An Analysis of Numerical Techniques for Mixed Fractional Integro-Differential Equations with a Symmetric Singular Kernel

Abstract

In this study, we investigate a class of mixed fractional partial integro-differential equations (FrPI-DE) involving symmetric singular kernels. The considered model problem involves Caputo fractional derivatives and integral operators that describe spatial interactions in a bounded domain. For the purpose of analysis, the original problem is reformulated in the form of a nonlinear Volterra–Fredholm integral equation (NV-FIE). The existence and uniqueness of the solution are established by the Banach fixed point theorem. To compute numerical solutions, a modified Toeplitz matrix method (TMM) is proposed to handle the singular kernel efficiently. The method transforms the integral equation to a system of nonlinear algebraic equations, which can be solved numerically. The convergence properties of the resulting numerical scheme are analyzed and illustrate the effectiveness of the method by providing numerical examples involving logarithmic, Cauchy-type, and weakly singular kernels. Numerical results indicate that the proposed method provides highly accurate approximations and exhibits stable convergence behavior for different parameter values. Furthermore, these results confirm the effectiveness and reliability of the proposed method for solving fractional integro-differential equations that include symmetric singular kernels.

1. Introduction

In recent decades, fractional calculus has attracted increasing attention because it can be used to describe complex phenomena depending on memory and hereditary properties. Unlike classical integer-order calculus, fractional derivatives incorporate nonlocal operators that allow the present state of a system to depend on its past behavior. Because of this property, fractional calculus has been effectively applied in many fields of science and engineering. These applications demonstrate the effectiveness of fractional models in capturing the dynamics of processes that cannot be adequately represented by traditional integer-order models [1,2,3,4,5].

Among the various fractional models, fractional integro-differential equations (FIDEs) play a significant role because they combine fractional derivatives with integral operators that represent spatial interactions and long-range effects. Such equations naturally arise in many areas of applied mathematics and physics. In particular, fractional integro-differential equations have been used to model systems exhibiting anomalous diffusion and fractal dynamics, where long-memory effects and nonlocal interactions play an essential role. These models provide a powerful framework for studying complex dynamical systems in both theoretical and applied contexts.

One of the central research directions is the stability analysis of fractional integro-differential systems. Stability analysis is essential for understanding the qualitative behavior of fractional dynamical systems and for ensuring that numerical approximations remain reliable over time. Beyond stability, the investigation of the existence and uniqueness of solutions is a fundamental step in validating fractional mathematical models and ensuring their well-posedness. Several authors have investigated stability properties, existence, and uniqueness of solutions for different classes of fractional integro-differential equations and related models [6,7,8,9,10,11].

Since analytical solutions of fractional integro-differential equations are rarely available, the development of efficient numerical methods has become an important area of research. Various numerical techniques have been proposed to approximate the solutions of such equations, including discretization schemes, spectral methods, numerical approaches based on integral equation formulations, and others [12,13,14,15,16,17,18,19,20,21,22,23]. These methods have been successfully applied to a wide range of fractional problems and have demonstrated promising accuracy and convergence properties. However, many of the existing approaches may face computational difficulties or require specialized treatment when dealing with symmetric singular kernels. The presence of such kernels introduces additional mathematical and computational challenges, particularly when nonlinear terms and fractional operators are involved. Classical numerical methods may require special treatments for different kernel structures, which limits their applicability to more general classes of fractional integro-differential equations. Therefore, the development of numerical methods capable of efficiently handling fractional integro-differential equations with singular kernels remains an important research problem.

Motivated by these challenges, the present study focuses on the numerical analysis of mixed fractional partial integro-differential equations with symmetric singular kernels. The main objective of this work is to develop a robust numerical framework based on a modified Toeplitz matrix method (TMM) that can efficiently handle the singular kernel structure and convert the problem into a system of nonlinear algebraic equations. This approach provides a stable and accurate numerical scheme while maintaining a unified treatment for different types of singular kernels. In addition, the existence and uniqueness of the solution are established using the Banach fixed-point theorem, and the convergence and stability of the numerical scheme are analyzed. Toeplitz matrix methods have previously been applied to various classes of integral equations due to their computational efficiency and simplicity in transforming integral systems into algebraic systems [24].

In this work, we will consider the following fractional partial integro-differential equation (FrPI-DE) with a kind of symmetric singular kernel:

$$\begin{array}{cc}\hfill \left({\mu }_0{\displaystyle \frac{{\partial }^{\alpha +1}}{\partial t^{\alpha +1}}}+{\mu }_1{\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}+{\mu }_2\right)\phi (x,t)& =g(x,t)+\lambda {\int }_0^{t}F(t,\tau )\mathrm{d}\tau {\int }_{\mathsf{\Omega }}k\left(\right|x-y\left|\right)\gamma (y,\tau ,\phi (y,\tau ))\mathrm{d}y;0<\alpha <1,\hfill \end{array}$$

(1)

under the initial conditions

$$\phi (x,0)=v_1\left(x\right),{\displaystyle \frac{{\partial }^{\alpha }\phi (x,0)}{\partial t^{\alpha }}}=v_2\left(x\right),$$

(2)

where ${\mu }_0,{\mu }_1$ and ${\mu }_2$ are constant parameters, with Caputo fractional derivatives ${\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}$ order $\alpha$, and $\phi (x,t)\in L_2\left[\mathsf{\Omega }\right]\times C[0,T]$, where $L_2\left[\mathsf{\Omega }\right]\times C[0,T]$ is a Banach space. $g(x,t)$ is a known continuous function in $L_2\left[\mathsf{\Omega }\right]\times C[0,T]$. In addition, $F(t,\tau )$ and $\gamma (y,\tau ,\phi (y,\tau \left)\right)$ are known continuous functions. The kernel $k\left(\right|x-y\left|\right)$, in general, has a symmetric singular term.

