrightarrow{\gamma }(\varkappa ,\mathfrak{t})=\left(\frac{\partial \overrightarrow{\gamma }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\gamma }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\gamma }\right)(\varkappa ,\mathfrak{t})+\mathbb{S}(\varkappa ,\mathfrak{t})\overrightarrow{\gamma }(\varkappa -1,\mathfrak{t})=\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\mathrm{\Lambda }}}_1\cup {\tilde{\mathrm{\Lambda }}}_7.$$

(10)

Case 1:

$$\begin{array}{c}\hspace{1em}\overrightarrow{\mathfrak{L}}\overrightarrow{\gamma }(\varkappa ,\mathfrak{t})=\hfill \\ \left\{\begin{array}{}\mathrm{(11)}& {\overrightarrow{\mathfrak{L}}}_1\overrightarrow{\gamma }(\varkappa ,\mathfrak{t})\hfill & =\left(\frac{\partial \overrightarrow{\gamma }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\gamma }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\gamma }\right)(\varkappa ,\mathfrak{t})=\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})-\mathbb{S}(\varkappa ,\mathfrak{t}){\overrightarrow{\varpi }}_L(\varkappa -1,\mathfrak{t})\mathrm{on}{\tilde{\Lambda }}_1\cup {\tilde{\Lambda }}_2\hfill \mathrm{(12)}& {\overrightarrow{\mathfrak{L}}}_2\overrightarrow{\gamma }(\varkappa ,\mathfrak{t})\hfill & =\left(\frac{\partial \overrightarrow{\gamma }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\gamma }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\gamma }\right)(\varkappa ,\mathfrak{t})+\mathbb{S}(\varkappa ,\mathfrak{t})\overrightarrow{\gamma }(\varkappa -1,\mathfrak{t})=\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\Lambda }}_3\cup {\tilde{\Lambda }}_4.\hfill \end{array}\right.\hfill \end{array}$$

Case 2:

$$\begin{array}{c}\hspace{1em}\overrightarrow{\mathfrak{L}}\overrightarrow{\gamma }(\varkappa ,\mathfrak{t})=\hfill \\ \left\{\begin{array}{}\mathrm{(13)}& {\overrightarrow{\mathfrak{L}}}_1\overrightarrow{\gamma }(\varkappa ,\mathfrak{t})\hfill & =\left(\frac{\partial \overrightarrow{\gamma }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\gamma }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\gamma }\right)(\varkappa ,\mathfrak{t})=\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})-\mathbb{S}(\varkappa ,\mathfrak{t}){\overrightarrow{\varpi }}_L(\varkappa -1,\mathfrak{t})\mathrm{on}{\tilde{\Lambda }}_5\hfill \mathrm{(14)}& {\overrightarrow{\mathfrak{L}}}_2\overrightarrow{\gamma }(\varkappa ,\mathfrak{t})\hfill & =\left(\frac{\partial \overrightarrow{\gamma }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\gamma }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\gamma }\right)(\varkappa ,\mathfrak{t})+\mathbb{S}(\varkappa ,\mathfrak{t})\overrightarrow{\gamma }(\varkappa -1,\mathfrak{t})=\overrightarrow{\mathfrak{f}}(\varkappa ,\mathfrak{t})\mathrm{on}{\tilde{\Lambda }}_6\cup {\tilde{\Lambda }}_7.\hfill \end{array}\right.\hfill \end{array}$$

$$\overrightarrow{\xi }\overrightarrow{\gamma }(\varkappa ,\mathfrak{t})=\overrightarrow{\xi }{\overrightarrow{\vartheta }}_0(\varkappa ,\mathfrak{t}),\overrightarrow{\gamma }(\varkappa ,0)={\overrightarrow{\vartheta }}_o(\varkappa ,0),$$

(15)

$$\overrightarrow{\gamma }(0,\mathfrak{t})={\mathbb{R}}^{-1}(0,\mathfrak{t})[\overrightarrow{\mathfrak{f}}(0,\mathfrak{t})-\mathbb{S}(0,\mathfrak{t}){\overrightarrow{\varpi }}_L(-1,\mathfrak{t})],$$

$$\overrightarrow{\gamma }(\mathfrak{y}+,\mathfrak{t})={\mathbb{R}}^{-1}(\mathfrak{y},\mathfrak{t})[\overrightarrow{\mathfrak{f}}(\mathfrak{y}+,\mathfrak{t})-\mathbb{S}(\mathfrak{y},\mathfrak{t}){\overrightarrow{\varpi }}_L(\mathfrak{y}-1,\mathfrak{t})].$$

$\overrightarrow{\varsigma }={({\varsigma }_1(\varkappa ,\mathfrak{t}),{\varsigma }_2(\varkappa ,\mathfrak{t}))}^T$ represents the solution of the following equation.

$$\overrightarrow{\mathfrak{L}}\overrightarrow{\varsigma }(\varkappa ,\mathfrak{t})=\left(\frac{\partial \overrightarrow{\varsigma }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\varsigma }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\varsigma }\right)(\varkappa ,\mathfrak{t})+\mathbb{S}(\varkappa ,\mathfrak{t})\overrightarrow{\varsigma }(\varkappa -1,\mathfrak{t})=\overrightarrow{0}\mathrm{on}{\tilde{\mathrm{\Lambda }}}_1\cup {\tilde{\mathrm{\Lambda }}}_7.$$

(16)

Case 1:

$$\begin{array}{cc}\hfill \overrightarrow{\mathfrak{L}}\overrightarrow{\varsigma }(\varkappa ,\mathfrak{t})& =\left\{\begin{array}{}\mathrm{(17)}& {\overrightarrow{\mathfrak{L}}}_1\overrightarrow{\varsigma }(\varkappa ,\mathfrak{t})\hfill & =\left(\frac{\partial \overrightarrow{\varsigma }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\varsigma }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\varsigma }\right)(\varkappa ,\mathfrak{t})=\overrightarrow{0}\mathrm{on}{\tilde{\Lambda }}_1\cup {\tilde{\Lambda }}_2\hfill \mathrm{(18)}& {\overrightarrow{\mathfrak{L}}}_2\overrightarrow{\varsigma }(\varkappa ,\mathfrak{t})\hfill & =\left(\frac{\partial \overrightarrow{\varsigma }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\varsigma }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\varsigma }\right)(\varkappa ,\mathfrak{t})+\mathbb{S}(\varkappa ,\mathfrak{t})\overrightarrow{\varsigma }(\varkappa -1,\mathfrak{t})=\overrightarrow{0}\mathrm{on}{\tilde{\Lambda }}_3\cup {\tilde{\Lambda }}_4.\hfill \end{array}\right.\hfill \end{array}$$

Case 2:

$$\begin{array}{cc}\hfill \overrightarrow{\mathfrak{L}}\overrightarrow{\varsigma }(\varkappa ,\mathfrak{t})& =\left\{\begin{array}{}\mathrm{(19)}& {\overrightarrow{\mathfrak{L}}}_1\overrightarrow{\varsigma }(\varkappa ,\mathfrak{t})\hfill & =\left(\frac{\partial \overrightarrow{\varsigma }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\varsigma }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\varsigma }\right)(\varkappa ,\mathfrak{t})=\overrightarrow{0}\mathrm{on}{\tilde{\Lambda }}_5\hfill \mathrm{(20)}& {\overrightarrow{\mathfrak{L}}}_2\overrightarrow{\varsigma }(\varkappa ,\mathfrak{t})\hfill & =\left(\frac{\partial \overrightarrow{\varsigma }}{\partial \mathfrak{t}}+\mathbb{E}\frac{\partial \overrightarrow{\varsigma }}{\partial \varkappa }+\mathbb{R}\overrightarrow{\varsigma }\right)(\varkappa ,\mathfrak{t})+\mathbb{S}(\varkappa ,\mathfrak{t})\overrightarrow{\varsigma }(\varkappa -1,\mathfrak{t})=\overrightarrow{0}\mathrm{on}{\tilde{\Lambda }}_6\cup {\tilde{\Lambda }}_7.\hfill \end{array}\right.\hfill \end{array}$$

with $\overrightarrow{\varsigma }=\overrightarrow{\vartheta }-\overrightarrow{\gamma }$,

$$\overrightarrow{\xi }\overrightarrow{\varsigma }(\varkappa ,\mathfrak{t})=\overrightarrow{\xi }(\overrightarrow{\vartheta }-\overrightarrow{\gamma })(\varkappa ,\mathfrak{t}),\overrightarrow{\varsigma }(\varkappa ,0)=\overrightarrow{\vartheta }(\varkappa ,0)-\overrightarrow{\gamma }(\varkappa ,0),$$

(21)

$$\overrightarrow{\varsigma }(0,\mathfrak{t})=\overrightarrow{\vartheta }(0,\mathfrak{t})-\overrightarrow{\gamma }(0,\mathfrak{t}),$$

$$\overrightarrow{\varsigma }(\mathfrak{y}+,\mathfrak{t})=\overrightarrow{\varsigma }(\mathfrak{y}-,\mathfrak{t})-\left[\overrightarrow{\gamma }\right](\mathfrak{y},\mathfrak{t}).$$

Functions ${\Psi }_{0,\iota },{\Psi }_{1,\iota },\iota =1,2$, known as layer functions, are introduced and associated with the solution ${\vartheta }_{\iota }$ as follows

$${\Psi }_{\wp ,\iota }\left(\varkappa \right)={\mathfrak{e}}^{-\alpha (\varkappa -\wp )/{\epsilon }_{\iota }},\wp =0,1,\mathfrak{y},1+\mathfrak{y},\iota =1,2\mathrm{on}(\varkappa ,t)\in \overline{\tilde{\mathrm{\Lambda }}}$$

(22)

where

$$\begin{array}{ccccc}\hfill \mathbf{Case}\mathbf{1}\mathbf{:}{\Psi }_{0,\iota }\left(\varkappa \right)& ={\mathfrak{e}}^{-\alpha \varkappa /{\epsilon }_{\iota }}\hfill & \hfill \mathrm{on}{\overline{\tilde{\mathrm{\Lambda }}}}_1,\mathbf{Case}\mathbf{2}\mathbf{:}{\Psi }_{0,\iota }\left(\varkappa \right)& ={\mathfrak{e}}^{-\alpha \varkappa /{\epsilon }_{\iota }}\hfill & \hfill \mathrm{on}{\overline{\tilde{\mathrm{\Lambda }}}}_5,\\ \hfill {\Psi }_{\mathfrak{y},\iota }\left(\varkappa \right)& ={\mathfrak{e}}^{-\alpha (\varkappa -\mathfrak{y})/{\epsilon }_{\iota }}\hfill & \hfill \mathrm{on}{\overline{\tilde{\mathrm{\Lambda }}}}_2,{\Psi }_{1,\iota }\left(\varkappa \right)& ={\mathfrak{e}}^{-\alpha (\varkappa -1)/{\epsilon }_{\iota }}\hfill & \hfill \mathrm{on}{\overline{\tilde{\mathrm{\Lambda }}}}_6,\\ \hfill {\Psi }_{1,\iota }\left(\varkappa \right)& ={\mathfrak{e}}^{-\alpha (\varkappa -1)/{\epsilon }_{\iota }}\hfill & \hfill \mathrm{on}{\overline{\tilde{\mathrm{\Lambda }}}}_3,{\Psi }_{\mathfrak{y},\iota }\left(\varkappa \right)& ={\mathfrak{e}}^{-\alpha (\varkappa -\mathfrak{y})/{\epsilon }_{\iota }}\hfill & \hfill \mathrm{on}{\overline{\tilde{\mathrm{\Lambda }}}}_7.\\ \hfill {\Psi }_{1+\mathfrak{y},\iota }\left(\varkappa \right)& ={\mathfrak{e}}^{-\alpha (\varkappa -(1+\mathfrak{y}\left)\right)/{\epsilon }_{\iota }}\hfill & \hfill \mathrm{on}{\overline{\tilde{\mathrm{\Lambda }}}}_4.\end{array}$$

5. Solution Bounds

Theorem 2. 

Let conditions (4) $and$ (5) hold for $\mathbb{R}(\varkappa ,\mathfrak{t})$ and $\mathbb{S}(\varkappa ,\mathfrak{t})$. Let $\overrightarrow{\vartheta }$ represent the solution of problems (1), (2), and (3). Then, for each $\iota ,\iota =1,2$, there exists a constant C such that, for all $(\varkappa ,\mathfrak{t})\in \overline{\tilde{\mathrm{\Lambda }}}\setminus \{(\mathfrak{y},\mathfrak{t}),(1+\mathfrak{y},\mathfrak{t})\}$

$$\begin{array}{cc}\hfill |\frac{{\partial }^{\ell }{\vartheta }_{\iota }}{\partial {\mathfrak{t}}^{\ell }}(\varkappa ,\mathfrak{t})|& \le C\left\{\Vert {\overrightarrow{\varpi }}_L\left(\mathfrak{t}\right)\Vert +\Vert {\overrightarrow{\varpi }}_B\left(\varkappa \right)\Vert +\sum _{q=0}^{\ell }{\Vert \frac{{\partial }^q\overrightarrow{\mathfrak{f}}}{\partial {\mathfrak{t}}^q}\Vert }_{{\widehat{\mathrm{\Lambda }}}^{*}}\right\},\ell =0,1,2\hfill \\ \hfill |\frac{\partial {\vartheta }_{\iota }}{\partial \varkappa }(\varkappa ,\mathfrak{t})|& \le C{\epsilon }_{\iota }^{-1}\left\{\Vert {\overrightarrow{\varpi }}_L\left(\mathfrak{t}\right)\Vert +\Vert {\overrightarrow{\varpi }}_B\left(\varkappa \right)\Vert +\sum _{q=0}^1{\Vert \frac{{\partial }^q\overrightarrow{\mathfrak{f}}}{\partial {\mathfrak{t}}^q}\Vert }_{{\widehat{\mathrm{\Lambda }}}^{*}}\right\}\hfill \\ \hfill |\frac{{\partial }^2{\vartheta }_{\iota }}{\partial \varkappa \partial \mathfrak{t}}(\varkappa ,\mathfrak{t})|& \le C{\epsilon }_{\iota }^{-1}\left\{\Vert {\overrightarrow{\varpi }}_L\left(\mathfrak{t}\right)\Vert +\Vert {\overrightarrow{\varpi }}_B\left(\varkappa \right)\Vert +\sum _{q=0}^2{\Vert \frac{{\partial }^q\overrightarrow{\mathfrak{f}}}{\partial {\mathfrak{t}}^q}\Vert }_{{\widehat{\mathrm{\Lambda }}}^{*}}\right\}\hfill \\ \hfill |\frac{{\partial }^2{\vartheta }_{\iota }}{\partial {\varkappa }^2}(\varkappa ,\mathfrak{t})|& \le C{\epsilon }_{\iota }^{-2}\left\{\Vert {\overrightarrow{\varpi }}_L\left(\mathfrak{t}\right)\Vert +\Vert {\overrightarrow{\varpi }}_B\left(\varkappa \right)\Vert +\Vert \frac{\partial \overrightarrow{\mathfrak{f}}}{\partial \varkappa }{\Vert }_{{\widehat{\mathrm{\Lambda }}}^{*}}+\sum _{q=0}^2{\Vert \frac{{\partial }^q\overrightarrow{\mathfrak{f}}}{\partial {\mathfrak{t}}^q}\Vert }_{{\widehat{\mathrm{\Lambda }}}^{*}}\right\}\hfill \end{array}$$

