A Certain Numerical Algorithm for Solving a Fractional Partial Model with a Neumann Constraint in a Hilbert Space

Abstract

This research examines a fractional partial advection–dispersion model, incorporating both mobile and immobile components, employing the Hilbert reproducing algorithm under an appropriate Neumann constraint condition. To effectively formulate the model while adhering to the specified constraints, two suitable Hilbert spaces are constructed, with the time-fractional Caputo derivative being utilized in the model’s formulation. Alongside the convergence analysis, a derived approximate solution formula is presented, and a systematic computational algorithm is developed to effectively implement the solution methodology. Numerical applications related to the proposed model are presented, complemented by tables and graphical illustrations. In conclusion, significant results are analyzed, and directions for future research are outlined.

1. Introduction

Mathematical modeling serves as a pivotal analytical framework for understanding complex real-world systems, enabling precise characterization of phenomena across diverse scientific and engineering disciplines. Recently, fractional calculus has emerged as a valuable mathematical tool for modeling systems that exhibit long memory effects and anomalous diffusion, providing a more precise representation than traditional differential models. Its increasing relevance stems from its ability to account for non-local dependencies and hereditary behaviors, making it particularly useful in cases where traditional derivatives fail to capture the full complexity of the underlying dynamics [1,2,3,4,5,6]. Among various formulations, the Caputo derivative is especially notable for its ability to preserve initial conditions while efficiently representing temporal evolution. Its applications have seen remarkable expansion across multiple disciplines. In electrical engineering, fractional-order circuits provide enhanced performance characteristics compared to their integer-order counterparts. Similarly, in fluid mechanics, fractional models enable a more precise description of viscoelastic behavior and anomalous transport phenomena. Additionally, they play a key role in statistical mechanics, signal processing, telecommunications, image processing, and biomedical engineering, aiding advanced modeling [7,8,9,10,11].

As mentioned earlier, there are various definitions for calculating fractional derivatives in the literature, including, but not limited to, the Riemann–Liouville, Grünwald–Letnikov, Crank–Nicholson, Hadamard, and Caputo derivatives [8,9,10,11,12]. Many of these definitions have been widely accepted in the scientific community due to their theoretical importance and practical applications, while others have faced some criticism. Caputo derivatives effectively model systems with memory and non-local interactions, offering more flexibility than traditional derivatives. They are crucial in capturing complex behaviors like anomalous diffusion in various scientific fields, including fluid dynamics and material science, by providing a more accurate representation of time-dependent phenomena. In this paper, we adopt the definition of the Time-Fractional Caputo Derivative (TFCD) due to its broad applicability and its ability to preserve hereditary characteristics, making it particularly well-suited for the proposed model. This definition is presented as follows:

$${\partial }_x^{\mu }G\left(y,x\right)=\left\{\begin{array}{l}{\displaystyle \frac{1}{\mathsf{\Gamma }\left(1-\mu \right)}}{\int }_0^x{\left(x-\xi \right)}^{-\mu }{\partial }_{\delta }^{1}G\left(y,\xi \right)d\xi ,\begin{array}{c} 0<\mu <1,\end{array}\\ {\partial }_x^{1}G\left(y,x\right), \mu =1.\end{array}\right.$$

(1)

The mobile–immobile advection–dispersion model (MIADM) describes solute transport in heterogeneous media, where solutes partially move in a mobile phase while another portion remains temporarily trapped in immobile zones. This model plays a crucial role in hydrology, environmental engineering, and porous media to predict contaminant dispersion and groundwater flow movements [13,14,15,16,17]. While this model employs ordinary derivatives to describe advection and dispersion, it often fails to account for anomalous diffusion and heterogeneous systems. To overcome these limitations, fractional calculus was integrated, forming the mobile–immobile fractional advection–dispersion model (MIFADM). The authors in [18] proposed a numerical framework based on the Crank–Nicolson difference scheme to solve the MIFADM, incorporating both Atangana–Baleanu–Caputo and Caputo fractional derivatives, while providing a comprehensive analysis of the system’s physical behavior. Focusing on solute transport dynamics, the authors in [19] applied a computational scheme for MIFADM, utilizing fractional derivatives to capture the mobile–immobile interactions and improve the accuracy of the model’s predictions in heterogeneous media. A Chebyshev spectral collocation approach has been introduced in [20] to solve variable-order space–time MIFADM using Caputo derivatives. This approach ensures spectral convergence within Chebyshev-weighted Sobolev spaces, offering higher computational efficiency than existing approaches while effectively handling Dirichlet and Neumann boundary conditions. The authors in [21] developed a high-order computational method for solving temporal MIFADM. They utilized the first-order finite difference technique with linear interpolation for time discretization and the Legendre polynomial-based collocation technique for spatial discretization, also providing a convergence analysis and stability discussion. An implicit finite difference scheme combined with a collocation method was used to implement a novel numerical method to deal with the time-variable MIFADM in [22]. This scheme addresses challenges in simulating solute transport with fractional derivatives in both time and space. For additional computational and numerical techniques to address MIFADM, we refer to [23,24,25] and the references cited therein.

Our discussion investigates the approximate solutions for MIFADM by adapting the Reproducing Hilbert space algorithm (RHA) in conjunction with the TFCD. Specifically, we conduct simulations to solve the following model while incorporating the terms of the Neumann constraint condition (NCCs):

$${ℇ}_1{\partial }_x^{\mu }G\left(y,x\right)+{ℇ}_2{\partial }_x^{1}G\left(y,x\right)+{ℇ}_3{\partial }_y^{1}G\left(y,x\right)-{ℇ}_4{\partial }_y^{2}G\left(y,x\right)+\mathcal{F}\left(G\left(y,x\right)\right)=\mathcal{K}\left(y,x\right),$$

(2)

associated with

$$\begin{array}{l}G\left(y,0\right)=0,\\ G_y\left(0,x\right)=0,\\ G_y\left(1,x\right)=0.\end{array}$$

(3)

Here, $\mathbb{A}:=\left[\mathrm{0,1}\right]$, $\mathcal{K}\in C\left(\mathbb{A}\times \mathbb{A}\to \mathbb{R}\right)$, $\mathcal{F}\in C\left(\mathbb{R}\to \mathbb{R}\right)$, $\mathrm{G}\in C\left(\mathbb{A}\times \mathbb{A}\to \mathbb{R}\right)$, and ${\mathrm{ℇ}}_1,{\mathrm{ℇ}}_2,{\mathrm{ℇ}}_3,{\mathrm{ℇ}}_4\in {\mathbb{R}}^{+}$.

Typically, NCCs define the behavior of a function at a domain’s boundary by specifying its normal derivative, often as a constant or zero. This is particularly useful for solving physical problems, as it allows the function to be determined based on the given data and the differential equation that governs the model. For example, in heat transfer problems, the NCCs can be used to specify the amount of heat that is transferred across the boundary [26,27,28]. Indeed, this type of constraint can be used to exploit the symmetries of a region and can be imposed by selecting flux/source and filling in the boundary flux/source section.