Furthermore, we assume the following assumptions:

(i)

The kernel of position $k\left(\right|x-y\left|\right)$ satisfies ${\left[{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}{k}^2\left(\right|x-y\left|\right)\mathrm{d}x\mathrm{d}y\right]}^{{\displaystyle \frac{1}{2}}}\le C$, C is a constant.

(ii)

The functions $v_1\left(x\right)$ and $v_2\left(x\right)$ satisfy $|v_1\left(x\right)|\le B_1,\left|v_2\left(x\right)\right|\le B_2,$$B_1$ and $B_2$ are constants.

(iii)

The function $F(t,\tau )$ satisfies $\left|F\right(t,\tau \left)\right|\le D;\forall$$t,\tau \in [0,T],0<\tau \le t\le T\le 1$, D is a constant.

(iv)

The norm of the continous function $g(x,t)$ is defined as:

$${\parallel g(x,t)\parallel }_{L_2\left[\mathsf{\Omega }\right]\times C[0,T]}=\underset{0<t\le T}{max}\left|{\int }_0^t{\left[{\int }_{\mathsf{\Omega }}{g}^2(x,\tau )\mathrm{d}x\right]}^{{\displaystyle \frac{1}{2}}}\mathrm{d}\tau \right|=\mathcal{G},\mathcal{G}\mathrm{is}\mathrm{a}\mathrm{constant}.$$

(v)

The function $\gamma (y,\tau ,\phi (y,\tau \left)\right)$ adheres to the following conditions:

$$\begin{array}{c}\hfill \parallel \gamma (y,\tau ,\phi (y,\tau \left)\right)\parallel =\underset{0<t\le T}{max}\left|{\int }_0^t{\left[{\int }_{\mathsf{\Omega }}{\gamma }^2(y,\tau ,\phi (y,\tau ))\mathrm{d}y\right]}^{{\displaystyle \frac{1}{2}}}\mathrm{d}\tau \right|\le {\xi }_1\parallel \phi (y,\tau )\parallel \le \xi \parallel \phi (y,\tau )\parallel \\ \hfill \parallel \gamma (y,\tau ,{\phi }_1(y,\tau ))-\gamma (y,\tau ,{\phi }_2(y,\tau ))\parallel \le {\xi }_2\parallel {\phi }_1(y,\tau )-{\phi }_2(y,\tau )\parallel \le \xi \parallel {\phi }_1(y,\tau )-{\phi }_2(y,\tau )\parallel ,\end{array}$$

where $\xi >max\{{\xi }_1,{\xi }_2\},{\xi }_1,{\xi }_2$ and $\xi$ are constants.

The basic structure of this article is organized as follows: Section 2 establishes the formulation of the nonlinear Volterra–Fredholm integral. In Section 3, we introduce the convergence of a general solution for the nonlinear Volterra–Fredholm integral equation and the stability of the error. In Section 4, using the Quadrature method, a nonlinear Volterra–Fredholm integral equation leads to a system of nonlinear Fredholm integral equations. In Section 5, the modified Toeplitz matrix method (TMM) on an integral equation yields a nonlinear algebraic system. The convergence of the nonlinear algebraic system is discussed in Section 6, while Section 7 solves various illustrative examples by using the program Wolfram Mathematica 11 to confirm the efficiency of the approach. Finally, Section 8 offers concluding remarks and outlines directions for future work.

2. Formulation and Existence–Uniqueness Analysis

In this section, a fractional partial integro-differential equation with a symmetric singular kernel is transformed into an equivalent nonlinear Volterra–Fredholm integral equation with a symmetric singular kernel. The existence and uniqueness of the solution are then established under appropriate assumptions.

2.1. Formulation of the Nonlinear Volterra–Fredholm Integral Equation

By integrating Equation (1), we obtain

$$\begin{array}{cc}\hfill {\mu }_0\left({\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}\phi (x,t)-{\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}\phi (x,0)\right)& + {\mu }_1{I}^{1-\alpha }\phi (x,t)+{\mu }_2{\int }_0^t\phi (x,\tau )\mathrm{d}\tau ={\int }_0^{t}g(x,\tau )\mathrm{d}\tau \hfill \\ & + \lambda {\int }_0^t{\int }_{\mathsf{\Omega }}(t-\tau )F(t,\tau )k\left(\right|x-y\left|\right)\gamma (y,\tau ,\phi (y,\tau ))\mathrm{d}y\mathrm{d}\tau .\hfill \end{array}$$

Using conditions (2), we obtain

$$\begin{array}{cc}\hfill {\mu }_0\left({\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}\phi (x,t)-v_2\left(x\right)\right)& + {\mu }_1{I}^{1-\alpha }\phi (x,t)+{\mu }_2{\int }_0^t\phi (x,\tau )\mathrm{d}\tau ={\int }_0^{t}g(x,\tau )\mathrm{d}\tau \hfill \\ \hfill & + \lambda {\int }_0^t{\int }_{\mathsf{\Omega }}(t-\tau )F(t,\tau )k\left(\right|x-y\left|\right)\gamma (y,\tau ,\phi (y,\tau ))\mathrm{d}y\mathrm{d}\tau .\hfill \end{array}$$

(3)