for Case 1 and for all $(\varkappa ,\mathfrak{t})\in \overline{\tilde{\mathrm{\Lambda }}}\setminus \left\{(\mathfrak{y},\mathfrak{t})\right\},$

$$\begin{array}{cc}\hfill |\frac{{\partial }^{\ell }{\vartheta }_{\iota }}{\partial {\mathfrak{t}}^{\ell }}(\varkappa ,\mathfrak{t})|& \le C\left\{\Vert {\overrightarrow{\varpi }}_L\left(\mathfrak{t}\right)\Vert +\Vert {\overrightarrow{\varpi }}_B\left(\varkappa \right)\Vert +\sum _{q=0}^{\ell }{\Vert \frac{{\partial }^q\overrightarrow{\mathfrak{f}}}{\partial {\mathfrak{t}}^q}\Vert }_{{\widehat{\mathrm{\Lambda }}}^{★}}\right\},\ell =0,1,2\hfill \\ \hfill |\frac{\partial {\vartheta }_{\iota }}{\partial \varkappa }(\varkappa ,\mathfrak{t})|& \le C{\epsilon }_{\iota }^{-1}\left\{\Vert {\overrightarrow{\varpi }}_L\left(\mathfrak{t}\right)\Vert +\Vert {\overrightarrow{\varpi }}_B\left(\varkappa \right)\Vert +\sum _{q=0}^1{\Vert \frac{{\partial }^q\overrightarrow{\mathfrak{f}}}{\partial {\mathfrak{t}}^q}\Vert }_{{\widehat{\mathrm{\Lambda }}}^{★}}\right\}\hfill \\ \hfill |\frac{{\partial }^2{\vartheta }_{\iota }}{\partial \varkappa \partial \mathfrak{t}}(\varkappa ,\mathfrak{t})|& \le C{\epsilon }_{\iota }^{-1}\left\{\Vert {\overrightarrow{\varpi }}_L\left(\mathfrak{t}\right)\Vert +\Vert {\overrightarrow{\varpi }}_B\left(\varkappa \right)\Vert +\sum _{q=0}^2{\Vert \frac{{\partial }^q\overrightarrow{\mathfrak{f}}}{\partial {\mathfrak{t}}^q}\Vert }_{{\widehat{\mathrm{\Lambda }}}^{★}}\right\}\hfill \\ \hfill |\frac{{\partial }^2{\vartheta }_{\iota }}{\partial {\varkappa }^2}(\varkappa ,\mathfrak{t})|& \le C{\epsilon }_{\iota }^{-2}\left\{\Vert {\overrightarrow{\varpi }}_L\left(\mathfrak{t}\right)\Vert +\Vert {\overrightarrow{\varpi }}_B\left(\varkappa \right)\Vert +\Vert \frac{\partial \overrightarrow{\mathfrak{f}}}{\partial \varkappa }{\Vert }_{{\widehat{\mathrm{\Lambda }}}^{★}}+\sum _{q=0}^2{\Vert \frac{{\partial }^q\overrightarrow{\mathfrak{f}}}{\partial {\mathfrak{t}}^q}\Vert }_{{\widehat{\mathrm{\Lambda }}}^{★}}\right\}\mathit{for}\mathit{Case}\mathit{2}.\hfill \end{array}$$

Proof. 

The bounds on the solutions are derived by employing steps and techniques analogous to those of Lemma 3.1 in [25]. □

The succeeding lemmas deal with finding bounds on $\overrightarrow{\gamma },\overrightarrow{\varsigma }$ and their derivatives.

Lemma 3. 

Let conditions (4) $and$ (5) hold for $\mathbb{R}(\varkappa ,\mathfrak{t})$ and $\mathbb{S}(\varkappa ,\mathfrak{t})$. Then, $\overrightarrow{\gamma }$ and its derivatives satisfy the following bounds for all $(\varkappa ,\mathfrak{t})\in \overline{\tilde{\mathrm{\Lambda }}}\setminus \{(\mathfrak{y},\mathfrak{t}),(1+\mathfrak{y},\mathfrak{t})\}$ for Case 1 and $(\varkappa ,\mathfrak{t})\in \overline{\tilde{\mathrm{\Lambda }}}\setminus \left\{(\mathfrak{y},\mathfrak{t})\right\}$ for Case 2

$$\begin{array}{cc}\hfill |\overrightarrow{\gamma }(\varkappa ,\mathfrak{t})|& \le C\hfill \\ \hfill |\frac{{\partial }^{\ell }\overrightarrow{\gamma }}{\partial {\mathfrak{t}}^{\ell }}(\varkappa ,\mathfrak{t})|& \le C,\mathit{for}\ell =1,2\hfill \\ \hfill |\frac{{\partial }^2\overrightarrow{\gamma }}{\partial \varkappa \partial \mathfrak{t}}(\varkappa ,\mathfrak{t})|& \le C\hfill \\ \hfill |\frac{{\partial }^{\ell }{\gamma }_{\iota }}{\partial {\varkappa }^{\ell }}(\varkappa ,\mathfrak{t})|& \le C{\epsilon }_{\iota }^{1-\ell },\mathit{for}\ell =1,2,\iota =1,2.\hfill \end{array}$$

Proof. 

The bounds on the $\overrightarrow{\gamma }$ and its derivatives are derived by employing steps and techniques analogous to those of Lemma 5.1 in [26]. □

Lemma 4. 

Let conditions (4) $and$ (5) hold for $\mathbb{R}(\varkappa ,\mathfrak{t})$ and $\mathbb{S}(\varkappa ,\mathfrak{t})$. Then, $\overrightarrow{\varsigma }$ and its derivatives satisfy the following bounds.

Case 1:  For all $(\varkappa ,\mathfrak{t})\in {\widehat{\mathrm{\Lambda }}}^{*},\iota =1,2$ and $\wp =0,\mathfrak{y},1,1+\mathfrak{y}$

$$\begin{array}{cc}\hfill |\frac{{\partial }^{\ell }{\varsigma }_{\iota }}{\partial {\mathfrak{t}}^{\ell }}(\varkappa ,\mathfrak{t})|& \le C{\Psi }_{\wp ,2}\left(\varkappa \right),\mathit{for}\ell =0,1,2\hfill \\ \hfill |\frac{\partial {\varsigma }_{\iota }}{\partial \varkappa }(\varkappa ,\mathfrak{t})|& \le C\sum _{q=\iota }^2\frac{{\Psi }_{\wp ,q}\left(\varkappa \right)}{{\epsilon }_q}\hfill \\ \hfill |\frac{{\partial }^2{\varsigma }_{\iota }}{\partial {\varkappa }^2}(\varkappa ,\mathfrak{t})|& \le C{\epsilon }_{\iota }^{-1}\sum _{q=1}^2\frac{{\Psi }_{\wp ,q}\left(\varkappa \right)}{{\epsilon }_q}.\hfill \end{array}$$

Case 2:  For all $(\varkappa ,\mathfrak{t})\in {\widehat{\mathrm{\Lambda }}}^{★},\iota =1,2$ and $\wp =0,1,\mathfrak{y}$

$$\begin{array}{cc}\hfill |\frac{{\partial }^{\ell }{\varsigma }_{\iota }}{\partial {\mathfrak{t}}^{\ell }}(\varkappa ,\mathfrak{t})|& \le C{\Psi }_{\wp ,2}\left(\varkappa \right),\mathit{for}\ell =0,1,2\hfill \\ \hfill |\frac{\partial {\varsigma }_{\iota }}{\partial \varkappa }(\varkappa ,\mathfrak{t})|& \le C\sum _{q=\iota }^2\frac{{\Psi }_{\wp ,q}\left(\varkappa \right)}{{\epsilon }_q}\hfill \\ \hfill |\frac{{\partial }^2{\varsigma }_{\iota }}{\partial {\varkappa }^2}(\varkappa ,\mathfrak{t})|& \le C{\epsilon }_{\iota }^{-1}\sum _{q=1}^2\frac{{\Psi }_{\wp ,q}\left(\varkappa \right)}{{\epsilon }_q}.\hfill \end{array}$$

Proof. 

The bounds on the $\overrightarrow{\varsigma }$ and its derivatives are derived by employing steps and techniques analogous to those of Lemma 5.2 in [26]. □

5.1. Sharper Estimates

Definition 1. 

For each $\iota ,j,1\le \iota \ne j\le 2$, $\wp =0,\mathfrak{y},1,1+\mathfrak{y}$ for Case 1 and $\wp =0,1,\mathfrak{y}$ for Case 2, the unique point ${\tilde{\varkappa }}^{*}$ in $(0,2]$ is defined by

$${\displaystyle \frac{{\Psi }_{\wp ,\iota }(\wp +{\tilde{\varkappa }}^{*})}{{\epsilon }_{\iota }}}={\displaystyle \frac{{\Psi }_{\wp ,j}(\wp +{\tilde{\varkappa }}^{*})}{{\epsilon }_j}}.$$

(23)

Definition 2. 

For all $\iota ,j$ such that $1\le \iota <j\le 2$, there exist points ${\tilde{\varkappa }}^{*}$ that are uniquely defined and satisfy the inequalities presented below for Case 1 and Case 2.

Case 1:

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\Psi }_{0,\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}& >{\displaystyle \frac{{\Psi }_{0,j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in [0,{\tilde{\varkappa }}^{*}),{\displaystyle \frac{{\Psi }_{0,\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}<{\displaystyle \frac{{\Psi }_{0,j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in ({\tilde{\varkappa }}^{*},\mathfrak{y}),\hfill \\ \hfill {\displaystyle \frac{{\Psi }_{\mathfrak{y},\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}& >{\displaystyle \frac{{\Psi }_{\mathfrak{y},j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in (\mathfrak{y},\mathfrak{y}+{\tilde{\varkappa }}^{*}),{\displaystyle \frac{{\Psi }_{\mathfrak{y},\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}<{\displaystyle \frac{{\Psi }_{\mathfrak{y},j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in (\mathfrak{y}+{\tilde{\varkappa }}^{*},1),\hfill \\ \hfill {\displaystyle \frac{{\Psi }_{1,\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}& >{\displaystyle \frac{{\Psi }_{1,j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in (1,1+{\tilde{\varkappa }}^{*}),{\displaystyle \frac{{\Psi }_{1,\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}<{\displaystyle \frac{{\Psi }_{1,j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in (1+{\tilde{\varkappa }}^{*},1+\mathfrak{y}),\hfill \\ \hfill {\displaystyle \frac{{\Psi }_{1+\mathfrak{y},\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}& >{\displaystyle \frac{{\Psi }_{1+\mathfrak{y},j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in (1+\mathfrak{y},1+\mathfrak{y}+{\tilde{\varkappa }}^{*}),\hfill \\ \hfill {\displaystyle \frac{{\Psi }_{1+\mathfrak{y},\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}& <{\displaystyle \frac{{\Psi }_{1+\mathfrak{y},j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in (1+\mathfrak{y}+{\tilde{\varkappa }}^{*},2].\hfill \end{array}$$

Case 2:

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\Psi }_{0,\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}& >{\displaystyle \frac{{\Psi }_{0,j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in [0,{\tilde{\varkappa }}^{*}),{\displaystyle \frac{{\Psi }_{0,\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}<{\displaystyle \frac{{\Psi }_{0,j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in ({\tilde{\varkappa }}^{*},1),\hfill \\ \hfill {\displaystyle \frac{{\Psi }_{1,\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}& >{\displaystyle \frac{{\Psi }_{1,j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in (1,1+{\tilde{\varkappa }}^{*}),{\displaystyle \frac{{\Psi }_{1,\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}<{\displaystyle \frac{{\Psi }_{1,j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in (1+{\tilde{\varkappa }}^{*},\mathfrak{y}),\hfill \\ \hfill {\displaystyle \frac{{\Psi }_{\mathfrak{y},\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}& >{\displaystyle \frac{{\Psi }_{\mathfrak{y},j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in (\mathfrak{y},\mathfrak{y}+{\tilde{\varkappa }}^{*}),{\displaystyle \frac{{\Psi }_{\mathfrak{y},\iota }\left(\varkappa \right)}{{\epsilon }_{\iota }}}<{\displaystyle \frac{{\Psi }_{\mathfrak{y},j}\left(\varkappa \right)}{{\epsilon }_j}}\mathit{for}\varkappa \in (\mathfrak{y}+{\tilde{\varkappa }}^{*},2].\hfill \end{array}$$

Lemma 5. 

Let conditions (4) $and$ (5) hold for $\mathbb{R}(\varkappa ,\mathfrak{t})$ and $\mathbb{S}(\varkappa ,\mathfrak{t})$. Then, for each $\iota =1,2,$ there exists a decomposition

$${\varsigma }_{\iota }=\sum _{\mathfrak{m}=1}^2{\varsigma }_{\iota ,\mathfrak{m}}$$

for which the following estimates hold for each $\mathfrak{m}$.