Based on the previous discussion, the numerical investigation of MIFADM under NCCs based on TFCD is still rare and, to our knowledge, is absent from existing databases. This study introduces the promising numerical algorithm, RHA, that utilizes specialized function spaces to effectively approximate solutions. By leveraging advanced mathematical theories, this algorithm offers significant potential for solving complex problems with increased efficiency, exhibiting exceptional versatility in handling nonlinear operators while inherently preserving boundary condition enforcement through its functional construction; see [29,30,31,32,33] and [34,35], respectively. Notably, its foundation in functional analysis ensures rigorous theoretical guarantees regarding solution existence and convergence properties. Unlike traditional numerical schemes, this algorithm eliminates the need for spatial meshing, linearization, specialized singularity treatments, or complex preconditioning strategies. These attributes make it particularly suitable for challenging scenarios involving non-local fractional operators and multiscale phenomena. However, the proposed algorithm has certain limitations, such as the computational cost associated with higher-order approximations, which may become significant when dealing with complex systems or large-scale problems. Additionally, the algorithm’s accuracy may be affected by the choice of fractional parameters or basis functions, which may require careful consideration for optimal results in some contexts. Despite these challenges, the proposed algorithm remains easy to understand, flexible, and a powerful tool for handling a wide range of fractional models.

After introducing the model and its significance, we now review the paper’s structure in detail. In the first part, the appropriate spaces are defined and constructed. The second part develops both analytical and numerical solutions. The third part discusses the convergence theory. The fourth part outlines the algorithm steps and provides an applied example, along with related tables and graphics. Finally, the conclusion paragraph of the research is presented.

2. Appropriate Hilbert Spaces

Here, we present the shape and structure of the spaces, in addition to some mathematical details. Denote by $\mathrm{D}=\mathbb{A}\otimes \mathbb{A}$, ${‖G‖}_{\zeta }^2={⟨G\left(\delta ,y\right),G\left(\delta ,y\right)⟩}_{\zeta }$ and ${\partial }_y^3{\partial }_x^{3}G, {\partial }_y^1{\partial }_x^{1}G\in L^2\left(\mathrm{D}\right)$. Then, we construct ${\widehat{\mathrm{\wp }}}_{\left(\mathrm{1,2}\right)}\left(\mathbb{A}\right)=\{\eta :\eta \in AC\left(\mathbb{A}\right)\}$ with

$${⟨{\eta }_1\left(x\right),{\eta }_2\left(x\right)⟩}_{{\widehat{\mathrm{\wp }}}_{\left(\mathrm{1,2}\right)}}={\eta }_1\left(0\right){\eta }_2\left(0\right)+{\int }_{\mathbb{A}}{\eta }_1^{\prime }\left(y\right){\eta }_2^{\prime }\left(y\right)dy,$$

(4)

and ${\mathrm{\wp }}_{\left(\mathrm{2,3}\right)}\left(\mathbb{A}\right)=\left\{\eta :\eta ,{\eta }^{\prime },{\eta }^{\prime \prime }\in AC\left(\mathbb{A}\right)\wedge {\eta }^{\prime }\left(0\right)={\eta }^{\prime }\left(1\right)=0\right\},$ where

$${⟨{\eta }_1\left(y\right),{\eta }_2\left(y\right)⟩}_{{\mathrm{\wp }}_{\left(\mathrm{2,3}\right)}}={\displaystyle \sum _{\lambda =0}^1}{\eta }_1^{\left(\lambda \right)}\left(0\right){\eta }_2^{\left(\lambda \right)}\left(0\right)+{\eta }_1\left(1\right){\eta }_2\left(1\right)+{\int }_{\mathbb{A}}{\eta }_1^{\prime \prime \prime }\left(y\right){\eta }_2^{\prime \prime \prime }\left(y\right)dy.$$

(5)

In what follows, we have the following useful definition:

Definition 1

([30]). Denote by $\Theta \left(D\right)=\left\{G:{\partial }_y^2{\partial }_x^{2}G\left(y,x\right)\in C\left(D\right)\wedge G\left(y,0\right)=G\left(0,x\right)=G\left(1,x\right)=0\right\}$, where

$$\begin{array}{l}{⟨G_1\left(y,x\right),G_2\left(y,x\right)⟩}_{\Theta }\\ ={\displaystyle \sum _{\lambda =0}^1}{〈{\partial }_x^{\lambda }{G}_1\left(y,0\right),{\partial }_x^{\lambda }{G}_2\left(y,0\right)〉}_{{\wp }_{\left(\mathrm{2,3}\right)}}\\ +{\int }_{\mathbb{A}}\left[{\displaystyle \sum _{\lambda =0}^1}{\partial }_y^{\lambda }{\partial }_x^2{G}_1\left(0,x\right){\partial }_y^{\lambda }{\partial }_x^2{G}_2\left(0,x\right)+{\partial }_x^2{G}_1\left(1,x\right){\partial }_x^2{G}_2\left(1,x\right)\right]dx\\ +{\int }_{\mathbb{A}}{\int }_{\mathbb{A}}{\partial }_y^3{\partial }_x^2{G}_1\left(y,x\right){\partial }_y^3{\partial }_x^2{G}_2\left(y,x\right)dydx.\end{array}$$

(6)

In what follows, we recall some results that will be useful in the sequel.

Theorem 1

([30]). In $\Theta \left(D\right)$, we have ${\mathcal{E}}_{\left(\alpha ,\beta \right)}\left(y,x\right)={\mathcal{Z}}_{\alpha }^{\left\{3\right\}}\left(y\right){\mathcal{Z}}_{\beta }^{\left\{2\right\}}\left(x\right)$, such that

$${\mathcal{Z}}_{\alpha }^{\left\{3\right\}}\left(y\right)={\displaystyle \frac{1}{120}}\left\{\begin{array}{c}I\left(y,\alpha \right),y\le \alpha ,\\ I\left(\alpha ,y\right),y>\alpha ,\end{array}\right.$$

(7)

and

$${\mathcal{Z}}_{\beta }^{\left\{2\right\}}\left(x\right)=\left\{\begin{array}{c}\beta x+{\displaystyle \frac{1}{2}}\beta x^2-{\displaystyle \frac{1}{6}}{x}^3,x\le \beta ,\\ \beta x+{\displaystyle \frac{1}{2}}{\beta }^{2}x-{\displaystyle \frac{1}{6}}{\beta }^3,x>\beta .\end{array}\right.$$

(8)

Further, we have

$$I\left(y,\alpha \right)={\left(1-\alpha \right)}^3{y}^3\left[6y^2{\alpha }^2+3y\alpha \left(y-5\alpha \right)+10{\alpha }^2-5y\alpha +y^2.\right]$$

(9)

Definition 2

([30]). By $\mathfrak{B}\left(D\right),$ we denote the set of functions such that $\mathfrak{B}\left(D\right)=\left\{G:G\in C\left(D\right)\right\}$, where

$$\begin{array}{l}{⟨G_1\left(y,x\right),G_2\left(y,x\right)⟩}_{\mathfrak{B}}\\ =\langle G_1\left(y,0\right),G_2\left(y,0\right){\rangle }_{{\widehat{\wp }}_{\left(\mathrm{1,2}\right)}}\\ +{\int }_{\mathbb{A}}{\partial }_x^1{G}_1\left(0,x\right){\partial }_x^1{G}_2\left(0,x\right)dx\\ +{\int }_{\mathbb{A}}{\int }_{\mathbb{A}}{\partial }_y^1{\partial }_x^1{G}_1\left(y,x\right){\partial }_y^1{\partial }_x^1{G}_2\left(y,x\right)dydx.\end{array}$$

(10)

Theorem 2

([30]). In $\mathfrak{B}\left(D\right)$, we have ${\mathcal{G}}_{\left(\alpha ,\beta \right)}\left(y,x\right)={\widehat{\mathcal{Z}}}_{\alpha }^{\left\{1\right\}}\left(y\right){\mathcal{Z}}_{\beta }^{\left\{1\right\}}\left(x\right)$, such that

$${\widehat{\mathcal{Z}}}_{\alpha }^{\left\{1\right\}}\left(y\right)=1+\mathit{min}\left(y,\alpha \right),$$