By utilizing the Caputo fractional integral formula

$$I_a^{\alpha }\phi \left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_a^x\phi \left(y\right){(x-t)}^{\alpha -1}\mathrm{d}t,$$

(4)

and Cauchy’s formula for repeated integration,

$${\int }_0^t{\int }_0^{{\tau }_{\ell -1}}\cdots {\int }_0^{{\tau }_2}{\int }_0^{{\tau }_1}\phi \left(\tau \right)\mathrm{d}\tau \mathrm{d}{\tau }_1\cdots \mathrm{d}{\tau }_{\ell -2}\mathrm{d}{\tau }_{\ell -1}={\displaystyle \frac{1}{\mathsf{\Gamma }(\ell )}}{\int }_0^t{(t-\tau )}^{\ell -1}\phi \left(\tau \right)\mathrm{d}\tau ,$$

(5)

in the integration of Equation (3), and by applying the conditions (2), we obtain

$$\begin{array}{cc}\hfill {\mu }_0\phi (x,t)& +{\int }_0^t\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right)\phi (x,\tau )\mathrm{d}\tau =G(x,t)+\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\int }_{\mathsf{\Omega }}{(t-\tau )}^{\alpha +1}F(t,\tau )k\left(\right|x-y\left|\right)\gamma (y,\tau ,\phi (y,\tau ))\mathrm{d}y\mathrm{d}\tau ,\hfill \end{array}$$

(6)

where

$$G(x,t)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha }g(x,\tau )\mathrm{d}\tau +{\mu }_0{v}_1\left(x\right)+{\displaystyle \frac{{\mu }_0{t}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}}{v}_2\left(x\right).$$

(7)

Equation (6) is called the nonlinear Volterra–Fredholm integral equation (NV-FIE) with a symmetric singular kernel, and is equivalent to the FrPI-DE (1) under the conditions (2).

2.2. Existence and Uniqueness of the Solution

The existence and uniqueness analysis in this section ensure the mathematical well-formedness of the proposed fractional integro-differential model. Establishing these properties is essential for validating the reliability of the numerical method developed later, as convergence and stability analyses depend on the uniqueness of the solution. Moreover, the theoretical results provide a rigorous justification for applying the modified Toeplitz matrix method to nonlinear problems involving singular kernels.

Now, we use the Banach fixed-point theorem [25] to prove that Equation (6) has a unique solution. We will demonstrate this through the following theorem.

Theorem 1. 

If assumptions (i)–(v) are satisfied, then Equation (6) has a unique solution $\phi (x,t)$ in the Banach space $L_2\left[\mathsf{\Omega }\right]\times C[0,T], for0<T\le 1$, provided that

$$\left|\lambda \right|<{\displaystyle \frac{|{\mu }_0|\mathsf{\Gamma }(\alpha +1)-[|{\mu }_1|T\mathsf{\Gamma }(\alpha +1)+|{\mu }_2\left|T^{\alpha +1}\right]}{\xi DC{T}^{\alpha +2}}},(0<\alpha <1).$$

(8)

Proof. 

To discuss the existence and uniqueness of the solution for Equation (6), we do the following:

First, Equation (6) is rewritten in operator form as

$$\overline{V}\phi (x,t)={\displaystyle \frac{1}{{\mu }_0}}[G(x,t)+V_1\phi (x,t)+V_2\phi (x,t)],$$

(9)

where

$$\begin{array}{cc}\hfill V_1\phi (x,t)& ={\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\int }_{\mathsf{\Omega }}{(t-\tau )}^{\alpha +1}F(t,\tau )k\left(\right|x-y\left|\right)\gamma (y,\tau ,\phi (y,\tau ))\mathrm{d}y\mathrm{d}\tau ,\hfill \\ \hfill V_2\phi (x,t)& =-{\int }_0^t\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right)\phi (x,\tau )\mathrm{d}\tau .\hfill \end{array}$$

(10)

Next, we show that the operator $\overline{V}$ is bounded.Taking norms on both sides of Equation (9) yields

$$\parallel \overline{V}\phi (x,t)\parallel \le {\displaystyle \frac{1}{|{\mu }_0|}}[\parallel G(x,t)\parallel +\parallel V_1\phi (x,t)\parallel +\parallel V_2\phi (x,t)\parallel ].$$

(11)

From Equations (7), (9), and (10), assumptions (i)–(v), and standard inequalities, we have