Case 1:  For all $(\varkappa ,\mathfrak{t})\in {\widehat{\mathrm{\Lambda }}}^{*}$ and $\wp =0,\mathfrak{y},1,1+\mathfrak{y}$

$$\begin{array}{cc}\hfill |\frac{\partial {\varsigma }_{\iota ,1}}{\partial \varkappa }(\varkappa ,\mathfrak{t})|& \le C{\epsilon }_{\iota }^{-1}{\Psi }_{\wp ,m}\left(\varkappa \right)\hfill \\ \hfill |\frac{{\partial }^2{\varsigma }_{\iota ,2}}{\partial {\varkappa }^2}(\varkappa ,\mathfrak{t})|& \le C{\epsilon }_{\iota }^{-1}{\epsilon }_2^{-1}{\Psi }_{\wp ,2}\left(\varkappa \right).\hfill \end{array}$$

Case 2:  For all $(\varkappa ,\mathfrak{t})\in {\widehat{\mathrm{\Lambda }}}^{★}$ and $\wp =0,1,\mathfrak{y}$

$$\begin{array}{cc}\hfill |\frac{\partial {\varsigma }_{\iota ,1}}{\partial \varkappa }(\varkappa ,\mathfrak{t})|& \le C{\epsilon }_{\iota }^{-1}{\Psi }_{\wp ,m}\left(\varkappa \right)\hfill \\ \hfill |\frac{{\partial }^2{\varsigma }_{\iota ,2}}{\partial {\varkappa }^2}(\varkappa ,\mathfrak{t})|& \le C{\epsilon }_{\iota }^{-1}{\epsilon }_2^{-1}{\Psi }_{\wp ,2}\left(\varkappa \right).\hfill \end{array}$$

Proof. 

The required estimates are derived by employing steps and techniques analogous to those of Lemma 10.3 in [26]. □

5.2. Domain Discretization

Temporal domain is meticulously discretized into a uniform mesh ${\overline{\tilde{\mathrm{\Lambda }}}}_{\mathfrak{t}}^M$ composed of M mesh intervals on ${\Omega }_{\mathfrak{t}}$. Now, to capture the intricate layer behavior of the solutions, the spatial domain ${\Omega }_{\varkappa }$ is discretized into a piecewise-uniform Shishkin mesh ${\overline{\tilde{\mathrm{\Lambda }}}}_{\varkappa }^N$ composed of N mesh intervals.

For Case 1, the interval ${\Omega }_{\varkappa }$ is partitioned into 12 sub-intervals, outlined as follows

$$[0,{\eta }_1]\cup ({\eta }_1,{\eta }_2]\cup ({\eta }_2,\mathfrak{y}]\cup (\mathfrak{y},\mathfrak{y}+{\tau }_1]\cup (\mathfrak{y}+{\tau }_1,\mathfrak{y}+{\tau }_2]\cup (\mathfrak{y}+{\tau }_2,1]\cup [1,1+{\eta }_1]\cup$$

$$(1+{\eta }_1,1+{\eta }_2]\cup (1+{\eta }_2,1+\mathfrak{y}]\cup (1+\mathfrak{y},1+\mathfrak{y}+{\tau }_1]\cup (1+\mathfrak{y}+{\tau }_1,1+\mathfrak{y}+{\tau }_2]\cup (1+\mathfrak{y}+{\tau }_2,2].$$

The transition parameters ${\eta }_1,{\eta }_2,{\tau }_1,{\tau }_2$ are defined as

$$\begin{array}{cccc}\hfill {\eta }_1=& min\left\{\frac{{\eta }_2}{2},\frac{{\epsilon }_1}{\delta }lnN\right\},\hfill & \hfill {\eta }_2& =min\left\{\frac{\mathfrak{y}}{2},\frac{{\epsilon }_2}{\delta }lnN\right\},\hfill \end{array}$$

(24)

$$\begin{array}{cccc}\hfill {\tau }_1=& min\left\{\frac{{\tau }_2}{2},\frac{{\epsilon }_1}{\delta }lnN\right\},\hfill & \hfill {\tau }_2& =min\left\{\frac{1-\mathfrak{y}}{2},\frac{{\epsilon }_2}{\delta }lnN\right\}.\hfill \end{array}$$

(25)

The sub-intervals $({\eta }_2,\mathfrak{y}],(\mathfrak{y}+{\tau }_2,1],(1+{\eta }_2,1+\mathfrak{y}]$ and $(1+\mathfrak{y}+{\tau }_2,2]$ are discretized using a uniform mesh consisting of $\frac{N}{8}$ mesh points. A uniform mesh consisting of $\frac{N}{16}$ mesh points is deployed on each of the sub-intervals $[0,{\eta }_1],({\eta }_1,{\eta }_2],(\mathfrak{y},\mathfrak{y}+{\tau }_1],(\mathfrak{y}+{\tau }_1,\mathfrak{y}+{\tau }_2],[1,1+{\eta }_1],(1+{\eta }_1,1+{\eta }_2],(1+\mathfrak{y},1+\mathfrak{y}+{\tau }_1]$, and $(1+\mathfrak{y}+{\tau }_1,1+\mathfrak{y}+{\tau }_2]$.

For Case 2, the interval ${\Omega }_{\varkappa }$ is partitioned into 9 distinct sub-intervals, outlined as follows

$$[0,{\eta }_1]\cup ({\eta }_1,{\eta }_2]\cup ({\eta }_2,1]\cup [1,1+{\tau }_1]\cup (1+{\tau }_1,1+{\tau }_2]\cup (1+{\tau }_2,\mathfrak{y}]\cup (\mathfrak{y},\mathfrak{y}+{\rho }_1]\cup (\mathfrak{y}+{\rho }_1,\mathfrak{y}+{\rho }_2]\cup (\mathfrak{y}+{\rho }_2,2].$$

The transition parameters ${\eta }_1,{\eta }_2,{\tau }_1,{\tau }_2,{\rho }_1,{\rho }_2$ are defined as

$$\begin{array}{cccc}\hfill {\eta }_1=& min\left\{\frac{{\eta }_2}{2},\frac{{\epsilon }_1}{\delta }lnN\right\},\hfill & \hfill {\eta }_2& =min\left\{\frac{1}{2},\frac{{\epsilon }_2}{\delta }lnN\right\},\hfill \end{array}$$

(26)

$$\begin{array}{cccc}\hfill {\tau }_1=& min\left\{\frac{{\tau }_2}{2},\frac{{\epsilon }_1}{\delta }lnN\right\},\hfill & \hfill {\tau }_2& =min\left\{\frac{\mathfrak{y}-1}{2},\frac{{\epsilon }_2}{\delta }lnN\right\},\hfill \end{array}$$

(27)

$$\begin{array}{cccc}\hfill {\rho }_1=& min\left\{\frac{{\rho }_2}{2},\frac{{\epsilon }_1}{\delta }lnN\right\},\hfill & \hfill {\rho }_2& =min\left\{\frac{2-\mathfrak{y}}{2},\frac{{\epsilon }_2}{\delta }lnN\right\}.\hfill \end{array}$$

(28)

The sub-intervals $({\eta }_2,1],(1+{\tau }_2,\mathfrak{y}]$ and $(\mathfrak{y}+{\rho }_2,2]$ are discretized using a uniform mesh consisting of $\frac{N}{6}$ mesh points. A uniform mesh consisting of $\frac{N}{12}$ mesh points is deployed on each of the sub-intervals $[0,{\eta }_1],({\eta }_1,{\eta }_2],[1,1+{\tau }_1],(1+{\tau }_1,1+{\tau }_2],(\mathfrak{y},\mathfrak{y}+{\rho }_1]$, and $(\mathfrak{y}+{\rho }_1,\mathfrak{y}+{\rho }_2].$

Also, ${\mathfrak{H}}_j^{-}={\varkappa }_j-{\varkappa }_{j-1}$, ${\mathfrak{H}}_j^{+}={\varkappa }_{j+1}-{\varkappa }_j$, $J=\{{\varkappa }_j:{\mathfrak{H}}_j^{-}\ne {\mathfrak{H}}_j^{+}\}\mathrm{and}{\mu }_{\kappa }={\mathfrak{t}}_{\kappa }-{\mathfrak{t}}_{\kappa -1}$.

For problems (1)–(3), a classical layer-resolving finite difference scheme is developed using the aforementioned discretization.

$${\overrightarrow{\mathfrak{L}}}^{M,N}\overrightarrow{G}=(D_{\mathfrak{t}}^{-}\overrightarrow{G}+\mathbb{E}{D}_{\varkappa }^{-}\overrightarrow{G}+\mathbb{R}\overrightarrow{G})({\varkappa }_j,{\mathfrak{t}}_{\kappa })+\mathbb{S}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\overrightarrow{G}({\varkappa }_j-1,{\mathfrak{t}}_{\kappa })=\overrightarrow{\mathfrak{f}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\mathrm{on}{\tilde{\mathrm{\Lambda }}}^{M,N},$$

(29)

$${\overrightarrow{\xi }}^{M,N}\overrightarrow{G}({\varkappa }_j,{\mathfrak{t}}_{\kappa })=(\overrightarrow{G}-\mathbb{E}{D}_{\varkappa }^{+}\overrightarrow{G})({\varkappa }_j,{\mathfrak{t}}_{\kappa })={\overrightarrow{\varpi }}_L({\varkappa }_j,{\mathfrak{t}}_{\kappa })\mathrm{on}\mathrm{\Lambda },\overrightarrow{G}({\varkappa }_j,0)={\overrightarrow{\varpi }}_B\left({\varkappa }_j\right)\mathrm{on}{\overline{\tilde{\mathrm{\Lambda }}}}_{\varkappa }^N.$$

(30)

The problem represented in (29) is reformulated for Case 1 and Case 2 as follows:

Case 1:

$${\overrightarrow{\mathfrak{L}}}_1^{M,N}\overrightarrow{G}=\left(D_{\mathfrak{t}}^{-}\overrightarrow{G}+\mathbb{E}{D}_{\varkappa }^{-}\overrightarrow{G}+\mathbb{R}\overrightarrow{G}\right)({\varkappa }_j,{\mathfrak{t}}_{\kappa })=\overrightarrow{\mathfrak{f}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })-\mathbb{S}({\varkappa }_j,{\mathfrak{t}}_{\kappa }){\overrightarrow{\varpi }}_L({\varkappa }_j-1,{\mathfrak{t}}_{\kappa })\mathrm{on}{\tilde{\mathrm{\Lambda }}}_1^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_2^{M,N},$$

(31)

$${\overrightarrow{\mathfrak{L}}}_2^{M,N}\overrightarrow{G}=\left(D_{\mathfrak{t}}^{-}\overrightarrow{G}+E{D}_{\varkappa }^{-}\overrightarrow{G}+\mathbb{R}\overrightarrow{G}\right)({\varkappa }_j,{\mathfrak{t}}_{\kappa })+\mathbb{S}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\overrightarrow{G}({\varkappa }_j-1,{\mathfrak{t}}_{\kappa })=\overrightarrow{\mathfrak{f}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\mathrm{on}{\tilde{\mathrm{\Lambda }}}_3^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_4^{M,N},$$

(32)

$${\overrightarrow{\xi }}^{M,N}\overrightarrow{G}({\varkappa }_j,{\mathfrak{t}}_{\kappa })=(\overrightarrow{G}-E{D}_{\varkappa }^{+}\overrightarrow{G})({\varkappa }_j,{\mathfrak{t}}_{\kappa })={\overrightarrow{\varpi }}_L({\varkappa }_j,{\mathfrak{t}}_{\kappa })\mathrm{on}\mathrm{\Lambda },\overrightarrow{G}({\varkappa }_j,0)={\overrightarrow{\varpi }}_B\left({\varkappa }_j\right)\mathrm{on}{\overline{\tilde{\mathrm{\Lambda }}}}_{\varkappa }^N,$$

$${\epsilon }_{\iota }(D_{\varkappa }^{+}-D_{\varkappa }^{-})G_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })=\left[{\mathfrak{f}}_{\iota }\right]({\varkappa }_j,{\mathfrak{t}}_{\kappa }),\mathrm{where}j=\frac{N}{4},\frac{N}{2},\frac{3N}{4}.$$

Case 2:

$${\overrightarrow{\mathfrak{L}}}_1^{M,N}\overrightarrow{G}=\left(D_{\mathfrak{t}}^{-}\overrightarrow{G}+\mathbb{E}{D}_{\varkappa }^{-}\overrightarrow{G}+\mathbb{R}\overrightarrow{G}\right)({\varkappa }_j,{\mathfrak{t}}_{\kappa })=\overrightarrow{\mathfrak{f}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })-\mathbb{S}({\varkappa }_j,{\mathfrak{t}}_{\kappa }){\overrightarrow{\varpi }}_L({\varkappa }_j-1,{\mathfrak{t}}_{\kappa })\mathrm{on}{\tilde{\mathrm{\Lambda }}}_5^{M,N},$$

(33)

$${\overrightarrow{\mathfrak{L}}}_2^{M,N}\overrightarrow{G}=\left(D_{\mathfrak{t}}^{-}\overrightarrow{G}+\mathbb{E}{D}_{\varkappa }^{-}\overrightarrow{G}+\mathbb{R}\overrightarrow{G}\right)({\varkappa }_j,{\mathfrak{t}}_{\kappa })+\mathbb{S}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\overrightarrow{G}({\varkappa }_j-1,{\mathfrak{t}}_{\kappa })=\overrightarrow{\mathfrak{f}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\mathrm{on}{\tilde{\mathrm{\Lambda }}}_6^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_7^{M,N},$$

(34)

$${\overrightarrow{\xi }}^{M,N}\overrightarrow{G}({\varkappa }_j,{\mathfrak{t}}_{\kappa })=(\overrightarrow{G}-\mathbb{E}{D}_{\varkappa }^{+}\overrightarrow{G})({\varkappa }_j,{\mathfrak{t}}_{\kappa })={\overrightarrow{\varpi }}_L({\varkappa }_j,{\mathfrak{t}}_{\kappa })\mathrm{on}\mathrm{\Lambda },\overrightarrow{G}({\varkappa }_j,0)={\overrightarrow{\varpi }}_B\left({\varkappa }_j\right)\mathrm{on}{\overline{\tilde{\mathrm{\Lambda }}}}_{\varkappa }^N,$$

$${\epsilon }_{\iota }(D_{\varkappa }^{+}-D_{\varkappa }^{-})G_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })=\left[{\mathfrak{f}}_{\iota }\right]({\varkappa }_j,{\mathfrak{t}}_{\kappa }),\mathrm{where}j=\frac{N}{3},\frac{2N}{3}.$$

where

$$\begin{array}{cc}\hfill D_{\mathfrak{t}}^{-}{G}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })& ={\displaystyle \frac{G_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })-G_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa -1})}{{\mathfrak{t}}_{\kappa }-{\mathfrak{t}}_{\kappa -1}}}\hfill \\ \hfill D_{\varkappa }^{-}{G}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })& ={\displaystyle \frac{G_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })-G_{\iota }({\varkappa }_{j-1},{\mathfrak{t}}_{\kappa })}{{\varkappa }_j-{\varkappa }_{j-1}}}\hfill \\ \hfill D_{\varkappa }^{+}{G}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })& ={\displaystyle \frac{G_{\iota }({\varkappa }_{j+1},{\mathfrak{t}}_{\kappa })-G_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })}{{\varkappa }_{j+1}-{\varkappa }_j}}\hfill \end{array}$$

for $1\le \iota \le 2,1\le j\le N\mathrm{and}1\le \kappa \le M.$

Lemma 6. 