(11)

and

$${\mathcal{Z}}_{\beta }^{\left\{1\right\}}\left(x\right)=1+\min \left(\beta ,x.\right)$$

(12)

To start with the RHA, assume that ${\left\{\left(y_r,x_r\right)\right\}}_{r=1}^{\infty }$ is a countable dense in $D$. Also, assume that

$$\mathcal{H}\left[G\right]\left(y,x\right)={ℇ}_1{\partial }_x^{\mu }G\left(y,x\right)+{ℇ}_2{\partial }_x^{1}G\left(y,x\right)+{ℇ}_3{\partial }_y^{1}G\left(y,x\right)-{ℇ}_4{\partial }_y^{2}G\left(y,x\right),$$

(13)

and

$$\mathcal{H}\begin{array}{c}:\mathsf{\Theta }\left(\mathrm{D}\right)\to \mathfrak{B}\left(\mathrm{D}\right),\end{array}$$

(14)

$$\begin{array}{c}\mathcal{H}\left[G\right]\left(y,x\right)=\mathrm{G}\left(y,x,G\left(y,x\right)\right)\equiv \mathcal{K}\left(y,x\right)-\mathcal{F}\left(G\left(y,x.\right)\right)\end{array}$$

(15)

3. Numerical Solution and Convergence

In this part, by taking into account [30], assume that ${‖G_{k-1}‖}_{\mathsf{\Theta }}<\infty$ and $\mathrm{G}\left(y,x,G\left(y,x\right)\right)$ is continuous. Then, the main part to be clarified here is that if ${‖G_{k-1}-G‖}_{\mathsf{\Theta }}\to 0$ and $\left(y_k,x_k\right)\to \left(\alpha ,\beta \right)$ as $k\to \infty$, then as $k\to \infty$, and $G$ satisfies the condition that $G\left(y_k,x_k,G_{k-1}\left(y_k,x_k\right)\right)\to G\left(\alpha ,\beta ,G\left(\alpha ,\beta \right)\right)$.

Suppose that $G\in C\left(\mathrm{D}\right)$ and ${\left\{{\overline{\psi }}_{\mathrm{r}}\left(y,x\right)\right\}}_{\mathrm{r}=1}^{\infty }$ is an orthonormal system. Then, ${\left\{{\psi }_{\mathrm{r}}\left(y,x\right)\right\}}_{\mathrm{r}=1}^{\infty }$ is clearly complete in $\mathsf{\Theta }\left(\mathrm{D}\right)$ and ${\psi }_{\mathrm{r}}\left(y,x\right)={\left.{\mathcal{H}}_{\left(\alpha ,\beta \right)}\left[\mathcal{E}\right]\left(y,x\right)\right|}_{\left(\alpha ,\beta \right)=\left(y_{\mathrm{r}},x_{\mathrm{r}}\right)}.$ However, let us consider the following result.

Theorem 3.

In $\Theta \left(D\right)$, we have ${\left\{{\psi }_r\left(y,x\right)\right\}}_{r=1}^{\infty }$ is complete, and

$${\psi }_r\left(y,x\right)={\left.{\mathcal{H}}_{\left(\alpha ,\beta \right)}\left[\mathcal{E}\right]\left(y,x\right)\right|}_{\left(\alpha ,\beta \right)=\left(y_r,x_r.\right)}$$

(16)

Proof. 

First, from the previous analysis, we have

$$\begin{array}{l}{\psi }_{\mathrm{r}}\left(y,x\right)={\mathcal{H}}^{\ast }\left[{\psi }_{\mathrm{r}}\right]\left(y,x\right)\\ ={⟨{\mathcal{H}}^{\ast }\left[{\psi }_{\mathrm{r}}\right]\left(\alpha ,\beta \right),{\mathcal{E}}_{\left(y,x\right)}\left(\alpha ,\beta \right)⟩}_{\mathsf{\Theta }}\\ ={⟨{\psi }_{\mathrm{r}}\left(\alpha ,\beta \right),{\mathcal{H}}_{\left(\alpha ,\beta \right)}\left[{\mathcal{E}}_{\left(y,x\right)}\right]\left(\alpha ,\beta \right)⟩}_{\mathfrak{B}}\\ ={\left.{\mathcal{H}}_{\left(\alpha ,\beta \right)}\left[{\mathcal{E}}_{\left(y,x\right)}\right]\left(\alpha ,\beta \right)\right|}_{\left(\alpha ,\beta \right)=\left(y_{\mathrm{r}},x_{\mathrm{r}}\right)}\\ ={\left.{\mathcal{H}}_{\left(\alpha ,\beta \right)}\left[{\mathcal{E}}_{\left(\alpha ,\beta \right)}\right]\left(y,x\right)\right|}_{\left(\alpha ,\beta \right)=\left(y_{\mathrm{r}},x_{\mathrm{r}}\right)}.\end{array}$$

(17)

Or, equivalently, ${\psi }_{\mathrm{r}}\left(y,x\right)\in \mathsf{\Theta }\left(\mathrm{D}\right)$. So, for any $G\in \mathsf{\Theta }\left(\mathrm{D}\right)$, it follows that if ${⟨G\left(y,x\right),{\psi }_{\mathrm{r}}\left(y,x\right)⟩}_{\mathsf{\Theta }}=0$ with $\mathrm{r}=\mathrm{1,2},\dots$, then

$$\begin{array}{l}{⟨\sigma \left(y,x\right),{\psi }_{\mathrm{r}}\left(y,x\right)⟩}_{\mathsf{\Theta }}={⟨G\left(y,x\right),{\mathcal{H}}^{\ast }\left[{\psi }_{\mathrm{r}}\right]\left(y,x\right)⟩}_{\mathsf{\Theta }}\\ ={⟨\mathcal{H}\left[G\right]\left(y,x\right),{\psi }_{\mathrm{r}}\left(x\right)⟩}_{\mathfrak{B}}\\ =\mathcal{H}\left[G\right]\left(y_{\mathrm{r}},x_{\mathrm{r}}\right)\\ =0.\end{array}$$

(18)

However, by the existence of ${\mathcal{H}}^{-1}$, we infer that $\mathcal{H}\left[\sigma \right]\left(y,x\right)=0$ or $G=0$. This completes the proof of the theorem. □

Theorem 4.

If ${\mathcal{S}}_r={\displaystyle \sum _{j=1}^r}{\mathcal{o}}_{rj}G\left(y_j,x_j,G\left(y_j,x_j\right)\right)$, then we have

$$\begin{array}{c}G\left(y,x\right)={\displaystyle \sum _{r=1}^{\infty }}{\mathcal{S}}_r{\overline{\psi }}_r\left(y,x\right).\end{array}$$

(19)

Proof. 