$$\begin{array}{cc}\hfill \parallel G(x,t)\parallel & = \parallel {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha }g(x,\tau )\mathrm{d}\tau +{\mu }_0{v}_1\left(x\right)+{\displaystyle \frac{{\mu }_0{t}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}}{v}_2\left(x\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}∥{\int }_0^t{(t-\tau )}^{\alpha }g(x,\tau )\mathrm{d}\tau ∥+\parallel {\mu }_0{v}_1\left(x\right)\parallel +\parallel {\displaystyle \frac{{\mu }_0{t}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}}{v}_2\left(x\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{\mathcal{G}}{\mathsf{\Gamma }\left(\alpha \right)}}∥{\int }_0^t{(t-\tau )}^{\alpha }\mathrm{d}\tau ∥+|{\mu }_0|\parallel v_1\left(x\right)\parallel +{\displaystyle \frac{|{\mu }_0|}{\mathsf{\Gamma }(\alpha +1)}}\parallel t^{\alpha }{v}_2\left(x\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{\mathcal{G}}{\mathsf{\Gamma }\left(\alpha \right)}}∥{\displaystyle \frac{t^{\alpha +1}}{\alpha +1}}∥+|{\mu }_0|\parallel v_1\left(x\right)\parallel +{\displaystyle \frac{|{\mu }_0|}{\mathsf{\Gamma }(\alpha +1)}}\underset{0<t\le T}{max}{\int }_0^t{\left[{\int }_{\mathsf{\Omega }}{\left[{\tau }^{\alpha }{v}_2\left(x\right)\right]}^2\mathrm{d}x\right]}^{{\displaystyle \frac{1}{2}}}\mathrm{d}\tau \hfill \\ \hfill & \le {\displaystyle \frac{\mathcal{G}}{\mathsf{\Gamma }\left(\alpha \right)}}\underset{0<t\le T}{max}{\int }_0^t{\tau }^{\alpha +1}\mathrm{d}\tau +|{\mu }_0|\parallel v_1\left(x\right)\parallel +{\displaystyle \frac{|{\mu }_0|}{\mathsf{\Gamma }(\alpha +1)}}\underset{0<t\le T}{max}{\int }_0^t{\tau }^{\alpha }\mathrm{d}\tau {\left[{\int }_{\mathsf{\Omega }}{\left[v_2\left(x\right)\right]}^2\mathrm{d}x\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le {\displaystyle \frac{\mathcal{G}{T}^{\alpha +2}}{\mathsf{\Gamma }(\alpha +1)}}+|{\mu }_0|\parallel v_1\left(x\right)\parallel +{\displaystyle \frac{|{\mu }_0|T^{\alpha +1}}{\mathsf{\Gamma }(\alpha +2)}}\parallel v_2\left(x\right)\parallel .\hfill \end{array}$$

Hence, we conclude that

$$\parallel G(x,t)\parallel \le {\displaystyle \frac{\mathcal{G}{T}^{\alpha +2}}{\mathsf{\Gamma }(\alpha +1)}}+|{\mu }_0|\parallel v_1\left(x\right)\parallel +{\displaystyle \frac{|{\mu }_0|T^{\alpha +1}}{\mathsf{\Gamma }(\alpha +2)}}\parallel v_2\left(x\right)\parallel =P,$$

(12)

where $P={\displaystyle \frac{\mathcal{G}{T}^{\alpha +2}}{\mathsf{\Gamma }(\alpha +1)}}+|{\mu }_0|\parallel v_1\left(x\right)\parallel +{\displaystyle \frac{|{\mu }_0|T^{\alpha +1}}{\mathsf{\Gamma }(\alpha +2)}}\parallel v_2\left(x\right)\parallel$ is a constant.

Also,

$$\begin{array}{cc}\hfill \parallel V_1\phi (x,t)\parallel & =∥{\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\int }_{\mathsf{\Omega }}{(t-\tau )}^{\alpha +1}F(t,\tau )k\left(\right|x-y\left|\right)\gamma (y,\tau ,\phi (y,\tau ))\mathrm{d}y\mathrm{d}\tau ∥\hfill \\ \hfill & \le {\displaystyle \frac{\left|\lambda \right|D}{\mathsf{\Gamma }\left(\alpha \right)}}∥{\int }_0^t{(t-\tau )}^{\alpha +1}{\int }_{\mathsf{\Omega }}k\left(\right|x-y\left|\right)\gamma (y,\tau ,\phi (y,\tau ))\mathrm{d}y\mathrm{d}\tau ∥\hfill \\ \hfill & \le {\displaystyle \frac{\left|\lambda \right|D}{\mathsf{\Gamma }\left(\alpha \right)}}∥{\int }_0^t{(t-\tau )}^{\alpha +1}{\left({\int }_{\mathsf{\Omega }}{k}^2\left(\right|x-y\left|\right)\mathrm{d}y\right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_{\mathsf{\Omega }}{\gamma }^2(y,\tau ,\phi (y,\tau ))\mathrm{d}y\right)}^{{\displaystyle \frac{1}{2}}}\mathrm{d}\tau ∥\hfill \\ \hfill & \le {\displaystyle \frac{\left|\lambda \right|D}{\mathsf{\Gamma }\left(\alpha \right)}}\underset{0<t\le T}{max}{\int }_0^t{\left[{\int }_{\mathsf{\Omega }}{\left\{{\int }_0^t{(t-\tau )}^{\alpha +1}{\left({\int }_{\mathsf{\Omega }}{k}^2\mathrm{d}y\right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_{\mathsf{\Omega }}{\gamma }^2\mathrm{d}y\right)}^{{\displaystyle \frac{1}{2}}}\mathrm{d}\tau \right\}}^2\mathrm{d}x\right]}^{{\displaystyle \frac{1}{2}}}\mathrm{d}\tau \hfill \\ \hfill & \le {\displaystyle \frac{\left|\lambda \right|D}{\mathsf{\Gamma }\left(\alpha \right)}}{\left[{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}{k}^2\mathrm{d}x\mathrm{d}y\right]}^{{\displaystyle \frac{1}{2}}}\underset{0<t\le T}{max}{\int }_0^t{\left[{\left\{{\int }_0^t{(t-\tau )}^{\alpha +1}{\left({\int }_{\mathsf{\Omega }}{\gamma }^2\mathrm{d}y\right)}^{{\displaystyle \frac{1}{2}}}\mathrm{d}\tau \right\}}^2\right]}^{{\displaystyle \frac{1}{2}}}\mathrm{d}\tau \hfill \\ \hfill & \le {\displaystyle \frac{\xi \left|\lambda \right|DC{T}^{\alpha +2}}{\mathsf{\Gamma }(\alpha +1)}}\parallel \phi (x,t)\parallel .\hfill \end{array}$$