Let conditions (4) $and$ (5) hold for $\mathbb{R}(\varkappa ,\mathfrak{t})$ and $\mathbb{S}(\varkappa ,\mathfrak{t})$. Let us consider  $\overrightarrow{\Theta }$ to be any mesh function such that ${\overrightarrow{\xi }}^{M,N}\overrightarrow{\Theta }(0,{\mathfrak{t}}_{\kappa })\ge \overrightarrow{0},\overrightarrow{\Theta }({\varkappa }_j,0)\ge \overrightarrow{0},{\overrightarrow{\mathfrak{L}}}_1^{M,N}\overrightarrow{\Theta }\ge \overrightarrow{0}$ on ${\tilde{\mathrm{\Lambda }}}_1^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_2^{M,N}$,${\overrightarrow{\mathfrak{L}}}_2^{M,N}\overrightarrow{\Theta }\ge \overrightarrow{0}$ on ${\tilde{\mathrm{\Lambda }}}_3^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_4^{M,N},(D_{\varkappa }^{+}-D_{\varkappa }^{-})\overrightarrow{\Theta }({\varkappa }_j,{\mathfrak{t}}_{\kappa })\le \overrightarrow{0},j=\frac{N}{4},\frac{N}{2},\frac{3N}{4}$ in Case 1 and ${\overrightarrow{\mathfrak{L}}}_1^{M,N}\overrightarrow{\Theta }\ge \overrightarrow{0}$ on ${\tilde{\mathrm{\Lambda }}}_5^{M,N}$,${\overrightarrow{\mathfrak{L}}}_2^{M,N}\overrightarrow{\Theta }\ge \overrightarrow{0}$ on ${\tilde{\mathrm{\Lambda }}}_6^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_7^{M,N},(D_{\varkappa }^{+}-D_{\varkappa }^{-})\overrightarrow{\Theta }({\varkappa }_j,{\mathfrak{t}}_{\kappa })\le \overrightarrow{0},j=\frac{N}{3},\frac{2N}{3}$ in Case 2. Then, it implies that $\overrightarrow{\Theta }\ge \overrightarrow{0}$ on ${\overline{\tilde{\mathrm{\Lambda }}}}^{M,N}.$

Proof. 

For $\acute{\iota },\acute{j},\acute{\kappa }$, let ${\displaystyle {\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})=\underset{\iota ,j,\kappa }{min}{\Theta }_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })}$ and consider the case where the lemma does not hold. Then, ${\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})<0.$ From the stipulated hypotheses, it is simple to establish that $({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})\notin {\Omega }^{M,N},{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})-{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}-1},{\mathfrak{t}}_{\acute{\kappa }})\le 0,{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}+1},{\mathfrak{t}}_{\acute{\kappa }})-{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})\ge 0$, and ${\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})-{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }-1})\le 0.$

If $({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})=(0,{\mathfrak{t}}_{\acute{\kappa }})$, then

$$\begin{array}{c}\hfill {\overrightarrow{\xi }}^{M,N}\overrightarrow{\Theta }(0,{\mathfrak{t}}_{\acute{\kappa }})=\overrightarrow{\Theta }(0,{\mathfrak{t}}_{\acute{\kappa }})-E{D}_{\varkappa }^{+}\overrightarrow{\Xi }(0,{\mathfrak{t}}_{\acute{\kappa }})<0,\end{array}$$

which is a contradiction. For ${\mathfrak{t}}_{\acute{\kappa }}=0,\overrightarrow{\Theta }({\varkappa }_{\acute{j}},0)<0$, which leads to a contradiction.

Case 1: For $({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})\in {\tilde{\mathrm{\Lambda }}}_1^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_2^{M,N},$

$$\begin{array}{cc}\hfill {\left({\overrightarrow{\mathfrak{L}}}_1^{M,N}\overrightarrow{\Theta }\right)}_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})& =D_{\mathfrak{t}}^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+{\epsilon }_{\acute{\iota }}{D}_{\varkappa }^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+\sum _{\ell =1}^2{\mathfrak{r}}_{\acute{\iota }\ell }({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}){\Theta }_{\ell }({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})<0,\hfill \end{array}$$

which contradicts our assumption. For $({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})\in {\tilde{\mathrm{\Lambda }}}_3^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_4^{M,N},$ it follows that

$$\begin{array}{cc}\hfill & {\left({\overrightarrow{\mathfrak{L}}}_2^{M,N}\overrightarrow{\Theta }\right)}_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})\hfill \\ \hfill & =D_{\mathfrak{t}}^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+{\epsilon }_{\acute{\iota }}{D}_{\varkappa }^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+\sum _{\ell =1}^2{\mathfrak{r}}_{\acute{\iota }\ell }({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}){\Theta }_{\ell }({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+{\mathfrak{s}}_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}){\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}}-1,{\mathfrak{t}}_{\acute{\kappa }})\hfill \\ \hfill & \le D_{\mathfrak{t}}^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+{\epsilon }_{\acute{\iota }}{D}_{\varkappa }^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+\sum _{\ell =1}^2{\mathfrak{r}}_{\acute{\iota }\ell }({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}){\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+{\mathfrak{s}}_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}){\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})<0,\hfill \end{array}$$

which contradicts our assumption.

Case 2: For $({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})\in {\tilde{\mathrm{\Lambda }}}_5^{M,N},$

$$\begin{array}{cc}\hfill {\left({\overrightarrow{\mathfrak{L}}}_1^{M,N}\overrightarrow{\Theta }\right)}_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})& =D_{\mathfrak{t}}^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+{\epsilon }_{\acute{\iota }}{D}_{\varkappa }^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+\sum _{\ell =1}^2{\mathfrak{r}}_{\acute{\iota }\ell }({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}){\Theta }_{\ell }({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})<0,\hfill \end{array}$$

which contradicts our assumption. For $({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})\in {\tilde{\mathrm{\Lambda }}}_6^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_7^{M,N},$ it follows that

$$\begin{array}{cc}\hfill & {\left({\overrightarrow{\mathfrak{L}}}_2^{M,N}\overrightarrow{\Theta }\right)}_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})\hfill \\ \hfill & =D_{\mathfrak{t}}^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+{\epsilon }_{\acute{\iota }}{D}_{\varkappa }^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+\sum _{\ell =1}^2{\mathfrak{r}}_{\acute{\iota }\ell }({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}){\Theta }_{\ell }({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+{\mathfrak{s}}_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}){\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}}-1,{\mathfrak{t}}_{\acute{\kappa }})\hfill \\ \hfill & \le D_{\mathfrak{t}}^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+{\epsilon }_{\acute{\iota }}{D}_{\varkappa }^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+\sum _{\ell =1}^2{\mathfrak{r}}_{\acute{\iota }\ell }({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}){\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})+{\mathfrak{s}}_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}){\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})<0,\hfill \end{array}$$

which contradicts our assumption.

If ${\varkappa }_{\acute{j}},\acute{j}=\frac{N}{4},\frac{N}{2},\frac{3N}{4}$ for Case 1, then

$$D_{\varkappa }^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})\le 0\le D_{\varkappa }^{+}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}).$$

Also,

$$D_{\varkappa }^{+}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})\le D_{\varkappa }^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}),$$

so

$${\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}-1},{\mathfrak{t}}_{\acute{\kappa }})={\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})={\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}+1},{\mathfrak{t}}_{\acute{\kappa }}).$$

Then, ${\left({\overrightarrow{\mathfrak{L}}}^{M,N}\overrightarrow{\Theta }\right)}_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})<0,$ which contradicts our assumption.

If ${\varkappa }_{\acute{j}},\acute{j}=\frac{N}{3},\frac{2N}{3}$ for Case 2, then

$$D_{\varkappa }^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})\le 0\le D_{\varkappa }^{+}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}).$$

Also,

$$D_{\varkappa }^{+}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})\le D_{\varkappa }^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }}),$$

so

$${\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}-1},{\mathfrak{t}}_{\acute{\kappa }})={\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})={\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}+1},{\mathfrak{t}}_{\acute{\kappa }}).$$

Then, ${\left({\overrightarrow{\mathfrak{L}}}^{M,N}\overrightarrow{\Theta }\right)}_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})<0,$ which contradicts our assumption. This implies that ${\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})\ge 0.$ The proof of the lemma is complete. □

Lemma 7. 

Let conditions (4) $and$ (5) hold for $\mathbb{R}(\varkappa ,\mathfrak{t})$ and $\mathbb{S}(\varkappa ,\mathfrak{t})$. Let us consider  $\overrightarrow{\Theta }$ to be any mesh function, and then, for Case 1

$$|\overrightarrow{\Theta }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|\le max\{\Vert {\overrightarrow{\xi }}^{M,N}\overrightarrow{\Theta }(0,{\mathfrak{t}}_{\kappa }),\Vert \overrightarrow{\Theta }({\varkappa }_j,0)\Vert ,\frac{1}{\delta }\Vert {\overrightarrow{\mathfrak{L}}}_1^{M,N}\overrightarrow{\Theta }{\Vert }_{{\tilde{\mathrm{\Lambda }}}_1^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_2^{M,N}},\frac{1}{\delta }\Vert {\overrightarrow{\mathfrak{L}}}_2^{M,N}\overrightarrow{\Theta }{\Vert }_{{\tilde{\mathrm{\Lambda }}}_3^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_4^{M,N}}\},$$

for $0\le j\le N,0\le \kappa \le M$, and, for Case 2,

$$|\overrightarrow{\Theta }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|\le max\{\Vert {\overrightarrow{\xi }}^{M,N}\overrightarrow{\Theta }(0,{\mathfrak{t}}_{\kappa }),\Vert \overrightarrow{\Theta }({\varkappa }_j,0)\Vert ,\frac{1}{\delta }\Vert {\overrightarrow{\mathfrak{L}}}_1^{M,N}\overrightarrow{\Theta }{\Vert }_{{\tilde{\mathrm{\Lambda }}}_5^{M,N}},\frac{1}{\delta }\Vert {\overrightarrow{\mathfrak{L}}}_2^{M,N}\overrightarrow{\Theta }{\Vert }_{{\tilde{\mathrm{\Lambda }}}_6^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_7^{M,N}}\},$$

for $0\le j\le N,0\le \kappa \le M.$

Proof. 

Define barrier functions

$$\begin{array}{cc}\hfill {\overrightarrow{\theta }}^{\pm }({\varkappa }_j,{\mathfrak{t}}_{\kappa })=& max\{\Vert {\overrightarrow{\xi }}^{M,N}\overrightarrow{\Theta }(0,{\mathfrak{t}}_{\kappa }),\Vert \overrightarrow{\Theta }({\varkappa }_j,0)\Vert ,\frac{1}{\delta }\Vert {\overrightarrow{\mathfrak{L}}}_1^{M,N}\overrightarrow{\Theta }{\Vert }_{{\tilde{\mathrm{\Lambda }}}_1^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_2^{M,N}},\frac{1}{\delta }\Vert {\overrightarrow{\mathfrak{L}}}_2^{M,N}\overrightarrow{\Theta }{\Vert }_{{\tilde{\mathrm{\Lambda }}}_3^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_4^{M,N}}\}\hfill \\ \hfill & \pm \overrightarrow{\Theta }({\varkappa }_j,{\mathfrak{t}}_{\kappa }).\hfill \end{array}$$

It is evident that ${\overrightarrow{\xi }}^{M,N}{\overrightarrow{\theta }}^{\pm }(0,{\mathfrak{t}}_{\kappa })\ge \overrightarrow{0}$ on $\mathrm{\Lambda },{\overrightarrow{\theta }}^{\pm }({\varkappa }_j,0)\ge \overrightarrow{0}$  and also  ${\overrightarrow{\mathfrak{L}}}_1^{M,N}{\overrightarrow{\theta }}^{\pm }\ge \overrightarrow{0}$  on  ${\tilde{\mathrm{\Lambda }}}_1^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_2^{M,N},{\overrightarrow{\mathfrak{L}}}_2^{M,N}{\overrightarrow{\theta }}^{\pm }\ge \overrightarrow{0}$  on  ${\tilde{\mathrm{\Lambda }}}_3^{M,N}\cup {\tilde{\mathrm{\Lambda }}}_4^{M,N}$, and, for $j=\frac{N}{4},\frac{N}{2},\frac{3N}{4}$, it follows that

$$D_{\varkappa }^{+}{\theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})-D_{\varkappa }^{-}{\theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})=D_{\varkappa }^{+}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})-D_{\varkappa }^{-}{\Theta }_{\acute{\iota }}({\varkappa }_{\acute{j}},{\mathfrak{t}}_{\acute{\kappa }})=0.$$

Hence, from the result of Lemma 6, it follows that ${\overrightarrow{\theta }}^{\pm }\ge \overrightarrow{0}$ on ${\overline{\tilde{\mathrm{\Lambda }}}}^{M,N}$ for Case 1. By applying analogous procedure used in Case 1, the required result ${\overrightarrow{\theta }}^{\pm }\ge \overrightarrow{0}$ on ${\overline{\tilde{\mathrm{\Lambda }}}}^{M,N}$ for Case 2 is obtained. The proof of the lemma is complete. □

6. Discrete Solution Components and Error Estimates

The discrete solution $\overrightarrow{G}({\varkappa }_j,{\mathfrak{t}}_{\kappa })$ of (29) is decomposed into $\overrightarrow{{\rm Y}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })+\overrightarrow{\zeta }({\varkappa }_j,{\mathfrak{t}}_{\kappa })$, where $\overrightarrow{{\rm Y}}$ and $\overrightarrow{\zeta }$ are discrete smooth and discrete singular components, respectively.