Fundamentally, ${\displaystyle \sum _{\mathrm{r}=1}^{\infty }}{\mathcal{S}}_{\mathrm{r}}{\overline{\psi }}_{\mathrm{r}}\left(y,x\right)$ is the Fourier expansion on ${\left\{{\overline{\psi }}_{\mathrm{r}}\left(y,x\right)\right\}}_{\mathrm{r}=1}^{\mathrm{\infty }}$. Thus, we have

$$\begin{array}{l}G\left(y,x\right)={\displaystyle \sum _{r=1}^{\infty }}{⟨G\left(y,x\right),{\overline{\psi }}_r\left(y,x\right)⟩}_{\Theta }{\overline{\psi }}_r\left(y,x\right)\\ ={\displaystyle \sum _{r=1}^{\infty }}\langle G\left(y,x\right),{\displaystyle \sum _{j=1}^r}{\mathcal{o}}_{rj}{\psi }_j\left(y,x\right){\rangle }_{\Theta }{\overline{\psi }}_r\left(y,x\right)\\ ={\displaystyle \sum _{r=1}^{\infty }}{\displaystyle \sum _{j=1}^r}{\mathcal{o}}_{rj}{⟨G\left(y,x\right),{\mathcal{H}}^{*}\left[{\sigma }_j\right]\left(y,x\right)⟩}_{\Theta }{\overline{\psi }}_r\left(y,x\right)\\ ={\displaystyle \sum _{r=1}^{\infty }}{\displaystyle \sum _{j=1}^r}{\mathcal{o}}_{rj}{⟨\mathcal{H}\left[G\right]\left(y,x\right),{\mathcal{G}}_{\left(y_j,x_j\right)}\left(y,x\right)⟩}_{\mathfrak{B}}{\overline{\psi }}_r\left(y,x\right)\\ ={\displaystyle \sum _{r=1}^{\infty }}{\displaystyle \sum _{j=1}^r}{\mathcal{o}}_{rj}\mathcal{H}\left[G\right]\left(y_j,x_j\right){\overline{\psi }}_r\left(y,x\right)\\ ={\displaystyle \sum _{r=1}^{\infty }}{\displaystyle \sum _{j=1}^r}{\mathcal{o}}_{rj}G\left(y_j,x_j,G\left(y_j,x_j\right)\right){\overline{\psi }}_r\left(y,x\right)\\ ={\displaystyle \sum _{r=1}^{\infty }}{\mathcal{S}}_r{\overline{\psi }}_r\left(y,x\right). \end{array}$$

(20)

This completes the proof of the theorem. □

Corollary 1.

The $k$-approximation fulfills the equation

$$G_k\left(y,x\right)={\displaystyle \sum _{r=1}^k}{\mathcal{S}}_r{\overline{\psi }}_r\left(y,x\right).$$

(21)

Theorem 5.

If ${‖G_{k-1}-G‖}_{\Theta }\to 0$ and $\left(y_k,x_k\right)\to \left(\alpha ,\beta \right)$ as $k\to \infty$, then we have

$$G\left(y_k,x_k,G_{k-1}\left(y_k,x_k\right)\right)\to G\left(\alpha ,\beta ,G\left(\alpha ,\beta \right)\right)\mathrm{as} k\to \infty .$$

(22)

Proof. 

Applying the identities

$$\begin{array}{l}{G}_{k-1}\left(\alpha ,\beta \right)={⟨G_{k-1}\left(y,x\right),{\mathcal{S}}_{\left(\alpha ,\beta \right)}\left(y,x\right)⟩}_{\mathsf{\Theta }},\\ G\left(\alpha ,\beta \right)={⟨G\left(y,x\right),{\mathcal{S}}_{\left(\alpha ,\beta \right)}\left(y,x\right)⟩}_{\mathsf{\Theta }},\\ \nabla G\left(\mathrm{a},\mathrm{b}\right)={\partial }_{y}G\left(\mathrm{a},\mathrm{b}\right)e_1+{\partial }_{x}G\left(\mathrm{a},\mathrm{b}\right)e_2,\\ e_1=⟨\mathrm{1,0}⟩,\\ e_2=⟨\mathrm{0,1}⟩,\end{array}$$

(23)

we reach to the inequalities

$$\begin{array}{r}\left|G_{k-1}\left(\alpha ,\beta \right)-G\left(\alpha ,\beta \right)\right|=\left|{⟨G_{k-1}\left(y,x\right)-G\left(y,x\right),{\mathcal{S}}_{\left(\alpha ,\beta \right)}\left(y,x\right)⟩}_{\mathsf{\Theta }}\right|\\ \le {‖G_{k-1}-G‖}_{\mathsf{\Theta }}{‖{\mathcal{S}}_{\left(\alpha ,\beta \right)}\left(y,x\right)‖}_{\mathsf{\Theta }}\le \mathrm{\aleph }{‖G_{k-1}-G‖}_{\mathsf{\Theta }}\end{array}.$$

(24)

and

$$\begin{array}{c}\left|\nabla G_{k-1}\left(\zeta ,\delta \right)\right|=\sqrt{{\left({\partial }_y{G}_{k-1}\left(\zeta ,\delta \right)\right)}^2+{\left({\partial }_x{G}_{k-1}\left(\zeta ,\delta \right)\right)}^2}\\ \le \sqrt{{\mathrm{\aleph }}_1^2+{\mathrm{\aleph }}_2^2}{‖G_{k-1}‖}_{\mathsf{\Theta }}.\end{array}$$

(25)

Merging (24) and (25) reveals that

$$\begin{array}{l}\left|G_{k-1}\left(y_k,x_k\right)-G\left(\alpha ,\beta \right)\right|=\left|G_{k-1}\left(y_k,x_k\right)-G_{k-1}\left(\alpha ,\beta \right)+G_{k-1}\left(\alpha ,\beta \right)-G\left(\alpha ,\beta \right)\right|\\ \le \left|G_{k-1}\left(y_k,x_k\right)-G_{k-1}\left(\alpha ,\beta \right)\right|+\left|G_{k-1}\left(\alpha ,\beta \right)-G\left(\alpha ,\beta \right)\right|\\ \le \left|\nabla G_{k-1}\left(\zeta ,\delta \right)\right|\left|\left(y_k,x_k\right)-\left(\alpha ,\beta \right)\right|+\left|G_{k-1}\left(\alpha ,\beta \right)-G\left(\alpha ,\beta \right)\right|\\ \le \sqrt{{\mathrm{\aleph }}_1^2+{\mathrm{\aleph }}_2^2}{‖G_{k-1}‖}_{\mathsf{\Theta }}\left|\left(y_k,x_k\right)-\left(\alpha ,\beta \right)\right|+\mathrm{\aleph }{‖G_{k-1}-G‖}_{\mathsf{\Theta }}.\end{array}$$

(26)

Note that when ${‖G_{k-1}-G‖}_{\mathsf{\Theta }}\to 0$, $\left(y_k,x_k\right)\to \left(\alpha ,\beta \right)$ as $k\to \mathrm{\infty }$, and $as {‖G_{k-1}‖}_{\mathsf{\Theta }}<\infty$, we have $\left|G_{k-1}\left(y_k,x_k\right)-G\left(\alpha ,\beta \right)\right|\to 0$ as $k\to \mathrm{\infty }$. Hence, the continuum hypothesis of $G$ shows that $G\left(y_k,x_k,G_{k-1}\left(y_k,x_k\right)\right)\to G\left(\alpha ,\beta ,G\left(\alpha ,\beta \right)\right)$ as $k\to \mathrm{\infty }$. The proof of the theorem is therefore completed. □

Theorem 6.

We have $G_k\left(y,x\right)\to G\left(y,x\right)$ uniformly as $k\to \infty$.

Proof. 