Based on this, we conclude that

$$\parallel V_1\phi (x,t)\parallel \le {\displaystyle \frac{\xi \left|\lambda \right|DC{T}^{\alpha +2}}{\mathsf{\Gamma }(\alpha +1)}}\parallel \phi (x,t)\parallel ={\beta }_1\parallel \phi (x,t)\parallel ,$$

(13)

where ${\beta }_1={\displaystyle \frac{\xi \left|\lambda \right|DC{T}^{\alpha +2}}{\mathsf{\Gamma }(\alpha +1)}}.$

Similarly,

$$\begin{array}{cc}\hfill \parallel V_2\phi (x,t)\parallel & =∥-{\int }_0^t\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right)\phi (x,\tau )\mathrm{d}\tau ∥\hfill \\ \hfill & \le \underset{0<t\le T}{max}{\int }_0^t{\left[{\int }_{\mathsf{\Omega }}{\left\{{\int }_0^t\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right)\phi (x,\tau )\mathrm{d}\tau \right\}}^2\mathrm{d}x\right]}^{{\displaystyle \frac{1}{2}}}\mathrm{d}\tau \hfill \\ \hfill & \le \underset{0<t\le T}{max}{\int }_0^t{\int }_0^t{\left[{\int }_{\mathsf{\Omega }}{\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right)}^2{\phi }^2(x,\tau )\mathrm{d}x\right]}^{{\displaystyle \frac{1}{2}}}\mathrm{d}\tau \mathrm{d}\tau \hfill \\ \hfill & \le \underset{0<t\le T}{max}{\int }_0^t\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right)\mathrm{d}\tau ·\underset{0<t\le T}{max}{\int }_0^t{\left\{{\int }_{\mathsf{\Omega }}{\phi }^2(x,\tau )\mathrm{d}x\right\}}^{{\displaystyle \frac{1}{2}}}\mathrm{d}\tau \hfill \\ \hfill & \le \left[|{\mu }_1|T+{\displaystyle \frac{|{\mu }_2|T^{\alpha +1}}{\mathsf{\Gamma }(\alpha +1)}}\right]\parallel \phi (x,t)\parallel .\hfill \end{array}$$

From the above, we obtain

$$\parallel V_2\phi (x,t)\parallel \le \left[|{\mu }_1|T+{\displaystyle \frac{|{\mu }_2|T^{\alpha +1}}{\mathsf{\Gamma }(\alpha +1)}}\right]\parallel \phi (x,t)\parallel ={\beta }_2\parallel \phi (x,t)\parallel ,$$

(14)

where ${\beta }_2=\left[|{\mu }_1|T+{\displaystyle \frac{|{\mu }_2|T^{\alpha +1}}{\mathsf{\Gamma }(\alpha +1)}}\right].$

Now, from inequalities (12), (13), and (14) with (11), we get

$$\parallel \overline{V}\phi (x,t)\parallel \le {\displaystyle \frac{P}{|{\mu }_0|}}+\beta \parallel \phi (x,t)\parallel ,$$

$$\beta ={\displaystyle \frac{{\beta }_1+{\beta }_2}{|{\mu }_0|}}={\displaystyle \frac{1}{|{\mu }_0|}}\left[{\displaystyle \frac{\xi \left|\lambda \right|DC{T}^{\alpha +2}}{\mathsf{\Gamma }(\alpha +1)}}+\left|{\mu }_1\right|T+{\displaystyle \frac{|{\mu }_2|T^{\alpha +1}}{\mathsf{\Gamma }(\alpha +1)}}\right]<1,$$

whenever

$$\left|\lambda \right|<{\displaystyle \frac{|{\mu }_0|\mathsf{\Gamma }(\alpha +1)-[|{\mu }_1|T\mathsf{\Gamma }(\alpha +1)+|{\mu }_2\left|T^{\alpha +1}\right]}{\xi DC{T}^{\alpha +2}}}.$$

Therefore, the integral operator $\overline{V}$ is bounded.

Next, we prove that $\overline{V}$ is continuous. Considering two functions ${\phi }_1$ and ${\phi }_2$ in the space $L_2\left[\mathsf{\Omega }\right]\times C[0,T], for0<T\le 1$, which satisfy Equation (9) as follow:

$$\overline{V}{\phi }_1={\displaystyle \frac{1}{{\mu }_0}}[G(x,t)+V_1{\phi }_1+V_2{\phi }_1],$$

(15)

$$\overline{V}{\phi }_2={\displaystyle \frac{1}{{\mu }_0}}[G(x,t)+V_1{\phi }_2+V_2{\phi }_2].$$

(16)

From Equations (15) and (16), we obtain

$$\overline{V}[{\phi }_1-{\phi }_2]={\displaystyle \frac{1}{{\mu }_0}}[V_1[{\phi }_1-{\phi }_2]+V_2[{\phi }_1-{\phi }_2]].$$

As was shown in the boundedness proof, we can establish that

$$\parallel \overline{V}[{\phi }_1-{\phi }_2]\parallel \le \beta \parallel {\phi }_1-{\phi }_2\parallel .$$

Since $\beta <1,$ the operator $\overline{V}$ is continuous.