$${\overrightarrow{\mathfrak{L}}}^{M,N}\overrightarrow{{\rm Y}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })=\left(D_{\mathfrak{t}}^{-}\overrightarrow{{\rm Y}}+\mathbb{E}{D}_{\varkappa }^{-}\overrightarrow{{\rm Y}}+\mathbb{R}\overrightarrow{{\rm Y}}\right)({\varkappa }_j,{\mathfrak{t}}_{\kappa })+\mathbb{S}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\overrightarrow{{\rm Y}}({\varkappa }_j-1,{\mathfrak{t}}_{\kappa })=\overrightarrow{\mathfrak{f}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\mathrm{on}{\tilde{\mathrm{\Lambda }}}^{M,N},$$

(35)

$${\overrightarrow{\mathfrak{L}}}^{M,N}\overrightarrow{\zeta }({\varkappa }_j,{\mathfrak{t}}_{\kappa })=\left(D_{\mathfrak{t}}^{-}\overrightarrow{\zeta }+\mathbb{E}{D}_{\varkappa }^{-}\overrightarrow{\zeta }+\mathbb{R}\overrightarrow{\zeta }\right)({\varkappa }_j,{\mathfrak{t}}_{\kappa })+\mathbb{S}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\overrightarrow{\zeta }({\varkappa }_j-1,{\mathfrak{t}}_{\kappa })=\overrightarrow{0}\mathrm{on}{\tilde{\mathrm{\Lambda }}}^{M,N}.$$

(36)

The problems represented in (35) and (36) are reformulated for Case 1 and Case 2 as follows:

Case 1:

$$\begin{array}{cc}\hfill {\overrightarrow{\mathfrak{L}}}^{M,N}\overrightarrow{{\rm Y}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })& =\left\{\begin{array}{cc}{\overrightarrow{\mathfrak{L}}}_1^{M,N}\overrightarrow{{\rm Y}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\hfill & =\left(D_{\mathfrak{t}}^{-}\overrightarrow{{\rm Y}}+\mathbb{E}{D}_{\varkappa }^{-}\overrightarrow{{\rm Y}}+\mathbb{R}\overrightarrow{{\rm Y}}\right)({\varkappa }_j,{\mathfrak{t}}_{\kappa })\hfill \\ \mathrm{(37)}& \hfill & =\overrightarrow{\mathfrak{f}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })-\mathbb{S}({\varkappa }_j,{\mathfrak{t}}_{\kappa }){\overrightarrow{\varpi }}_L({\varkappa }_j-1,{\mathfrak{t}}_{\kappa })\mathrm{on}{\tilde{\Lambda }}_1^{M,N}\cup {\tilde{\Lambda }}_2^{M,N},\hfill {\overrightarrow{\mathfrak{L}}}_2^{M,N}\overrightarrow{{\rm Y}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\hfill & =\left(D_{\mathfrak{t}}^{-}\overrightarrow{{\rm Y}}+\mathbb{E}{D}_{\varkappa }^{-}\overrightarrow{{\rm Y}}+\mathbb{R}\overrightarrow{{\rm Y}}\right)({\varkappa }_j,{\mathfrak{t}}_{\kappa })+\mathbb{S}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\overrightarrow{{\rm Y}}({\varkappa }_j-1,{\mathfrak{t}}_{\kappa })\hfill \\ \mathrm{(38)}& \hfill & =\overrightarrow{\mathfrak{f}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\mathrm{on}{\tilde{\Lambda }}_3^{M,N}\cup {\tilde{\Lambda }}_4^{M,N},\hfill \end{array}\right.\hfill \end{array}$$

$${\overrightarrow{\xi }}^{M,N}\overrightarrow{{\rm Y}}({\varkappa }_j,t_{\kappa })=\overrightarrow{\xi }\overrightarrow{\gamma }({\varkappa }_j,{\mathfrak{t}}_{\kappa }),\overrightarrow{{\rm Y}}({\varkappa }_j,0)=\overrightarrow{\gamma }({\varkappa }_j,0),$$

(39)

$$\begin{array}{cc}\hfill & {\overrightarrow{\mathfrak{L}}}_1^{M,N}\overrightarrow{\zeta }=\overrightarrow{0}\mathrm{on}{\tilde{\Lambda }}_1^{M,N}\cup {\tilde{\Lambda }}_2^{M,N},{\overrightarrow{\mathfrak{L}}}_2^{M,N}\overrightarrow{\zeta }=\overrightarrow{0}\mathrm{on}{\tilde{\Lambda }}_3^{M,N}\cup {\tilde{\Lambda }}_4^{\mathbb{M},\mathbb{N}},\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & {\overrightarrow{\xi }}^{M,N}\overrightarrow{\zeta }({\varkappa }_j,{\mathfrak{t}}_{\kappa })=\overrightarrow{\xi }\overrightarrow{\varsigma }({\varkappa }_j,{\mathfrak{t}}_k),\overrightarrow{\zeta }({\varkappa }_j,0)=\overrightarrow{\varsigma }({\varkappa }_j,0),\hfill \end{array}$$

(41)

$$D_{\varkappa }^{-}\overrightarrow{\zeta }({\varkappa }_{{\varkappa }_j},{\mathfrak{t}}_{\kappa })+D_{\varkappa }^{-}\overrightarrow{{\rm Y}}({\varkappa }_{{\varkappa }_j},{\mathfrak{t}}_{\kappa })=D_{\varkappa }^{+}\overrightarrow{\zeta }({\varkappa }_{{\varkappa }_j},{\mathfrak{t}}_{\kappa })+D_{\varkappa }^{+}\overrightarrow{{\rm Y}}({\varkappa }_{{\varkappa }_j},{\mathfrak{t}}_{\kappa }),\mathrm{for}j=\frac{N}{4},\frac{N}{2},\frac{3N}{4}.$$

Case 2:

$$\begin{array}{cc}\hfill {\overrightarrow{\mathfrak{L}}}^{M,N}\overrightarrow{{\rm Y}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })& =\left\{\begin{array}{cc}{\overrightarrow{\mathfrak{L}}}_1^{M,N}\overrightarrow{{\rm Y}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\hfill & =\left(D_{\mathfrak{t}}^{-}\overrightarrow{{\rm Y}}+\mathbb{E}{D}_{\varkappa }^{-}\overrightarrow{{\rm Y}}+\mathbb{R}\overrightarrow{{\rm Y}}\right)({\varkappa }_j,{\mathfrak{t}}_{\kappa })\hfill \\ \mathrm{(42)}& \hfill & =\overrightarrow{\mathfrak{f}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })-\mathbb{S}({\varkappa }_j,{\mathfrak{t}}_{\kappa }){\overrightarrow{\varpi }}_L({\varkappa }_j-1,{\mathfrak{t}}_{\kappa })\mathrm{on}{\tilde{\Lambda }}_5^{M,N}\hfill {\overrightarrow{\mathfrak{L}}}_2^{M,N}\overrightarrow{{\rm Y}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\hfill & =\left(D_{\mathfrak{t}}^{-}\overrightarrow{{\rm Y}}+\mathbb{E}{D}_{\varkappa }^{-}\overrightarrow{{\rm Y}}+\mathbb{R}\overrightarrow{{\rm Y}}\right)({\varkappa }_j,{\mathfrak{t}}_{\kappa })+\mathbb{S}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\overrightarrow{{\rm Y}}({\varkappa }_j-1,{\mathfrak{t}}_{\kappa })\hfill \\ \mathrm{(43)}& \hfill & =\overrightarrow{\mathfrak{f}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })\mathrm{on}{\tilde{\Lambda }}_6^{M,N}\cup {\tilde{\Lambda }}_7^{M,N},\hfill \end{array}\right.\hfill \end{array}$$

$${\overrightarrow{\xi }}^{M,N}\overrightarrow{{\rm Y}}({\varkappa }_j,t_{\kappa })=\overrightarrow{\xi }\overrightarrow{\gamma }({\varkappa }_j,{\mathfrak{t}}_{\kappa }),\overrightarrow{{\rm Y}}({\varkappa }_j,0)=\overrightarrow{\gamma }({\varkappa }_j,0),$$

(44)

$$\begin{array}{cc}\hfill & {\overrightarrow{\mathfrak{L}}}_1^{M,N}\overrightarrow{\zeta }=\overrightarrow{0}\mathrm{on}{\tilde{\Lambda }}_5^{M,N},{\overrightarrow{\mathfrak{L}}}_2^{M,N}\overrightarrow{\zeta }=\overrightarrow{0}\mathrm{on}{\tilde{\Lambda }}_6^{M,N}\cup {\tilde{\Lambda }}_7^{M,N},\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & {\overrightarrow{\xi }}^{M,N}\overrightarrow{\zeta }({\varkappa }_j,{\mathfrak{t}}_{\kappa })=\overrightarrow{\xi }\overrightarrow{\varsigma }({\varkappa }_j,{\mathfrak{t}}_k),\overrightarrow{\zeta }({\varkappa }_j,0)=\overrightarrow{\varsigma }({\varkappa }_j,0),\hfill \end{array}$$

(46)

$$D_{\varkappa }^{-}\overrightarrow{\zeta }({\varkappa }_{{\varkappa }_j},{\mathfrak{t}}_{\kappa })+D_{\varkappa }^{-}\overrightarrow{{\rm Y}}({\varkappa }_{{\varkappa }_j},{\mathfrak{t}}_{\kappa })=D_{\varkappa }^{+}\overrightarrow{\zeta }({\varkappa }_{{\varkappa }_j},{\mathfrak{t}}_{\kappa })+D_{\varkappa }^{+}\overrightarrow{{\rm Y}}({\varkappa }_{{\varkappa }_j},{\mathfrak{t}}_{\kappa }),\mathrm{for}j=\frac{N}{3},\frac{2N}{3}.$$

The error at each point $({\varkappa }_j,{\mathfrak{t}}_{\kappa })\in {\tilde{\mathrm{\Lambda }}}^{M,N}$ is provided by $\overrightarrow{\mathfrak{e}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })=\overrightarrow{G}({\varkappa }_j,{\mathfrak{t}}_{\kappa })-\overrightarrow{\vartheta }({\varkappa }_j,{\mathfrak{t}}_{\kappa })$. For $j\ne \frac{N}{4},\frac{N}{2},\frac{3N}{4}$ in Case 1 and $j\ne \frac{N}{3},\frac{2N}{3}$ in Case 2, the local truncation errors ${\overrightarrow{\mathfrak{L}}}^{M,N}\overrightarrow{\mathfrak{e}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })$ can be expressed as follows:

$$\begin{array}{cc}\hfill {\overrightarrow{\mathfrak{L}}}^{M,N}\overrightarrow{\mathfrak{e}}({\varkappa }_j,{\mathfrak{t}}_{\kappa })& ={\overrightarrow{\mathfrak{L}}}^{M,N}(\overrightarrow{{\rm Y}}-\overrightarrow{\gamma })({\varkappa }_j,{\mathfrak{t}}_{\kappa })+{\overrightarrow{\mathfrak{L}}}^{M,N}(\overrightarrow{\zeta }-\overrightarrow{\varsigma })({\varkappa }_j,{\mathfrak{t}}_{\kappa }),\hfill \\ \hfill {\left({\overrightarrow{\mathfrak{L}}}^{M,N}\overrightarrow{\mathfrak{e}}\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })& =({\displaystyle \frac{\partial }{\partial \mathfrak{t}}}-D_{\mathfrak{t}}^{-}){\vartheta }_{\iota }+{\epsilon }_{\iota }({\displaystyle \frac{\partial }{\partial \varkappa }}-D_{\varkappa }^{-}){\vartheta }_{\iota },\hfill \\ \hfill {\left({\overrightarrow{\mathfrak{L}}}^{M,N}(\overrightarrow{{\rm Y}}-\overrightarrow{\gamma })\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })& =({\displaystyle \frac{\partial }{\partial \mathfrak{t}}}-D_{\mathfrak{t}}^{-}){\gamma }_{\iota }+{\epsilon }_{\iota }({\displaystyle \frac{\partial }{\partial \varkappa }}-D_{\varkappa }^{-}){\gamma }_{\iota },\hfill \\ \hfill {\left({\overrightarrow{\mathfrak{L}}}^{M,N}(\overrightarrow{\zeta }-\overrightarrow{\varsigma })\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })& =({\displaystyle \frac{\partial }{\partial \mathfrak{t}}}-D_{\mathfrak{t}}^{-}){\varsigma }_{\iota }+{\epsilon }_{\iota }({\displaystyle \frac{\partial }{\partial \varkappa }}-D_{\varkappa }^{-}){\varsigma }_{\iota }.\hfill \end{array}$$

$$\begin{array}{cc}\hfill |{\left({\overrightarrow{\mathfrak{L}}}^{M,N}(\overrightarrow{{\rm Y}}-\overrightarrow{\gamma })\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|& =|({\displaystyle \frac{\partial }{\partial \mathfrak{t}}}-D_{\mathfrak{t}}^{-}){\gamma }_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|+|{\epsilon }_{\iota }({\displaystyle \frac{\partial }{\partial \varkappa }}-D_{\varkappa }^{-}){\gamma }_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|,\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill |{\left({\overrightarrow{\mathfrak{L}}}^{M,N}(\overrightarrow{\zeta }-\overrightarrow{\varsigma })\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|& =|({\displaystyle \frac{\partial }{\partial \mathfrak{t}}}-D_{\mathfrak{t}}^{-}){\varsigma }_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|+|{\epsilon }_{\iota }({\displaystyle \frac{\partial }{\partial \varkappa }}-D_{\varkappa }^{-}){\varsigma }_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|.\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill |{\left({\overrightarrow{\xi }}^{M,N}(\overrightarrow{{\rm Y}}-\overrightarrow{\gamma })\right)}_{\iota }(0,{\mathfrak{t}}_{\kappa })|& =|{\epsilon }_{\iota }(\frac{\partial }{\partial \varkappa }-D_{\varkappa }^{+}){\gamma }_{\iota }(0,{\mathfrak{t}}_{\kappa })|,\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill |{\left({\overrightarrow{\xi }}^{M,N}(\overrightarrow{\zeta }-\overrightarrow{\varsigma })\right)}_{\iota }(0,{\mathfrak{t}}_{\kappa })|& =|{\epsilon }_{\iota }(\frac{\partial }{\partial \varkappa }-D_{\varkappa }^{+}){\varsigma }_{\iota }(0,{\mathfrak{t}}_{\kappa })|.\hfill \end{array}$$