As $G_k\left(y,x\right)={\displaystyle \sum _{\mathrm{r}=1}^k}{\mathcal{S}}_{\mathrm{r}}{\overline{\psi }}_{\mathrm{r}}\left(y,x\right)$, one may write

$$G_{k+1}\left(y,x\right)=G_k\left(y,x\right)+{\mathcal{S}}_{k+1}{\overline{\psi }}_{k+1}\left(y,x\right).$$

(27)

Therefore, the orthogonality of ${\left\{{\overline{\psi }}_{\mathrm{r}}\left(y,x\right)\right\}}_{\mathrm{r}=1}^{\mathrm{\infty }}$ informs (27) can be written as

$$\begin{array}{l}{‖G_{k+1}‖}_{\mathsf{\Theta }}^2={‖G_k‖}_{\mathsf{\Theta }}^2+{\mathcal{S}}_{k+1}^2\\ ={‖G_{k-1}‖}_{\mathsf{\Theta }}^2+{\mathcal{S}}_k^2+{\mathcal{S}}_{k+1}^2\\ =\dots \\ ={‖G_0‖}_{\mathsf{\Theta }}^2+{\displaystyle \sum _{\mathrm{r}=1}^k}{\mathcal{S}}_{\mathrm{r}}^2.\end{array}$$

(28)

Thus, ${‖G_{k+1}‖}_{\mathsf{\Theta }}\ge {‖G_k‖}_{\mathsf{\Theta }}$. By the fact that ${‖G_k‖}_{\mathsf{\Theta }}<\infty$, we conclude that ${‖G_k‖}_{\mathsf{\Theta }}$ is convergent and $\mathrm{\aleph }$ can be found, such that ${\displaystyle \sum _{\mathrm{r}=1}^{\infty }}{\mathcal{S}}_{\mathrm{r}}^2=\mathrm{\aleph }$. Hence, we have ${\left\{{\mathcal{S}}_{\mathrm{r}}^2\right\}}_{\mathrm{r}=1}^{\mathrm{\infty }}\in l^2$ and

$$\left(G_s\left(x\right)-G_{s-1}\left(x\right)\right)\perp \left(G_{s-1}\left(x\right)-G_{s-2}\left(x\right)\right)\perp \dots \perp \left(G_{k+1}\left(x\right)-G_k\left(x\right)\right).$$

(29)

Still, at $\zeta >k$, we attain

$${‖G_s-G_k‖}_{\mathsf{\Theta }}^2={‖G_s-G_{s-1}+G_{s-1}-\dots +G_{k+1}-G_k‖}_{\mathsf{\Theta }}^2={‖G_s-G_{s-1}‖}_{\mathsf{\Theta }}^2+{‖G_{s-1}-G_{s-2}‖}_{\mathsf{\Theta }}^2+\dots +{‖G_{k+1}-G_k‖}_{\mathsf{\Theta }}^2.$$

(30)

Likewise, ${‖G_s-G_{s-1}‖}_{\mathsf{\Theta }}^2={\mathcal{S}}_s^2$ and ${‖G_s-G_k‖}_{\mathsf{\Theta }}^2={\displaystyle \sum _{\mathrm{r}=k+1}^s}{\mathcal{S}}_{\mathrm{r}}^2\to 0$ as $s,k\to \mathrm{\infty }$. Using the completeness of $\mathsf{\Theta }\left(\mathrm{D}\right),$ we find $G\in \mathsf{\Theta }\left(\mathrm{D}\right)$ and $G_k\left(y,x\right)\to G\left(y,x\right)$ as $k\to \mathrm{\infty }$. Applying $\underset{k\to \infty }{\lim }\left(·\right)$ for the formula

$$G_k\left(y,x\right)={\displaystyle \sum _{\mathrm{r}=1}^k}{\mathcal{S}}_{\mathrm{r}}{\overline{\psi }}_{\mathrm{r}}\left(y,x\right)$$

(31)

yields that $G\left(y,x\right)={\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{\infty }}}{\mathcal{S}}_{\mathrm{r}}{\overline{\psi }}_{\mathrm{r}}\left(y,x\right)$. By taking into account the property $\mathcal{H}$, we obtain

$$\mathcal{H}\left[G\right]\left(y,x\right)={\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{\infty }}}{\mathcal{S}}_{\mathrm{r}}\mathcal{H}\left[{\overline{\psi }}_{\mathrm{r}}\right]\left(y,x\right)$$

(32)

Therefore, from (32), we derive

$$\begin{array}{l}\mathcal{H}\left[G\right]\left(y_{\mathsf{\gamma }},x_{\mathsf{\gamma }}\right)={\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{\infty }}}{\mathcal{S}}_{\mathrm{r}}{⟨\mathcal{H}\left[{\overline{\psi }}_{\mathrm{r}}\right]\left(y,x\right),{\sigma }_{\mathsf{\gamma }}\left(y,x\right)⟩}_{\mathfrak{B}}\\ ={\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{\infty }}}{\mathcal{S}}_{\mathrm{r}}{⟨{\overline{\psi }}_{\mathrm{r}}\left(y,x\right),{\mathcal{H}}^{\ast }\left[{\sigma }_{\mathsf{\gamma }}\right]\left(y,x\right)⟩}_{\mathsf{\Theta }}\\ ={\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{\infty }}}{\mathcal{S}}_{\mathrm{r}}{⟨{\overline{\psi }}_{\mathrm{r}}\left(y,x\right),{\psi }_{\mathsf{\gamma }}\left(y,x\right)⟩}_{\mathsf{\Theta }}.\end{array}$$

(33)

Immediately after calculations, (33) gives

$$\begin{array}{l}{\displaystyle \sum _{\gamma =1}^r}{\mathcal{o}}_{r\gamma }\mathcal{H}\left[G\right]\left(y_{\gamma },x_{\gamma }\right)={\displaystyle \sum _{r=1}^{\infty }}{\mathcal{S}}_r{⟨{\overline{\psi }}_r\left(y,x\right),{\displaystyle \sum _{\gamma =1}^r}{\mathcal{o}}_{r\gamma }{\psi }_{\gamma }\left(y,x\right)⟩}_{\Theta }\\ ={\displaystyle \sum _{r=1}^{\infty }}{\mathcal{S}}_r{⟨{\overline{\psi }}_r\left(y,x\right),{\overline{\psi }}_{\gamma }\left(y,x\right)⟩}_{\Theta }\\ ={\mathcal{S}}_{\gamma }.\end{array}$$

(34)

Hence, it follows that

$$\mathcal{H}\left[G\right]\left(y_{\gamma },x_{\gamma }\right)=\mathrm{G}\left(y_{\gamma },x_{\gamma },G_{\gamma -1}\left(y_{\gamma },x_{\gamma }\right)\right).$$

(35)

Applying the density of ${\left\{\left(y_{\mathrm{r}},x_{\mathrm{r}}\right)\right\}}_{\mathrm{r}=1}^{\infty }$ at each $\left(\alpha ,\beta \right)\in \mathrm{D}$, we obtain ${\left\{\left(y_{k_{\lambda }},x_{k_{\lambda }}\right)\right\}}_{\lambda =1}^{\infty }\to \left(\alpha ,\beta \right)$ as $\lambda \to \mathrm{\infty }$. Furthermore,

$$\mathcal{H}\left[G\right]\left(y_{k_{\lambda }},x_{k_{\lambda }}\right)=\mathrm{G}\left(y_{k_{\lambda }},x_{k_{\lambda }},G_{k_{\lambda }-1}\left(y_{k_{\lambda }},x_{k_{\lambda }}\right)\right).$$

(36)

By assuming $\lambda \to \mathrm{\infty }$, we, from (36), obtain $\mathcal{H}\left[G\right]\left(\alpha ,\beta \right)=\mathrm{G}\left(\alpha ,\beta ,G\left(\alpha ,\beta \right)\right).$ Therefore, $G\left(y,x\right)$ $\in \mathcal{H}\left[G\right]\left(y,x\right)$. However, ${\overline{\psi }}_{\mathrm{r}}\left(y,x\right)\in \Theta \left(\mathrm{D}\right)$, which assures $G\left(y,x\right)$ agrees well with the given NCC.