Hence, $\overline{V}$ is a contraction operator; it follows from the Banach fixed point theorem that $\overline{V}$ has a unique fixed point, which constitutes the unique solution to the Equation (6). □

3. Analysis of Solution Convergence and Error Stability

In this section, we analyze the convergence of the solution sequence generated by the fixed-point formulation of the nonlinear Volterra–Fredholm integral Equation (6). In particular, we examine the contraction property of the associated integral operator and the stability of the resulting iterative sequence. This analysis provides a rigorous theoretical foundation ensuring the well-posedness and stability of the problem.

To establish convergence, we construct a sequence of functions $\{{\phi }_0,{\phi }_1,\dots ,{\phi }_{n-1},{\phi }_n,\dots \}.$ We select two consecutive elements from this sequence, ${\phi }_{n-1}$ and ${\phi }_n,$ such that

$$\begin{array}{cc}\hfill {\mu }_0{\phi }_n(x,t)& =G(x,t)-{\int }_0^t\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right){\phi }_{n-1}(x,\tau )\mathrm{d}\tau +\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\int }_{\mathsf{\Omega }}{(t-\tau )}^{\alpha +1}F(t,\tau )k\left(\right|x-y\left|\right)\gamma (y,\tau ,{\phi }_{n-1}(y,\tau ))\mathrm{d}y\mathrm{d}\tau ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mu }_0{\phi }_{n-1}(x,t)& =G(x,t)-{\int }_0^t\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right){\phi }_{n-2}(x,\tau )\mathrm{d}\tau +\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\int }_{\mathsf{\Omega }}{(t-\tau )}^{\alpha +1}F(t,\tau )k\left(\right|x-y\left|\right)\gamma (y,\tau ,{\phi }_{n-2}(y,\tau ))\mathrm{d}y\mathrm{d}\tau .\hfill \end{array}$$

Subtracting the above equations yields

$$\begin{array}{cc}\hfill {\mu }_0({\phi }_n& (x,t)-{\phi }_{n-1}(x,t))=-\left[{\int }_0^t\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right)\left({\phi }_{n-1}(x,\tau )-{\phi }_{n-2}(x,\tau )\right)\mathrm{d}\tau \right]+\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\int }_{\mathsf{\Omega }}{(t-\tau )}^{\alpha }F(t,\tau )k\left(\right|x-y\left|\right)(\gamma (y,\tau ,{\phi }_{n-1}(y,\tau ))-\gamma (y,\tau ,{\phi }_{n-2}(y,\tau )))\mathrm{d}y\mathrm{d}\tau .\hfill \end{array}$$

Taking norms on both sides and applying assumptions (i)–(v) together with standard inequalities, we conclude

$$\begin{array}{c}\hfill \parallel {\phi }_n(x,t)-{\phi }_{n-1}(x,t)\parallel \le \beta \parallel {\phi }_{n-1}(x,t)-{\phi }_{n-2}(x,t)\parallel ,\beta <1.\end{array}$$

(17)

Let

$${\mathsf{\Xi }}_n(x,t)={\phi }_n(x,t)-{\phi }_{n-1}(x,t).$$

(18)

Then

$${\mathsf{\Xi }}_0(x,t)={\phi }_0(x,t)={\displaystyle \frac{G(x,t)}{{\mu }_0}},{\phi }_n(x,t)=\sum _{k=0}^n{\mathsf{\Xi }}_k(x,t).$$

(19)

From (17) and (18), we get

$$\parallel {\mathsf{\Xi }}_n(x,t)\parallel \le \beta \parallel {\mathsf{\Xi }}_{n-1}(x,t)\parallel .$$

(20)

For $n=1$ and using (12) and (19) in (20), we obtain

$$\parallel {\mathsf{\Xi }}_1(x,t)\parallel \le P\beta .$$

By induction, we conclude

$$\parallel {\mathsf{\Xi }}_n(x,t)\parallel \le P{\beta }^n,\beta <1.$$

(21)

where P is a positive constant. Since $0<\beta <1$, the sequence $\left\{{\phi }_n\right\}$ is Cauchy and therefore converges to a solution $\phi (x,t)$. Hence

$$\phi (x,t)=\underset{n\to \infty }{lim}{\phi }_n(x,t)=\sum _{k=0}^{\infty }{\mathsf{\Xi }}_k(x,t).$$

Next, we analyze the stability of the error. Let ${\phi }_n(x,t)$ be an approximation solution to (6). Then

$$\begin{array}{cc}\hfill {\mu }_0{\phi }_n(x,t)& +{\int }_0^t\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right){\phi }_n(x,\tau )\mathrm{d}\tau =G_n(x,t)+\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\int }_{\mathsf{\Omega }}{(t-\tau )}^{\alpha }F(t,\tau )k\left(\right|x-y\left|\right)\gamma (y,\tau ,{\phi }_n(y,\tau ))\mathrm{d}y\mathrm{d}\tau .\hfill \end{array}$$

(22)

From (6) and (22),we obtain

$$\begin{array}{cc}\hfill {\mu }_0(\phi & (x,t)-{\phi }_n(x,t))=G(x,t)-G_n(x,t)-\hfill \\ \hfill & \left[{\int }_0^t\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right)\left(\phi (x,\tau )-{\phi }_n(x,\tau )\right)\mathrm{d}\tau \right]+\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\int }_{\mathsf{\Omega }}{(t-\tau )}^{\alpha +1}F(t,\tau )k\left(\right|x-y\left|\right)(\gamma (y,\tau ,\phi (y,\tau ))-\gamma (y,\tau ,{\phi }_n(y,\tau )))\mathrm{d}y\mathrm{d}\tau .\hfill \end{array}$$

(23)