(50)

$$|({\displaystyle \frac{\partial }{\partial \mathfrak{t}}}-D_{\mathfrak{t}}^{-})\overrightarrow{\Phi }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|\le C({\mathfrak{t}}_{\kappa }-{\mathfrak{t}}_{\kappa -1})\underset{s\in [{\mathfrak{t}}_{\kappa -1},{\mathfrak{t}}_{\kappa }]}{max}|{\displaystyle \frac{{\partial }^2\overrightarrow{\Phi }}{\partial {\mathfrak{t}}^2}}({\varkappa }_j,s)|,$$

(51)

$$|({\displaystyle \frac{\partial }{\partial \mathfrak{t}}}-D_{\mathfrak{t}}^{+})\overrightarrow{\Phi }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|\le C({\mathfrak{t}}_{\kappa +1}-{\mathfrak{t}}_{\kappa })\underset{s\in [{\mathfrak{t}}_{\kappa },{\mathfrak{t}}_{\kappa +1}]}{max}|{\displaystyle \frac{{\partial }^2\overrightarrow{\Phi }}{\partial {\mathfrak{t}}^2}}({\varkappa }_j,s)|,$$

(52)

$$\begin{array}{cc}\hfill |({\displaystyle \frac{\partial }{\partial \varkappa }}-D_{\varkappa }^{-})\overrightarrow{\Phi }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|& \le \left\{\begin{array}{}\mathrm{(53)}& \hfill & C({\varkappa }_j-{\varkappa }_{j-1}){\displaystyle \underset{s\in [{\varkappa }_{j-1},{\varkappa }_j]}{max}}|{\displaystyle \frac{{\partial }^2\overrightarrow{\Phi }}{\partial {\varkappa }^2}}(s,{\mathfrak{t}}_{\kappa })|,\hfill \mathrm{(54)}& \hfill & C{\displaystyle \underset{s\in [{\varkappa }_{j-1},{\varkappa }_j]}{max}}|{\displaystyle \frac{\partial \overrightarrow{\Phi }}{\partial \varkappa }}(s,{\mathfrak{t}}_{\kappa })|,\hfill \end{array}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill |({\displaystyle \frac{\partial }{\partial \varkappa }}-D_{\varkappa }^{+})\overrightarrow{\Phi }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|& \le \left\{\begin{array}{}\mathrm{(55)}& \hfill & C({\varkappa }_{j+1}-{\varkappa }_j){\displaystyle \underset{s\in [{\varkappa }_j,{\varkappa }_{j+1}]}{max}}|{\displaystyle \frac{{\partial }^2\overrightarrow{\Phi }}{\partial {\varkappa }^2}}(s,{\mathfrak{t}}_{\kappa })|,\hfill \mathrm{(56)}& \hfill & C{\displaystyle \underset{s\in [{\varkappa }_j,{\varkappa }_{j+1}]}{max}}|{\displaystyle \frac{\partial \overrightarrow{\Phi }}{\partial \varkappa }}(s,{\mathfrak{t}}_{\kappa })|.\hfill \end{array}\right.\hfill \end{array}$$

Theorem 3. 

Let conditions (4) $and$ (5) hold for $\mathbb{R}(\varkappa ,\mathfrak{t})$ and $\mathbb{S}(\varkappa ,\mathfrak{t})$. Let $\overrightarrow{\gamma }$ represent the solution of (10) and let $\overrightarrow{{\rm Y}}$ represent the solution of (35). Then, for Case 1,

$$|{\left({\overrightarrow{\mathfrak{L}}}^{M,N}(\overrightarrow{{\rm Y}}-\overrightarrow{\gamma })\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|\le C(M^{-1}+N^{-1}),j\ne \frac{N}{4},\frac{N}{2},\frac{3N}{4}$$

and, for Case 2,

$$|{\left({\overrightarrow{\mathfrak{L}}}^{M,N}(\overrightarrow{{\rm Y}}-\overrightarrow{\gamma })\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|\le C(M^{-1}+N^{-1}),j\ne \frac{N}{3},\frac{2N}{3}.$$

Proof. 

From the expressions (49) and (55), and using Lemma 3, for $\iota =1,2,$ it follows that

$$\begin{array}{cc}\hfill |{\left({\overrightarrow{\xi }}^{M,N}(\overrightarrow{{\rm Y}}-\overrightarrow{\gamma })\right)}_{\iota }(0,{\mathfrak{t}}_{\kappa })|& =|{\epsilon }_{\iota }({\displaystyle \frac{\partial }{\partial \varkappa }}-D_{\varkappa }^{+}){\gamma }_{\iota }(0,{\mathfrak{t}}_{\kappa })|\hfill \\ \hfill & \le C{\epsilon }_{\iota }({\varkappa }_1-{\varkappa }_0)\underset{s\in [{\varkappa }_0,{\varkappa }_1]}{max}|\frac{{\partial }^2{\gamma }_{\iota }}{\partial {\varkappa }^2}(s,{\mathfrak{t}}_{\kappa })|\hfill \\ \hfill & \le C{\epsilon }_{\iota }{N}^{-1}C{\epsilon }_{\iota }^{-1}\le C{N}^{-1}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill |{\left({\overrightarrow{\mathfrak{L}}}^{M,N}\left(\overrightarrow{{\rm Y}}-\overrightarrow{\gamma }\right)\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|& \le |({\displaystyle \frac{\partial }{\partial \mathfrak{t}}}-D_{\mathfrak{t}}^{-}){\gamma }_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|+|{\epsilon }_{\iota }({\displaystyle \frac{\partial }{\partial \varkappa }}-D_{\varkappa }^{-}){\gamma }_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|\hfill \\ \hfill & \le C{\mu }_{\kappa }\underset{s\in [{\mathfrak{t}}_{\kappa -1},{\mathfrak{t}}_{\kappa }]}{max}|{\displaystyle \frac{{\partial }^2{\gamma }_{\iota }}{\partial {\mathfrak{t}}^2}}({\varkappa }_j,s)|+C{\epsilon }_{\iota }{\mathfrak{H}}_j\underset{s\in [{\varkappa }_{j-1},{\varkappa }_j]}{max}|{\displaystyle \frac{{\partial }^2{\gamma }_{\iota }}{\partial {\varkappa }^2}}(s,{\mathfrak{t}}_{\kappa })|\hfill \\ \hfill & \le C{M}^{-1}+C{N}^{-1}{\epsilon }_{\iota }C{\epsilon }_{\iota }^{-1},\mathrm{using}\mathrm{Lemma}3\hfill \\ \hfill & \le C(M^{-1}+N^{-1}).\hfill \end{array}$$

The proof of the theorem is complete. □

Theorem 4. 

Let conditions (4) $and$ (5) hold for $\mathbb{R}(\varkappa ,\mathfrak{t})$ and $\mathbb{S}(\varkappa ,\mathfrak{t})$. Let $\overrightarrow{\varsigma }$ represent the solution of (16) and let $\overrightarrow{\zeta }$ represent the solution of (36). Then, for Case 1,

$$|{\left({\overrightarrow{\mathfrak{L}}}^{M,N}(\overrightarrow{\zeta }-\overrightarrow{\varsigma })\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|\le C(M^{-1}+N^{-1}lnN),j\ne \frac{N}{4},\frac{N}{2},\frac{3N}{4}$$

and, for Case 2,

$$|{\left({\overrightarrow{\mathfrak{L}}}^{M,N}(\overrightarrow{\zeta }-\overrightarrow{\varsigma })\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|\le C(M^{-1}+N^{-1}lnN),j\ne \frac{N}{3},\frac{2N}{3}.$$

Proof. 

The procedure outlined in Theorem 10.5 of [26] is employed in this theorem as it leads to a similar result. □

From the above results, it is concluded that, for Case 1, $j\ne \frac{N}{4}$ or $\frac{N}{2}$ or $\frac{3N}{4}$, and, for Case 2, $j\ne \frac{N}{3}$ or $\frac{2N}{3}$,

$$\begin{array}{c}\hfill |{\left({\overrightarrow{\mathfrak{L}}}^{M,N}\overrightarrow{\mathfrak{e}}\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|\le C(M^{-1}+N^{-1}lnN).\end{array}$$

Now, at the point $({\varkappa }_j,{\mathfrak{t}}_{\kappa }),j=\frac{N}{4}$ or $\frac{N}{2}$ or $\frac{3N}{4}$ for Case 1 and $j=\frac{N}{3}$ or $\frac{2N}{3}$ for Case 2, it follows that

$$\begin{array}{cc}\hfill |{\left({\overrightarrow{\mathfrak{L}}}^{M,N}\overrightarrow{\mathfrak{e}}\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })|& \le C{\epsilon }_{\iota }{\mathfrak{H}}^{+}\underset{[{\varkappa }_j,{\varkappa }_{j+1}]}{max}|{\vartheta }_{\iota }^{\prime \prime }\left(\nu \right)|+C{\epsilon }_{\iota }{\mathfrak{H}}^{-}\underset{[{\varkappa }_{j-1},{\varkappa }_j]}{max}|{\vartheta }_{\iota }^{\prime \prime }\left(\beta \right)|,\mathrm{where}\left\{\begin{array}{c}{\varkappa }_j<\nu <{\varkappa }_{j+1},\hfill \\ {\varkappa }_{j-1}<\beta <{\varkappa }_j\hfill \end{array}\right.\hfill \\ \hfill & \le C{N}^{-1}lnN.\hfill \end{array}$$

Theorem 5. 

Let $\overrightarrow{\vartheta }$ represent the solution of (1), (2), and (3) and $\overrightarrow{G}$ represent the solution of (29) and (30). Thus, for sufficiently large N,

$$||(\overrightarrow{G}-\overrightarrow{\vartheta })({\varkappa }_j,{\mathfrak{t}}_{\kappa })||\le C(M^{-1}+N^{-1}lnN).$$

Proof. 

For Case 1, consider the two mesh functions

$$\begin{array}{c}\hfill {\Theta }_{\iota }^{\pm }(x_j,{\mathfrak{t}}_{\kappa })=\left\{\begin{array}{cc}C(M^{-1}+N^{-1}lnN)(1+2{\varkappa }_j)\pm {\mathfrak{e}}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa }),\hfill & j\le \frac{N}{4}\mathrm{or}\frac{N}{2}\mathrm{or}\frac{3N}{4}\hfill \\ C(M^{-1}+N^{-1}lnN)(\mathfrak{y}+{\varkappa }_j)\pm {\mathfrak{e}}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa }),\hfill & j>\frac{N}{4}\mathrm{or}\frac{N}{2}\mathrm{or}\frac{3N}{4}\hfill \end{array}\right.\end{array}$$

and, for Case 2,

$$\begin{array}{c}\hfill {\Theta }_{\iota }^{\pm }({\varkappa }_j,{\mathfrak{t}}_{\kappa })=\left\{\begin{array}{cc}C(M^{-1}+N^{-1}lnN)(1+2{\varkappa }_j)\pm {\mathfrak{e}}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa }),\hfill & j\le \frac{N}{3}\mathrm{or}\frac{2N}{3}\hfill \\ C(M^{-1}+N^{-1}lnN)(\mathfrak{y}+{\varkappa }_j)\pm {\mathfrak{e}}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa }),\hfill & j>\frac{N}{3}\mathrm{or}\frac{2N}{3}\hfill \end{array}\right.\end{array}$$

where C is suitably chosen as sufficiently large constant. Hence, for Case 1, $j<\frac{N}{4}$ or $\frac{N}{2}$ or $\frac{3N}{4}$, and, for Case 2, $j<\frac{N}{3}$ or $\frac{2N}{3},$

$$\begin{array}{cc}\hfill {\left({\overrightarrow{\xi }}^{M,N}{\overrightarrow{\Theta }}^{\pm }\right)}_{\iota }(0,{\mathfrak{t}}_{\kappa })& ={\Theta }_{\iota }^{\pm }(0,{\mathfrak{t}}_{\kappa })-{\epsilon }_{\iota }{D}_{\varkappa }^{+}{\Theta }_{\iota }^{\pm }(0,{\mathfrak{t}}_{\kappa })\pm {\left({\overrightarrow{\xi }}^{M,N}\overrightarrow{\mathfrak{e}}\right)}_{\iota }(0,{\mathfrak{t}}_{\kappa })\hfill \\ \hfill & =C(M^{-1}+N^{-1}lnN)(1+2{\varkappa }_j)-{\epsilon }_{\iota }{D}_{\varkappa }^{+}C(M^{-1}+N^{-1}lnN)(1+2{\varkappa }_j)\hfill \\ \hfill & \pm C(M^{-1}+N^{-1}lnN)\ge 0\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\left({\overrightarrow{\mathfrak{L}}}^{M,N}{\overrightarrow{\Theta }}^{\pm }\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })& =D_{\mathfrak{t}}^{-}C(M^{-1}+N^{-1}lnN)(1+2{\varkappa }_j)+{\epsilon }_{\iota }{D}_{\varkappa }^{-}C(M^{-1}+N^{-1}lnN)(1+2{\varkappa }_j)\hfill \\ \hfill & +\sum _{\ell =1}^2{\mathfrak{r}}_{\iota \ell }({\varkappa }_j,{\mathfrak{t}}_{\kappa })C(M^{-1}+N^{-1}lnN)(1+2{\varkappa }_j)\hfill \\ \hfill & +{\mathfrak{s}}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })C(M^{-1}+N^{-1}lnN)(1+2{\varkappa }_{j-1})\pm C(M^{-1}+N^{-1}lnN)\ge 0\hfill \end{array}$$

and, for Case 1, $j>\frac{N}{4}$ or $\frac{N}{2}$ or $\frac{3N}{4}$, and, for Case 2, $j>\frac{N}{3}$ or $\frac{2N}{3},$

$$\begin{array}{cc}\hfill {\left({\overrightarrow{\xi }}^{M,N}{\overrightarrow{\Theta }}^{\pm }\right)}_{\iota }(0,{\mathfrak{t}}_{\kappa })& ={\Theta }_{\iota }^{\pm }(0,{\mathfrak{t}}_{\kappa })-{\epsilon }_{\iota }{D}_{\varkappa }^{+}{\Theta }_{\iota }^{\pm }(0,{\mathfrak{t}}_{\kappa })\pm {\left({\overrightarrow{\xi }}^{M,N}\overrightarrow{\mathfrak{e}}\right)}_{\iota }(0,{\mathfrak{t}}_{\kappa })\hfill \\ \hfill & =C(M^{-1}+N^{-1}lnN)(\mathfrak{y}+{\varkappa }_j)-{\epsilon }_{\iota }{D}_{\varkappa }^{+}C(M^{-1}+N^{-1}lnN)(\mathfrak{y}+{\varkappa }_j)\hfill \\ \hfill & \pm C(M^{-1}+N^{-1}lnN)\ge 0\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\left({\overrightarrow{\mathfrak{L}}}^{M,N}{\overrightarrow{\Theta }}^{\pm }\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })& =D_{\mathfrak{t}}^{-}C(M^{-1}+N^{-1}lnN)(\mathfrak{y}+{\varkappa }_j)+{\epsilon }_{\iota }{D}_{\varkappa }^{-}C(M^{-1}+N^{-1}lnN)(\mathfrak{y}+{\varkappa }_j)\hfill \\ \hfill & +\sum _{\ell =1}^2{\mathfrak{r}}_{\iota \ell }({\varkappa }_j,{\mathfrak{t}}_{\kappa })C(M^{-1}+N^{-1}lnN)(\mathfrak{y}+{\varkappa }_j)\hfill \\ \hfill & +{\mathfrak{s}}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })C(M^{-1}+N^{-1}lnN)(\mathfrak{y}+{\varkappa }_{j-1})\pm C(M^{-1}+N^{-1}lnN)\ge 0.\hfill \end{array}$$