Thus,

$${‖G\left(y,x\right)-G_k\left(y,x\right)‖}_{\mathsf{\Theta }}^2={\displaystyle \sum _{\mathrm{r}=k+1}^{\infty }}{\left({\displaystyle \sum _{j=1}^{\mathrm{r}}}{\mathcal{o}}_{\mathrm{r}j}\mathrm{G}\left(y_j,x_j,G\left(y_j,x_j\right)\right)\right)}^2.$$

(37)

The aforementioned Equation (37) reveals that ${\left\{G\left(y,x\right)-G_k\left(y,x\right)\right\}}_{k=1}^{\infty }\searrow$ in ${‖\cdot ‖}_{\mathsf{\Theta }}$. Once again, as ${\displaystyle \sum _{\mathrm{r}=1}^{\infty }}{\mathcal{S}}_{\mathrm{r}}{\overline{\psi }}_{\mathrm{r}}\left(y,x\right)<\infty$, we derive ${‖G\left(y,x\right)-G_k\left(y,x\right)‖}_{\mathsf{\Theta }}^2\to 0$ as $k\to \mathrm{\infty }$. This recognizes that ${\left\{G\left(y,x\right)-G_k\left(y,x\right)\right\}}_{k=1}^{\infty }$ in ${‖\cdot ‖}_{\mathsf{\Theta }}$. Hence, this gives ${\left\{G\left(y,x\right)-G_k\left(y,x\right)\right\}}_{k=1}^{\infty }\searrow$ in ${‖\cdot ‖}_{\mathsf{\Theta }}$, such that

$${‖G\left(y,x\right)-G_k\left(y,x\right)‖}_{\mathsf{\Theta }}\to 0\mathrm{as} k\to \infty$$

(38)

The proof of the theorem is therefore completed. □

4. Algorithms and Application

The computational results in this section were obtained using MATHEMATICA 11, whilst ${\mathrm{D}}_y={\displaystyle \frac{1}{\mathcal{P}}}$ and ${\mathrm{D}}_x={\displaystyle \frac{1}{\mathcal{Q}}}$ in $\mathbb{A},$ where $\mathcal{P}\times \mathcal{Q}\in \mathbb{N}$ or $\left(y_n,x_m\right)=\left(n{\mathrm{D}}_y,m{\mathrm{D}}_x\right)$ with $n=\mathrm{0,1},\cdots ,\mathcal{P}$ and $m=\mathrm{0,1},\cdots ,\mathcal{Q}$.

Find ${\mathcal{o}}_{\mathrm{r}j}$ with $\mathrm{r}=\mathrm{2,3},\dots$ and $j=\mathrm{1,2},\dots ,\mathrm{r}$ as follows: ${\mathcal{o}}_{11}={‖{\psi }_1‖}_{\mathsf{\Theta }}$, ${\left.{\mathcal{o}}_{\mathrm{r}\lambda }\right|}_{\mathrm{r}>\lambda }=-{\displaystyle \frac{1}{\sqrt{{\mathcal{b}}_{\mathrm{r}}\left(x\right)}}}{\displaystyle \sum _{p=\lambda }^{\mathrm{r}-1}}{\mathcal{a}}_{\mathrm{r},p}^2\left(x\right){\mathcal{o}}_{p\lambda }$, together with ${\left.{\mathcal{o}}_{\mathrm{r}\mathrm{r}}\right|}_{\mathrm{r}\ne 1}={\displaystyle \frac{1}{\sqrt{{\mathcal{b}}_{\mathrm{r}}\left(x\right)}}}$.

Application 1.

Consider the next MIFADM

$${\partial }_x^{\mu }G\left(y,x\right)+{\partial }_x^{1}G\left(y,x\right)+{\partial }_y^{1}G\left(y,x\right)-{\partial }_y^{2}G\left(y,x\right)+G^3\left(y,x\right)=\mathcal{K}\left(y,x\right),$$

(39)

associated with the NCC

$$\begin{array}{l}G\left(y,0\right)=0,\\ G_y\left(0,x\right)=0,\\ G_y\left(1,x\right)=0.\end{array}$$

(40)

In Application 1, $\mathcal{K}\left(y,x\right)$ is therefore determined as

$$\begin{array}{c}G\left(y,x\right)\end{array}=x^{\mu }\left(1-x\right)y^2{\left(1-y\right)}^2\exp \left(y\right).$$

(41)

However, indeed, $\left(y,x\right)\in \mathrm{D}$, $\mu \in \left(\left.\mathrm{0,1}\right]\right.$, $\mathcal{H}\left[G\right]\left(y,x\right)={\partial }_x^{\mu }G\left(y,x\right)+{\partial }_x^{1}G\left(y,x\right)+{\partial }_y^{1}G\left(y,x\right)-{\partial }_y^{2}G\left(y,x\right)$, and $G$ takes the shapes $G\left(y,x,G\left(y,x\right)\right)=\mathcal{K}\left(y,x\right)-G^3\left(y,x\right)$.

Application 2.

Consider the next MIFADM

$${\partial }_x^{\mu }G\left(y,x\right)+2{\partial }_x^{1}G\left(y,x\right)+2{\partial }_y^{1}G\left(y,x\right)-5{\partial }_y^{2}G\left(y,x\right)+G^2\left(y,x\right)=\mathcal{K}\left(y,x\right),$$

(42)

associated with the NCC

$$\begin{array}{l}G\left(y,0\right)=0,\\ G_y\left(0,x\right)=0,\\ G_y\left(1,x\right)=0.\end{array}$$

(43)

In Application 2, $\mathcal{K}\left(y,x\right)$ is determined as

$$\begin{array}{c}G\left(y,x\right)\end{array}=\pi x^{\mu +1}{\left(1-x\right)}^{3}y{\left(1-y\right)}^2\sin \left(y\right).$$

(44)

Indeed, $\left(y,x\right)\in \mathrm{D}$, $\mu \in \left(\left.\mathrm{0,1}\right]\right.$, $\mathcal{H}\left[G\right]\left(y,x\right)={\partial }_x^{\mu }G\left(y,x\right)+2{\partial }_x^{1}G\left(y,x\right)+2{\partial }_y^{1}G\left(y,x\right)-5{\partial }_y^{2}G\left(y,x\right)$, and $G$ takes the shapes $G\left(y,x,G\left(y,x\right)\right)=\mathcal{K}\left(y,x\right)-G^2\left(y,x\right)$.

The tabulated data concerning Applications 1 and 2 are listed in

Table 1. Results of error $\left|G_k\left(y_n,x_m\right)-G\left(y_n,x_m\right)\right|$ in Application 1 for $\mu \in \left\{\mathrm{0.7,0.8,0.9,1}\right\}$ through $\mathrm{D}$.