Suppose

$$R_n(x,t)=\phi (x,t)-{\phi }_n(x,t),r_n(x,t)=G(x,t)-G_n(x,t).$$

Hence, Equation (23) becomes

$$\begin{array}{cc}\hfill {\mu }_0{R}_n(x,t)& =r_n(x,t)-{\int }_0^t\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right)R_n(x,t)\mathrm{d}\tau +\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\int }_{\mathsf{\Omega }}{(t-\tau )}^{\alpha +1}F(t,\tau )k\left(\right|x-y\left|\right)(\gamma (y,\tau ,\phi (y,\tau ))-\gamma (y,\tau ,{\phi }_n(y,\tau )))\mathrm{d}y\mathrm{d}\tau .\hfill \end{array}$$

(24)

Equation (24) represents a mixed integral equation (MIE) for the error $R_n(x,t)$. As established in Section 2.2, this equation has a unique solution under the condition $\beta <1$.

It is worth noting that solutions of fractional integro-differential equations often exhibit limited regularity, particularly near the initial time due to the nonlocal nature of fractional operators. This reduced smoothness may influence the convergence rate of numerical schemes. In this work, the solution space $L_2\left[\mathsf{\Omega }\right]\times C[0,T]$ is adopted as a standard analytical framework commonly used in fractional calculus. Although idealized, this assumption allows rigorous analysis while remaining consistent with practical numerical observations.

4. System of Nonlinear Fredholm Integral Equations

Here, we apply the numerical technique presented in [26] to transform the nonlinear Volterra–Fredholm integral Equation (6) into a system of nonlinear Fredholm integral equations with a singular kernel. To achieve this, we partition the interval $[0,T]$ into M subintervals as

$$0=t_0<t_1<\cdots <t_m<\cdots <t_M=T,$$

where $t_m=m\mathsf{\Delta }t,m=0,1,\dots ,M,$ and $\mathsf{\Delta }t=T/M.$ Then Equation (6) can be written at $t=t_m$ as

$$\begin{array}{cc}\hfill {\mu }_0\phi (x,t_m)& +{\int }_0^{t_m}\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t_m-\tau )}^{\alpha }\right)\phi (x,\tau )\mathrm{d}\tau =G(x,t_m)+\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^{t_m}{\int }_{\mathsf{\Omega }}{(t_m-\tau )}^{\alpha +1}k\left(\right|x-y\left|\right)F(t_m,\tau )\gamma (y,\tau ,\phi (y,\tau ))\mathrm{d}y\mathrm{d}\tau ,\hfill \end{array}$$

(25)

where

$$G(x,t_m)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^{t_m}{(t_m-\tau )}^{\alpha }g(x,\tau )\mathrm{d}\tau +{\mu }_0{v}_1\left(x\right)+{\displaystyle \frac{{\mu }_0{t}_m^{\alpha }}{\mathsf{\Gamma }(1+\alpha )}}{v}_2\left(x\right).$$

To discretize the time integrals, we approximate them using a composite quadrature rule of the form

$${\int }_0^{t_m}f\left(\tau \right)d\tau \approx \sum _{n=0}^m{\omega }_{n}f\left(t_n\right),$$

where ${\omega }_n$ denotes the quadrature weights.

Thus, the terms for the Volterra integrals are as follows:

$$\begin{array}{cc}\hfill & {\int }_0^{t_m}{\int }_{\mathsf{\Omega }}{(t_m-\tau )}^{\alpha +1}k\left(\right|x-y\left|\right)F(t_m,\tau )\gamma (y,\tau ,\phi (y,\tau ))\mathrm{d}y\mathrm{d}\tau \approx \hfill \\ \hfill & \sum _{n=0}^m{\omega }_n{\int }_{\mathsf{\Omega }}{(t_m-t_n)}^{\alpha +1}k\left(\right|x-y\left|\right)F(t_m,t_n)\gamma (y,t_n,\phi (y,t_n))\mathrm{d}y,\hfill \\ \hfill & {\int }_0^{t_m}\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t_m-\tau )}^{\alpha }\right)\phi (x,\tau )\mathrm{d}\tau \approx \sum _{n=0}^m{\omega }_n\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t_m-t_n)}^{\alpha }\right)\phi (x,t_n),\hfill \\ \hfill & {\int }_0^{t_m}{(t_m-\tau )}^{\alpha }g(x,\tau )\mathrm{d}\tau \approx \sum _{n=0}^m{\omega }_n{(t_m-t_n)}^{\alpha }g(x,t_n).\hfill \end{array}$$

(26)

Using Equation (26) in Equation (25), we obtain

$$\begin{array}{cc}\hfill {\mu }_0\phi (x,t_m)& +\sum _{n=0}^m{\omega }_n\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{(t_m-t_n)}^{\alpha }\right)\phi (x,t_n)=G(x,t_m)+\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}\sum _{n=0}^m{\omega }_n{\int }_{\mathsf{\Omega }}{(t_m-t_n)}^{\alpha +1}k\left(\right|x-y\left|\right)F(t_m,t_n)\gamma (y,t_n,\phi (y,t_n))\mathrm{d}y,\hfill \end{array}$$

(27)

where

$$G(x,t_m)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}\sum _{n=0}^m{\omega }_n{(t_m-t_n)}^{\alpha }g(x,t_n)+{\mu }_0{v}_1\left(x\right)+{\displaystyle \frac{{\mu }_0{t}_m^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}}\left(v_2\left(x\right)\right).$$