For Case 1, $j=\frac{N}{4}$ or $\frac{N}{2}$ or $\frac{3N}{4}$, and, for Case 2, $j=\frac{N}{3}$ or $\frac{2N}{3},$

$$\begin{array}{cc}\hfill {\left({\overrightarrow{\mathfrak{L}}}^{M,N}{\overrightarrow{\Theta }}^{\pm }\right)}_{\iota }(x_{\iota },{\mathfrak{t}}_{\kappa })& =C(M^{-1}+N^{-1}lnN)\frac{(\mathfrak{y}+{\varkappa }_j+h^{+}-1-2{\varkappa }_j)}{h^{+}}\hfill \\ \hfill & -C(M^{-1}+N^{-1}lnN)\frac{(1+2{\varkappa }_j)-(1+2{\varkappa }_j-h^{-})}{h^{-}}\pm {\left({\overrightarrow{\mathfrak{L}}}^{M,N}\overrightarrow{\mathfrak{e}}\right)}_{\iota }({\varkappa }_j,{\mathfrak{t}}_{\kappa })\le 0.\hfill \end{array}$$

Thus, for sufficiently large N,

$$\begin{array}{c}\hfill ||(\overrightarrow{G}-\overrightarrow{\vartheta })||\le C(M^{-1}+N^{-1}lnN).\end{array}$$

The proof of the theorem is complete. □

7. Numerical Illustrations

To elucidate the proposed numerical scheme, numerical examples are introduced in this section to address two distinct cases resulting from the presence of discontinuous source term $(\mathfrak{y},\mathfrak{t})$ in the domain.

The following are some of the notations present in this section:

 

${\mathfrak{Y}}_{\epsilon }^N$: maximum point-wise two-mesh differences;

 

${\mathfrak{Y}}^N$: $\epsilon$—uniform maximum point-wise two-mesh differences;

 

${\mathfrak{p}}^N$: $\epsilon$—uniform order of local convergence;

 

${\mathfrak{p}}^{*}$: $\epsilon$—uniform order of convergence;

 

${\mathfrak{C}}_{{\mathfrak{p}}^{*}}^N$: $\epsilon$—uniform error constant.

M and N are the numbers of mesh points in temporal and spatial domains, respectively. Two-mesh algorithm [27] is applied to compute ${\mathfrak{p}}^{*}$ along with ${\mathfrak{C}}_{{\mathfrak{p}}^{*}}^N$.

The $\epsilon$—uniform maximum point-wise two-mesh differences value is obtained as follows:

$${\mathfrak{Y}}^N=\underset{\epsilon }{max}{\mathfrak{Y}}_{\epsilon }^N,where{\mathfrak{Y}}_{\epsilon }^N=||G^{M,N}-G^{M,2N}{||}_{{\tilde{\mathrm{\Lambda }}}^{M,N}}.$$

The $\epsilon$—uniform order of convergence is obtained as follows:

$${\mathfrak{p}}^{*}=\underset{N}{min}{\mathfrak{p}}^N,where{\mathfrak{p}}^N={log}_2\frac{{\mathfrak{Y}}^N}{{\mathfrak{Y}}^{2N}}.$$

The $\epsilon$—uniform error constant is obtained as follows:

$${\mathfrak{C}}_{{\mathfrak{p}}^{*}}^N=\frac{{\mathfrak{Y}}^N{N}^{{\mathfrak{p}}^{*}}}{1-2^{-{\mathfrak{p}}^{*}}}.$$

Example 1. 

Consider the following system of two singularly perturbed time-dependent delay initial value problems,

$$\begin{array}{cc}\hfill \frac{\partial {\vartheta }_1}{\partial \mathfrak{t}}(\varkappa ,\mathfrak{t})+{\epsilon }_1\frac{\partial {\vartheta }_1}{\partial \varkappa }(\varkappa ,\mathfrak{t})+(6+{\mathfrak{e}}^{\frac{\mathfrak{t}}{2}}){\vartheta }_1(\varkappa ,\mathfrak{t})-{\vartheta }_2(\varkappa ,\mathfrak{t})-{\vartheta }_1(\varkappa -1,\mathfrak{t})& ={\mathfrak{f}}_1(\varkappa ,\mathfrak{t}),\hfill \\ \hfill \frac{\partial {\vartheta }_2}{\partial \mathfrak{t}}(\varkappa ,\mathfrak{t})+{\epsilon }_2\frac{\partial {\vartheta }_2}{\partial \varkappa }(\varkappa ,\mathfrak{t})-0.5{\vartheta }_1(\varkappa ,\mathfrak{t})+(5+{\mathfrak{e}}^{\frac{-\mathfrak{t}}{2}}){\vartheta }_2(\varkappa ,\mathfrak{t})-0.5{\vartheta }_2(\varkappa -1,\mathfrak{t})& ={\mathfrak{f}}_2(\varkappa ,\mathfrak{t}),\hfill \end{array}$$

where

Case 1:$(\mathfrak{y},\mathfrak{t})$ presents in the interval $(0,1]\times (0,\mathbb{T}]$

$$\begin{array}{c}\hfill {\mathfrak{f}}_1(\varkappa ,\mathfrak{t})=\left\{\begin{array}{cc}3\hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (0,0.3)\times (0,\mathbb{T}],\hfill \\ 2\mathfrak{t}\hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (0.3,2]\times (0,\mathbb{T}],\hfill \end{array}\right.{\mathfrak{f}}_2(\varkappa ,\mathfrak{t})=\left\{\begin{array}{cc}2+2\mathfrak{t}\hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (0,0.3)\times (0,\mathbb{T}],\hfill \\ 5\hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (0.3,2]\times (0,\mathbb{T}],\hfill \end{array}\right.\end{array}$$

Case 2:$(\mathfrak{y},\mathfrak{t})$ presents in the interval $(1,2]\times (0,\mathbb{T}]$

$$\begin{array}{c}\hfill {\mathfrak{f}}_1(\varkappa ,\mathfrak{t})=\left\{\begin{array}{cc}3\hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (0,1.3)\times (0,\mathbb{T}],\hfill \\ 2\mathfrak{t}\hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (1.3,2]\times (0,\mathbb{T}],\hfill \end{array}\right.{\mathfrak{f}}_2(\varkappa ,\mathfrak{t})=\left\{\begin{array}{cc}2+2\mathfrak{t}\hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (0,1.3)\times (0,\mathbb{T}],\hfill \\ 5\hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (1.3,2]\times (0,\mathbb{T}],\hfill \end{array}\right.\end{array}$$

and, with Robin initial conditions in space and Dirichlet initial conditions in time

$$\begin{array}{cc}\hfill {\vartheta }_1(0,\mathfrak{t})-{\epsilon }_1\frac{\partial {\vartheta }_1}{\partial \varkappa }(0,\mathfrak{t})=1,& {\vartheta }_1(\varkappa ,0)=1,\hfill \\ \hfill {\vartheta }_2(0,\mathfrak{t})-{\epsilon }_2\frac{\partial {\vartheta }_2}{\partial \varkappa }(0,\mathfrak{t})=1,& {\vartheta }_2(\varkappa ,0)=1.\hfill \end{array}$$

The perturbation parameters ${\epsilon }_1,{\epsilon }_2$ are expressed in terms of $\sigma$ as ${\epsilon }_1=\frac{\sigma }{512}$ and ${\epsilon }_2=\frac{\sigma }{128}$, with $\sigma$ taking different values as outlined in

Table 1. For Case 1 (Example 1), values of ${\mathfrak{p}}^{*}$ and ${\mathfrak{C}}_{{\mathfrak{p}}^{*}}^N$ generated for ${\epsilon }_1=\frac{\sigma }{512},{\epsilon }_2=\frac{\sigma }{128}$.

$\mathit{\sigma }$	$\mathit{N}$: Number of Mesh Points
	96	192	384	768
1 × ${10}^{-2}$	1.0596 × ${10}^{-1}$	6.7301 × ${10}^{-2}$	4.0582 × ${10}^{-2}$	2.3597 × ${10}^{-2}$
1 × ${10}^{-4}$	1.0596 × ${10}^{-1}$	6.7301 × ${10}^{-2}$	4.0582 × ${10}^{-2}$	2.3597 × ${10}^{-2}$
1 × ${10}^{-6}$	1.0596 × ${10}^{-1}$	6.7301 × ${10}^{-2}$	4.0582 × ${10}^{-2}$	2.3597 × ${10}^{-2}$
1 × ${10}^{-8}$	1.0596 × ${10}^{-1}$	6.7301 × ${10}^{-2}$	4.0582 × ${10}^{-2}$	2.3597 × ${10}^{-2}$
1 × ${10}^{-10}$	1.0596 × ${10}^{-1}$	6.7301 × ${10}^{-2}$	4.0582 × ${10}^{-2}$	2.3597 × ${10}^{-2}$
${\mathfrak{Y}}^N$	1.0596 × ${10}^{-1}$	6.7301 × ${10}^{-2}$	4.0582 × ${10}^{-2}$	2.3597 × ${10}^{-2}$
${\mathfrak{p}}^N$	6.5482 × ${10}^{-1}$	7.2980 × ${10}^{-1}$	7.8222 × ${10}^{-1}$
${\mathfrak{C}}_{\mathfrak{p}}^N$	5.7685	5.7685	5.4764	5.0135
${\mathfrak{p}}^{*}=6.5482\times {10}^{-1}$
${\mathfrak{C}}_{{\mathfrak{p}}^{*}}^N=5.7685$

,

Table 2. Computation time in seconds for obtaining maximum point-wise errors of Example 1 (Case 1).

$\mathit{\sigma }$	N: Number of Mesh Points
	96	192	384	768
$1\times {10}^{-2}$	0.17	2.16	4.2	17.88
$1\times {10}^{-4}$	0.14	4.39	3.7	12.8
$1\times {10}^{-6}$	0.14	1.03	2.65	13
$1\times {10}^{-8}$	0.11	3	2.69	12.49
$1\times {10}^{-10}$	0.12	1.05	2.48	13.21

,

Table 3. For Case 2 (Example 1), values of ${\mathfrak{p}}^{*}$ and ${\mathfrak{C}}_{{\mathfrak{p}}^{*}}^N$ generated for ${\epsilon }_1=\frac{\sigma }{512},{\epsilon }_2=\frac{\sigma }{128}$.

$\mathit{\sigma }$	$\mathit{N}$: Number of Mesh Points
	96	192	384	768
1 $\times {10}^{-2}$	8.2939 $\times {10}^{-2}$	5.1802 $\times {10}^{-2}$	3.0901 $\times {10}^{-2}$	1.7851 $\times {10}^{-2}$
1 $\times {10}^{-4}$	8.2939 $\times {10}^{-2}$	5.1802 $\times {10}^{-2}$	3.0901 $\times {10}^{-2}$	1.7851 $\times {10}^{-2}$
1 $\times {10}^{-6}$	8.2939 $\times {10}^{-2}$	5.1802 $\times {10}^{-2}$	3.0901 $\times {10}^{-2}$	1.7851 $\times {10}^{-2}$
1 $\times {10}^{-8}$	8.2939 $\times {10}^{-2}$	5.1802 $\times {10}^{-2}$	3.0901 $\times {10}^{-2}$	1.7851 $\times {10}^{-2}$
1 $\times {10}^{-10}$	8.2939 $\times {10}^{-2}$	5.1802 $\times {10}^{-2}$	3.0901 $\times {10}^{-2}$	1.7851 $\times {10}^{-2}$
${\mathfrak{Y}}^N$	8.2939 $\times {10}^{-2}$	5.1802 $\times {10}^{-2}$	3.0901 $\times {10}^{-2}$	1.7851 $\times {10}^{-2}$
${\mathfrak{p}}^N$	6.7904 $\times {10}^{-1}$	7.4538 $\times {10}^{-1}$	7.9166 $\times {10}^{-1}$
${\mathfrak{C}}_{\mathfrak{p}}^N$	4.9011	4.9011	4.6808	4.3293
${\mathfrak{p}}^{*}=6.7904\times {10}^{-1}$
${\mathfrak{C}}_{{\mathfrak{p}}^{*}}^N=4.9011$

and

Table 4. Computation time in seconds for obtaining maximum point-wise errors of Example 1 (Case 2).