${\mathit{y}}_{\mathit{n}}$	${\mathit{x}}_{\mathit{m}}$	$\mathit{\mu }=0.7$	$\mathit{\mu }=0.8$	$\mathit{\mu }=0.9$	$\mathit{\mu }=1$
$0$	$0.2$	$8.106\times {10}^{-3}$	$1.174\times {10}^{-3}$	$2.707\times {10}^{-4}$	$5.227\times {10}^{-4}$
	$0.4$	$1.150\times {10}^{-3}$	$1.824\times {10}^{-3}$	$3.891\times {10}^{-4}$	$6.362\times {10}^{-4}$
	$0.6$	$1.655\times {10}^{-3}$	$2.359\times {10}^{-3}$	$5.241\times {10}^{-4}$	$7.634\times {10}^{-4}$
	$0.8$	$2.268\times {10}^{-3}$	$2.838\times {10}^{-3}$	$7.209\times {10}^{-4}$	$9.228\times {10}^{-4}$
	$1$	$3.114\times {10}^{-3}$	$3.460\times {10}^{-3}$	$9.498\times {10}^{-4}$	$1.963\times {10}^{-4}$
$0.2$	$0.2$	$1.278\times {10}^{-3}$	$7.632\times {10}^{-3}$	$2.286\times {10}^{-4}$	$1.511\times {10}^{-4}$
	$0.4$	$2.539\times {10}^{-3}$	$7.647\times {10}^{-3}$	$2.975\times {10}^{-4}$	$3.830\times {10}^{-4}$
	$0.6$	$4.732\times {10}^{-3}$	$7.597\times {10}^{-3}$	$3.784\times {10}^{-4}$	$5.873\times {10}^{-4}$
	$0.8$	$5.887\times {10}^{-3}$	$7.399\times {10}^{-3}$	$4.696\times {10}^{-4}$	$7.458\times {10}^{-4}$
	$1$	$7.941\times {10}^{-3}$	$7.615\times {10}^{-3}$	$5.765\times {10}^{-4}$	$9.568\times {10}^{-4}$
$0.4$	$0.2$	$1.713\times {10}^{-3}$	$6.410\times {10}^{-3}$	$7.725\times {10}^{-4}$	$1.963\times {10}^{-4}$
	$0.4$	$1.426\times {10}^{-3}$	$6.553\times {10}^{-3}$	$8.345\times {10}^{-4}$	$1.455\times {10}^{-4}$
	$0.6$	$1.946\times {10}^{-3}$	$5.974\times {10}^{-3}$	$1.808\times {10}^{-4}$	$1.600\times {10}^{-4}$
	$0.8$	$2.596\times {10}^{-3}$	$5.340\times {10}^{-3}$	$1.529\times {10}^{-4}$	$1.812\times {10}^{-4}$
	$1$	$3.344\times {10}^{-3}$	$4.269\times {10}^{-3}$	$1.555\times {10}^{-4}$	$2.749\times {10}^{-4}$
$0.6$	$0.2$	$4.552\times {10}^{-3}$	$2.521\times {10}^{-3}$	$2.618\times {10}^{-4}$	$2.288\times {10}^{-4}$
	$0.4$	$6.968\times {10}^{-3}$	$2.847\times {10}^{-3}$	$2.509\times {10}^{-4}$	$2.608\times {10}^{-4}$
	$0.6$	$8.128\times {10}^{-3}$	$6.502\times {10}^{-3}$	$3.239\times {10}^{-4}$	$3.107\times {10}^{-4}$
	$0.8$	$1.830\times {10}^{-3}$	$1.200\times {10}^{-3}$	$4.444\times {10}^{-4}$	$4.287\times {10}^{-4}$
	$1$	$1.438\times {10}^{-3}$	$3.245\times {10}^{-3}$	$5.341\times {10}^{-4}$	$5.142\times {10}^{-4}$
$0.8$	$0.2$	$2.278\times {10}^{-3}$	$6.818\times {10}^{-3}$	$7.143\times {10}^{-4}$	$6.235\times {10}^{-4}$
	$0.4$	$2.700\times {10}^{-3}$	$9.476\times {10}^{-3}$	$1.322\times {10}^{-4}$	$7.748\times {10}^{-4}$
	$0.6$	$3.326\times {10}^{-3}$	$1.318\times {10}^{-3}$	$1.930\times {10}^{-4}$	$9.983\times {10}^{-4}$
	$0.8$	$5.107\times {10}^{-3}$	$1.703\times {10}^{-3}$	$1.200\times {10}^{-4}$	$1.244\times {10}^{-4}$
	$1$	$7.662\times {10}^{-3}$	$2.218\times {10}^{-3}$	$2.617\times {10}^{-4}$	$1.510\times {10}^{-4}$
$1$	$0.2$	$1.488\times {10}^{-3}$	$3.902\times {10}^{-3}$	$3.384\times {10}^{-4}$	$1.782\times {10}^{-4}$
	$0.4$	$1.435\times {10}^{-3}$	$4.537\times {10}^{-3}$	$5.857\times {10}^{-4}$	$2.359\times {10}^{-4}$
	$0.6$	$2.504\times {10}^{-3}$	$5.922\times {10}^{-3}$	$7.210\times {10}^{-4}$	$2.700\times {10}^{-4}$
	$0.8$	$2.801\times {10}^{-3}$	$7.706\times {10}^{-3}$	$1.191\times {10}^{-4}$	$3.484\times {10}^{-4}$
	$1$	$4.446\times {10}^{-3}$	$9.637\times {10}^{-3}$	$1.631\times {10}^{-4}$	$4.306\times {10}^{-4}$

and

Table 2. Results of error $\left|G_k\left(y_n,x_m\right)-G\left(y_n,x_m\right)\right|$ in Application 2 for $\mu \in \left\{\mathrm{0.7,0.8,0.9,1}\right\}$ through $\mathrm{D}$.

${\mathit{y}}_{\mathit{n}}$	${\mathit{x}}_{\mathit{m}}$	$\mathit{\mu }=0.7$	$\mathit{\mu }=0.8$	$\mathit{\mu }=0.9$	$\mathit{\mu }=1$
$0$	$0.2$	$2.010\times {10}^{-3}$	$2.138\times {10}^{-3}$	$9486\times {10}^{-4}$	$7699\times {10}^{-4}$
	$0.4$	$4.303\times {10}^{-3}$	$7.001\times {10}^{-3}$	$2496\times {10}^{-4}$	$5067\times {10}^{-4}$
	$0.6$	$4.341\times {10}^{-3}$	$3.459\times {10}^{-3}$	$4.240\times {10}^{-4}$	$2889\times {10}^{-4}$
	$0.8$	$2.433\times {10}^{-3}$	$2.221\times {10}^{-3}$	$4076\times {10}^{-4}$	$6287\times {10}^{-4}$
	$1$	$8.855\times {10}^{-3}$	$1.841\times {10}^{-3}$	$2357\times {10}^{-4}$	$7077\times {10}^{-4}$
$0.2$	$0.2$	$2.377\times {10}^{-3}$	$7.069\times {10}^{-3}$	$3038\times {10}^{-4}$	$5167\times {10}^{-4}$
	$0.4$	$5.440\times {10}^{-3}$	$1.628\times {10}^{-3}$	$2930\times {10}^{-4}$	$2786\times {10}^{-4}$
	$0.6$	$1.394\times {10}^{-3}$	$2.542\times {10}^{-3}$	$6238\times {10}^{-4}$	$1.202\times {10}^{-4}$
	$0.8$	$2.494\times {10}^{-3}$	$2.906\times {10}^{-3}$	$1740\times {10}^{-4}$	$4.371\times {10}^{-4}$
	$1$	$3.082\times {10}^{-3}$	$2.591\times {10}^{-3}$	$3598\times {10}^{-4}$	$1.310\times {10}^{-4}$
$0.4$	$0.2$	$6.696\times {10}^{-3}$	$1.883\times {10}^{-3}$	$6877\times {10}^{-4}$	$3.951\times {10}^{-4}$
	$0.4$	$2.161\times {10}^{-3}$	$1.155\times {10}^{-3}$	$1234\times {10}^{-4}$	$1.027\times {10}^{-4}$
	$0.6$	$1.773\times {10}^{-3}$	$6.141\times {10}^{-3}$	$2103\times {10}^{-4}$	$2.473\times {10}^{-4}$
	$0.8$	$2.303\times {10}^{-3}$	$2.817\times {10}^{-3}$	$3431\times {10}^{-4}$	$5.582\times {10}^{-4}$
	$1$	$3.315\times {10}^{-3}$	$1.222\times {10}^{-3}$	$5395\times {10}^{-4}$	$1.191\times {10}^{-4}$
$0.6$	$0.2$	$4.991\times {10}^{-3}$	$4.713\times {10}^{-3}$	$8218\times {10}^{-4}$	$2.422\times {10}^{-4}$
	$0.4$	$6.652\times {10}^{-3}$	$1.676\times {10}^{-3}$	$1217\times {10}^{-4}$	$4.712\times {10}^{-4}$
	$0.6$	$8.082\times {10}^{-3}$	$5.546\times {10}^{-3}$	$1761\times {10}^{-4}$	$8.832\times {10}^{-4}$
	$0.8$	$1.099\times {10}^{-3}$	$1.721\times {10}^{-3}$	$2493\times {10}^{-4}$	$1.613\times {10}^{-4}$
	$1$	$1.883\times {10}^{-3}$	$5.039\times {10}^{-3}$	$3464\times {10}^{-4}$	$2.812\times {10}^{-4}$
$0.8$	$0.2$	$1.552\times {10}^{-3}$	$1.401\times {10}^{-3}$	$4730\times {10}^{-4}$	$4.808\times {10}^{-4}$
	$0.4$	$2.652\times {10}^{-3}$	$3.713\times {10}^{-3}$	$6361\times {10}^{-4}$	$8.018\times {10}^{-4}$
	$0.6$	$2.827\times {10}^{-3}$	$9.428\times {10}^{-3}$	$4361\times {10}^{-4}$	$1.307\times {10}^{-4}$
	$0.8$	$3.633\times {10}^{-3}$	$2.301\times {10}^{-3}$	$1104\times {10}^{-4}$	$2.087\times {10}^{-4}$
	$1$	$4.152\times {10}^{-3}$	$5.415\times {10}^{-3}$	$1429\times {10}^{-4}$	$3.269\times {10}^{-4}$
$1$	$0.2$	$5.303\times {10}^{-3}$	$1.233\times {10}^{-3}$	$1830\times {10}^{-4}$	$5.032\times {10}^{-4}$
	$0.4$	$6.451\times {10}^{-3}$	$2.723\times {10}^{-3}$	$2321\times {10}^{-4}$	$7.621\times {10}^{-4}$
	$0.6$	$7.283\times {10}^{-3}$	$5.832\times {10}^{-3}$	$2918\times {10}^{-4}$	$1.137\times {10}^{-4}$
	$0.8$	$3.008\times {10}^{-3}$	$1.218\times {10}^{-3}$	$3636\times {10}^{-4}$	$1.672\times {10}^{-4}$
	$1$	$1.112\times {10}^{-3}$	$2.479\times {10}^{-3}$	$4496\times {10}^{-4}$	$2.427\times {10}^{-4}$