$\mathit{\sigma }$	N: Number of Mesh Points
	96	192	384	768
$1\times {10}^{-2}$	0.18	1.26	3.25	15.31
$1\times {10}^{-4}$	0.17	1.19	2.98	16.03
$1\times {10}^{-6}$	0.23	1.14	3.14	15.86
$1\times {10}^{-8}$	0.23	1.18	3.21	15.64
$1\times {10}^{-10}$	0.14	1.25	3.72	15.43

. The values ${\mathfrak{p}}^{*}$ and $\mathfrak{C}{{\mathfrak{p}}^{*}}^N$ are calculated using the procedure outlined above, and the results are presented in Table 1 for Case 1 and Table 3 for Case 2. The values ${\mathfrak{p}}^{*}=0.65482$ and $\mathfrak{C}{{\mathfrak{p}}^{*}}^N=5.76854$ are obtained from Table 1 for Case 1 and the values ${\mathfrak{p}}^{*}=0.72069$ and $\mathfrak{C}{{\mathfrak{p}}^{*}}^N=5.7934$ are obtained from Table 3 for Case 2. From both tables, it is evident that, as N increases, the values of ${\mathfrak{p}}^N$ increase, whereas the values of $\mathfrak{C}{\mathfrak{p}}^N$ decrease. The computation times in seconds for obtaining the maximum point-wise errors for Case 1 are presented in Table 2, while those for Case 2 are provided in Table 4. Figure 1 represents the solutions of Case 1 of Example 1 for $M=16,N=192,{\epsilon }_1=\frac{\sigma }{512}$, and ${\epsilon }_2=\frac{\sigma }{128}$. The solutions ${\vartheta }_1$ and ${\vartheta }_2$ exhibit rapid transitions in the initial and interior regions of the domain due to the influence of perturbation parameters, discontinuous source terms, and delay terms. The solution components ${\vartheta }_1$ and ${\vartheta }_2$ form initial layers at $(0,t)$ near boundary region of the domain, while the interior layers emerge at $(1,t)$ due to the delay terms, as well as at $(0.3,t)$ and $(1.3,t)$ as a result of discontinuities in the source terms. Figure 2 represents the solutions of Case 2 of Example 1 for $M=16,N=288,{\epsilon }_1=\frac{\sigma }{512}$, and ${\epsilon }_2=\frac{\sigma }{128}$. The solution components ${\vartheta }_1$ and ${\vartheta }_2$ form initial layers at $(0,t)$ near boundary region of the domain, while interior layers emerge at $(1,t)$ due to the delay terms, and at $(1.3,t)$ as a result of discontinuities in the source terms.

Example 2. 

Consider the following system of two singularly perturbed time-dependent delay initial value problems,

$$\begin{array}{cc}\hfill \frac{\partial {\vartheta }_1}{\partial \mathfrak{t}}(\varkappa ,\mathfrak{t})+{\epsilon }_1\frac{\partial {\vartheta }_1}{\partial \varkappa }(\varkappa ,\mathfrak{t})+6{\mathfrak{e}}^{\varkappa }{\vartheta }_1(\varkappa ,\mathfrak{t})-0.5\varkappa {\vartheta }_2(\varkappa ,\mathfrak{t})-0.5{\vartheta }_1(\varkappa -1,\mathfrak{t})& ={\mathfrak{f}}_1(\varkappa ,\mathfrak{t}),\hfill \\ \hfill \frac{\partial {\vartheta }_2}{\partial \mathfrak{t}}(\varkappa ,\mathfrak{t})+{\epsilon }_2\frac{\partial {\vartheta }_2}{\partial \varkappa }(\varkappa ,\mathfrak{t})-(0.5+{\mathfrak{e}}^{-\varkappa }){\vartheta }_1(\varkappa ,\mathfrak{t})+(6+\varkappa ){\vartheta }_2(\varkappa ,\mathfrak{t})-{\vartheta }_2(\varkappa -1,\mathfrak{t})& ={\mathfrak{f}}_2(\varkappa ,\mathfrak{t}),\hfill \end{array}$$

where

Case 1:$(\mathfrak{y},\mathfrak{t})$ presents in the interval $(0,1]\times (0,\mathbb{T}]$

$$\begin{array}{c}\hfill {\mathfrak{f}}_1(\varkappa ,\mathfrak{t})=\left\{\begin{array}{cc}4\varkappa +2{\varkappa }^2\hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (0,0.3)\times (0,\mathbb{T}],\hfill \\ -\varkappa -2\mathfrak{t}\hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (0.3,2]\times (0,\mathbb{T}],\hfill \end{array}\right.{\mathfrak{f}}_2(\varkappa ,\mathfrak{t})=\left\{\begin{array}{cc}2.5+2\varkappa \hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (0,0.3)\times (0,\mathbb{T}],\hfill \\ 4.5\varkappa -\varkappa \hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (0.3,2]\times (0,\mathbb{T}],\hfill \end{array}\right.\end{array}$$

Case 2:$(\mathfrak{y},\mathfrak{t})$ presents in the interval $(1,2]\times (0,\mathbb{T}]$

$$\begin{array}{c}\hfill {\mathfrak{f}}_1(\varkappa ,\mathfrak{t})=\left\{\begin{array}{cc}4\varkappa +2{\varkappa }^2\hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (0,1.3)\times (0,\mathbb{T}],\hfill \\ -\varkappa -2\mathfrak{t}\hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (1.3,2]\times (0,\mathbb{T}],\hfill \end{array}\right.{\mathfrak{f}}_2(\varkappa ,\mathfrak{t})=\left\{\begin{array}{cc}2.5+2\varkappa \hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (0,1.3)\times (0,\mathbb{T}],\hfill \\ 4.5\varkappa -\varkappa \hfill & \mathrm{for}(\varkappa ,\mathfrak{t})\in (1.3,2]\times (0,\mathbb{T}],\hfill \end{array}\right.\end{array}$$

and, with Robin initial conditions in space and Dirichlet initial conditions in time,

$$\begin{array}{cc}\hfill {\vartheta }_1(0,\mathfrak{t})-{\epsilon }_1\frac{\partial {\vartheta }_1}{\partial \varkappa }(0,\mathfrak{t})=0.5,& {\vartheta }_1(\varkappa ,0)=1,\hfill \\ \hfill {\vartheta }_2(0,\mathfrak{t})-{\epsilon }_2\frac{\partial {\vartheta }_2}{\partial \varkappa }(0,\mathfrak{t})=1.5,& {\vartheta }_2(\varkappa ,0)=1.\hfill \end{array}$$

The perturbation parameters ${\epsilon }_1,{\epsilon }_2$ are expressed in terms of $\sigma$ as ${\epsilon }_1=\frac{\sigma }{64}$ and ${\epsilon }_2=\frac{\sigma }{16}$, with $\sigma$ taking different values as outlined in

Table 5. For Case 1 (Example 2), values of ${\mathfrak{p}}^{*}$ and ${\mathfrak{C}}_{{\mathfrak{p}}^{*}}^N$ generated for ${\epsilon }_1=\frac{\sigma }{64},{\epsilon }_2=\frac{\sigma }{16}$.

$\mathit{\sigma }$	$\mathit{N}$: Number of Mesh Points
	96	192	384	768
1 $\times {10}^{-2}$	5.8093 $\times {10}^{-2}$	3.6669 $\times {10}^{-2}$	2.2022 $\times {10}^{-2}$	1.2773 $\times {10}^{-2}$
1 $\times {10}^{-4}$	5.8093 $\times {10}^{-2}$	3.6669 $\times {10}^{-2}$	2.2022 $\times {10}^{-2}$	1.2773 $\times {10}^{-2}$
1 $\times {10}^{-6}$	5.8093 $\times {10}^{-2}$	3.6669 $\times {10}^{-2}$	2.2022 $\times {10}^{-2}$	1.2773 $\times {10}^{-2}$
1 $\times {10}^{-8}$	5.8093 $\times {10}^{-2}$	3.6669 $\times {10}^{-2}$	2.2022 $\times {10}^{-2}$	1.2773 $\times {10}^{-2}$
1 $\times {10}^{-10}$	5.8093 $\times {10}^{-2}$	3.6669 $\times {10}^{-2}$	2.2022 $\times {10}^{-2}$	1.2773 $\times {10}^{-2}$
${\mathfrak{Y}}^N$	5.8093 $\times {10}^{-2}$	3.6669 $\times {10}^{-2}$	2.2022 $\times {10}^{-2}$	1.2773 $\times {10}^{-2}$
${\mathfrak{p}}^N$	6.6375$\times {10}^{-1}$	7.3564 $\times {10}^{-1}$	7.8578 $\times {10}^{-1}$
${\mathfrak{C}}_{\mathfrak{p}}^N$	3.259	3.259	3.1006	2.8491
${\mathfrak{p}}^{*}=6.6375\times {10}^{-1}$
${\mathfrak{C}}_{{\mathfrak{p}}^{*}}^N=3.259$

,

Table 6. Computation time in seconds for obtaining maximum point-wise errors of Example 2 (Case 1).

$\mathit{\sigma }$	N: Number of Mesh Points
	96	192	384	768
$1\times {10}^{-2}$	0.17	1.67	3.91	28.18
$1\times {10}^{-4}$	0.14	5.15	4.24	16.62
$1\times {10}^{-6}$	0.18	1.41	3.89	17.75
$1\times {10}^{-8}$	0.12	4.79	4.03	18.58
$1\times {10}^{-10}$	0.27	1.19	8.79	33

,

Table 7. For Case 2 (Example 2), values of ${\mathfrak{p}}^{*}$ and ${\mathfrak{C}}_{{\mathfrak{p}}^{*}}^N$ generated for ${\epsilon }_1=\frac{\sigma }{64},{\epsilon }_2=\frac{\sigma }{16}$.

$\mathit{\sigma }$	$\mathit{N}$: Number of Mesh Points
	96	192	384	768
1 $\times {10}^{-2}$	4.4403 $\times {10}^{-2}$	2.7565 $\times {10}^{-2}$	1.638 $\times {10}^{-2}$	9.4407 $\times {10}^{-3}$
1 $\times {10}^{-4}$	4.4403 $\times {10}^{-2}$	2.7565 $\times {10}^{-2}$	1.638 $\times {10}^{-2}$	9.4407 $\times {10}^{-3}$
1 $\times {10}^{-6}$	4.4403 $\times {10}^{-2}$	2.7565 $\times {10}^{-2}$	1.638 $\times {10}^{-2}$	9.4407 $\times {10}^{-3}$
1 $\times {10}^{-8}$	4.4403 $\times {10}^{-2}$	2.7565 $\times {10}^{-2}$	1.638 $\times {10}^{-2}$	9.4407 $\times {10}^{-3}$
1 $\times {10}^{-10}$	4.4403 $\times {10}^{-2}$	2.7565 $\times {10}^{-2}$	1.638 $\times {10}^{-2}$	9.4407 $\times {10}^{-3}$
${\mathfrak{Y}}^N$	4.4403 $\times {10}^{-2}$	2.7565 $\times {10}^{-2}$	1.638 $\times {10}^{-2}$	9.4407 $\times {10}^{-3}$
${\mathfrak{p}}^N$	6.8773 $\times {10}^{-1}$	7.5091 $\times {10}^{-1}$	7.9498 $\times {10}^{-1}$
${\mathfrak{C}}_{\mathfrak{p}}^N$	2.7028	2.7028	2.5870	2.4017
${\mathfrak{p}}^{*}=6.8773\times {10}^{-1}$
${\mathfrak{C}}_{{\mathfrak{p}}^{*}}^N=2.7028$

and

Table 8. Computation time in seconds for obtaining maximum point-wise errors of Example 2 (Case 2).

$\mathit{\sigma }$	N: Number of Mesh Points
	96	192	384	768
$1\times {10}^{-2}$	0.19	1.28	3.91	16.98
$1\times {10}^{-4}$	0.25	1.25	3.81	17.11
$1\times {10}^{-6}$	0.31	2.46	6.97	32.2
$1\times {10}^{-8}$	0.31	2.21	6.93	28.63
$1\times {10}^{-10}$	0.15	1.34	3.62	15.55

. The values ${\mathfrak{p}}^{*}$ and $\mathfrak{C}{{\mathfrak{p}}^{*}}^N$ are calculated using the procedure outlined above, and the results are presented in Table 5 for Case 1 and Table 7 for Case 2. The values ${\mathfrak{p}}^{*}=0.66375$ and $\mathfrak{C}{{\mathfrak{p}}^{*}}^N=3.259$ are obtained from Table 5 for Case 1 and the values ${\mathfrak{p}}^{*}=0.68773$ and $\mathfrak{C}{{\mathfrak{p}}^{*}}^N=2.7028$ are obtained from Table 7 for Case 2. From both tables, it is evident that, as N increases, the values of ${\mathfrak{p}}^N$ increase, whereas the values of $\mathfrak{C}{\mathfrak{p}}^N$ decrease. The computation times in seconds for obtaining the maximum point-wise errors for Case 1 are presented in Table 6, while those for Case 2 are provided in Table 8. Figure 3 represents the solutions of Case 1 of Example 2 for $M=16,N=384,{\epsilon }_1=\frac{\sigma }{64}$, and ${\epsilon }_2=\frac{\sigma }{16}$. The solutions ${\vartheta }_1$ and ${\vartheta }_2$ exhibit rapid transitions in the initial and interior regions of the domain due to the influence of perturbation parameters, discontinuous source terms, and delay terms. The solution components ${\vartheta }_1$ and ${\vartheta }_2$ form initial layers at $(0,t)$ near the boundary region of the domain, while the interior layers emerge at $(1,t)$ due to the delay terms, as well as at $(0.3,t)$ and $(1.3,t)$ as a result of discontinuities in the source terms. Figure 4 represents the solutions of Case 2 of Example 2 for $M=16,N=384,{\epsilon }_1=\frac{\sigma }{64}$, and ${\epsilon }_2=\frac{\sigma }{16}$. The solution components ${\vartheta }_1$ and ${\vartheta }_2$ form initial layers at $(0,t)$ near the boundary region of the domain, while the interior layers emerge at $(1,t)$ due to the delay terms, and at $(1.3,t)$ as a result of discontinuities in the source terms. Figure 5 and Figure 6 present log-log plots of the maximum point-wise errors for Example 1 (Case 1 and Case 2) from Table 1 and Table 3, and for Example 2 (Case 1 and Case 2) from Table 5 and Table 7, respectively.

8. Conclusions

In this article, a classical layer-resolving numerical scheme was constructed for solving a system of two singularly perturbed time-dependent delay initial value problems with source terms that exhibit a jump at a specific point due to discontinuity and Robin initial conditions. This scheme incorporates a specially crafted piecewise uniform mesh to capture the layer behavior of the solutions in the initial and interior regions of the solution domain owing to the existence of delay and discontinuous source terms. The numerical experiments validated the computational outcomes, demonstrating strong concordance with the theoretical analysis. Additionally, the proposed method achieved first-order convergence. Moreover, the results clearly demonstrate that, as the number of mesh points increases, the order of convergence improves, while the error constant decreases. This demonstrates the robustness and efficiency of the proposed scheme. The results indicate that the proposed method remains stable and accurate across varying perturbation parameters, making it an efficient tool to solve complex equations. Future research will focus on extending the work to semilinear and nonlinear problems, with an emphasis on developing novel numerical methods to enhance computational efficiency.