, respectively, in which

$$\left|\begin{array}{c}\begin{array}{c}{G}_k\left(y_n,x_m\right)\end{array}\end{array}-G\left(y_n,x_m\right)\right|$$

(45)

symbolizes the absolute error over $\mathrm{D}$ at $\mu \in \left\{\mathrm{0.7,0.8,0.9,1}\right\}$. Hither, in Table 1 and Table 2, the closeness of the results to zero is an indicator of the accuracy of the presented RHA and its effectiveness in solving MIFADM.

The results of the RHA solution for Applications 1 and 2 are given in Figure 1 and Figure 2, respectively, by plotting the approximation $G_k\left(y_n,x_m\right)$ with $\mu \in \left\{\mathrm{0.25,0.5,0.75,1}\right\}$ over $\mathrm{D}$. Clearly, the Algorithms 1 and 2 result and the utilized model preserve the hereditary characteristics of the TFCD. A closer examination of the figures reveals that the graphics are roughly identical, as they have the same behavior, and this shows the continuity of the fractional derivative in preserving the hereditary properties of the solution.

Algorithm 1. Steps of the orthogonal systems

• Let ${\left\{\left(y_{\mathrm{r}},x_{\mathrm{r}}\right)\right\}}_{\mathrm{r}=1}^{\mathrm{\infty }}$ within $\mathrm{D}$. • Let ${\sigma }_{\mathrm{r}}\left(y,x\right)={\mathcal{G}}_{\left(y_{\mathrm{r}},x_{\mathrm{r}}\right)}\left(y,x\right)$. • Let ${\psi }_{\mathrm{r}}\left(y,x\right)={\mathcal{H}}^{\ast }\left[{\sigma }_{\mathrm{r}}\right]\left(y,x\right)$. • Build ${\left\{{\overline{\psi }}_{\mathrm{r}}\left(y,x\right)\right\}}_{\mathrm{r}=1}^{\infty }$ as ${\overline{\psi }}_{\mathrm{r}}\left(y,x\right)={\displaystyle \sum _{j=1}^{\mathrm{r}}}{\mathcal{o}}_{\mathrm{r}j}{\psi }_j\left(y,x\right)$. • Let ${\mathcal{a}}_{\mathrm{r},p}\left(x\right)={⟨{\psi }_{\mathrm{r}}\left(x\right),{\overline{\psi }}_p\left(x\right)⟩}_{\mathsf{\Theta }}$. • Let ${\mathcal{b}}_{\mathrm{r}}\left(x\right)={‖{\psi }_{\mathrm{r}}‖}_{\mathsf{\Theta }}^2-{\displaystyle \sum _{p=1}^{\mathrm{r}-1}}{\mathcal{a}}_{\mathrm{r},p}^2\left(x\right)$.

Algorithm 2. Steps of the RHA

• Let $k=\mathcal{P}\mathcal{Q}$ points in $\mathrm{D}$. • Let $\mathrm{r}=\mathrm{1,2},\cdots ,k$ and $j=\mathrm{1,2},\dots ,\mathrm{r}$. • Let ${\psi }_{\mathrm{r}}\left(y_n,x_m\right)={\left.{\mathcal{H}}_{\left(\alpha ,\beta \right)}\left[\mathcal{S}\right]\left(y,x\right)\right|}_{\left(\alpha ,\beta \right)=\left(y_n,x_m\right)}$. • Let ${\overline{\psi }}_{\mathrm{r}}\left(y_n,x_m\right)={\displaystyle \sum _{j=1}^{\mathrm{r}}}{\mathcal{o}}_{\mathrm{r}j}{\psi }_{\mathrm{r}}\left(y_n,x_m\right)$. • Let $G_0\left(y_1,x_1\right)$. • Let ${\mathcal{S}}_{\mathrm{r}}={\displaystyle \sum _{j=1}^{\mathrm{r}}}{\mathcal{o}}_{\mathrm{r}j}\mathrm{G}\left(y_n,x_m\right)$. • Let $G_{\mathrm{r}}\left(y_n,x_m\right)={\displaystyle \sum _{j=1}^{\mathrm{r}}}{{\mathcal{S}}_{\mathrm{r}}\overline{\psi }}_j\left(y_n,x_m\right)$.

5. Concluding Notes

This article presented the numerical solutions of MIFADM along with NCCs of two points using the RHA, based on the definition of TFCD. The forms of the exact and the approximate solution were derived in the Hilbert space, and the necessary convergence theory was provided. By calculating and scheduling the absolute error and drawing the approximate solutions for different values that represent the order of the TFCD, we conclude that the results are satisfactory and the algorithm demonstrates high accuracy and reliability. This work will be extended to higher-dimensional fractional models and nonclassical Neumann constraints. In parallel, efforts will focus on optimizing computational efficiency through adaptive basis selection. Moreover, future studies will examine the algorithm’s stability and convergence behavior under alternative fractional operators such as Riemann–Liouville and Atangana–Baleanu. Applications to more complex systems, including diffusion–wave equations and nonlinear fractional Schrödinger equations, will also be explored, broadening both the theoretical foundation and the practical applicability of the proposed method.