For simplicity, we introduce the notations

$${\phi }_m\left(x\right)=\phi (x,t_m),{\mathsf{\Lambda }}_{mn}=(t_m-t_n),F_{mn}=F(t_m,t_n),{\gamma }_n(y,{\phi }_n\left(y\right))=\gamma (y,t_n,\phi (y,t_n)).$$

Then the system becomes:

$$\begin{array}{cc}\hfill {\mu }_0{\phi }_m\left(x\right)& +\sum _{n=0}^m{\omega }_n\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{\mathsf{\Lambda }}_{mn}^{\alpha }\right){\phi }_n\left(x\right)=G_m\left(x\right)+\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}\sum _{n=0}^m{\omega }_n{\mathsf{\Lambda }}_{mn}^{\alpha +1}{F}_{mn}{\int }_{\mathsf{\Omega }}k\left(\right|x-y\left|\right){\gamma }_n(y,{\phi }_n\left(y\right))\mathrm{d}y,\hfill \end{array}$$

(28)

where

$$G_m\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}\sum _{n=0}^m{\omega }_n{\mathsf{\Lambda }}_{mn}^{\alpha }{g}_n\left(x\right)+{\mu }_0{v}_1\left(x\right)+{\displaystyle \frac{{\mu }_0{t}_m^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}}\left(v_2\left(x\right)\right).$$

Equation (28) can be rewritten as follows:

$$\begin{array}{cc}\hfill {\varpi }_m{\phi }_m\left(x\right)& =G_m\left(x\right)-\sum _{n=0}^{m-1}{\omega }_n\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{\mathsf{\Lambda }}_{mn}^{\alpha }\right){\phi }_n\left(x\right)+\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}\sum _{n=0}^m{\omega }_n{\mathsf{\Lambda }}_{mn}^{\alpha +1}{F}_{mn}{\int }_{\mathsf{\Omega }}k\left(\right|x-y\left|\right){\gamma }_n(y,{\phi }_n\left(y\right))\mathrm{d}y,\hfill \end{array}$$

(29)

where ${\varpi }_m={\mu }_0+{\omega }_m{\mu }_1.$

Equation (29) represents a finite system of Fredholm integral equations with a singular kernel.

5. The Modified Toeplitz Matrix Method

The classical Toeplitz matrix method has been widely used for the numerical solution of integral equations, particularly those with smooth convolution-type kernels. In its traditional formulation, the integral operator is discretized on a uniform grid, and the resulting coefficient matrix exhibits a Toeplitz structure due to the translation-invariant properties of the kernel. However, when the kernel contains singularities the direct application of the classical Toeplitz method may lead to reduced numerical accuracy and instability near the singular region.

To address this limitation, the present work introduces a modified Toeplitz matrix for fractional integro-differential equations with symmetric singular kernels. Unlike the traditional approach, the proposed method incorporates a local kernel decomposition and endpoint correction strategy, in which the integral over each subinterval is approximated using auxiliary functions determined through consistency conditions with constant and linear test functions. This procedure produces modified coefficients that explicitly capture the singular behavior of the kernel while preserving the computational advantages of the Toeplitz structure. As a result, the resulting discretized system maintains a Toeplitz-type matrix form but includes additional correction terms that significantly improve stability and accuracy. Moreover, the proposed framework allows different classes of weakly singular kernels to be treated within a unified numerical scheme, which distinguishes it from the conventional Toeplitz method that typically requires separate formulations for different singular kernels.

Now, we shall discuss the numerical solution of the finite system of Fredholm integral Equations (29) using the Toeplitz matrix method [24], assuming $\mathsf{\Omega }=[-1,1]$. The system can be written as

$$\begin{array}{cc}\hfill {\varpi }_m{\phi }_m\left(x\right)& =G_m\left(x\right)-\sum _{n=0}^{m-1}{\omega }_n\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{\mathsf{\Lambda }}_{mn}^{\alpha }\right){\phi }_n\left(x\right)+\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}\sum _{n=0}^m{\omega }_n{\mathsf{\Lambda }}_{mn}^{\alpha +1}{F}_{mn}{\int }_{-1}^{1}k\left(\right|x-y\left|\right){\gamma }_n(y,{\phi }_n\left(y\right))\mathrm{d}y.\hfill \end{array}$$

(30)

To approximate the spatial integral, we partition the interval $[-1,1]$ into $2N+1$ subintervals with step $\varrho ={\displaystyle \frac{1}{N}}$. Then

$${\int }_{-1}^{1}k\left(\right|x-y\left|\right){\gamma }_n(y,{\phi }_n\left(y\right))\mathrm{d}y=\sum _{k=-N}^{N-1}{\int }_{k\varrho }^{(k+1)\varrho }k\left(\right|x-y\left|\right){\gamma }_n(y,{\phi }_n\left(y\right))\mathrm{d}y.$$

(31)

Next, estimate the integral term on the right side by

$$\begin{array}{cc}\hfill {\int }_{k\varrho }^{(k+1)\varrho }k\left(\right|x-y\left|\right){\gamma }_n(y,{\phi }_n\left(y\right))\mathrm{d}y& ={\aleph }_k\left(x\right){\gamma }_n(k\varrho ,{\phi }_n\left(k\varrho \right))\hfill \\ \hfill & +{\Im }_k\left(x\right){\gamma }_n((k+1)\varrho ,{\phi }_n\left((k+1)\varrho \right))+O\left({\varrho }^2\right),\hfill \end{array}$$

(32)

where ${\aleph }_k\left(x\right)$ and ${\Im }_k\left(x\right)$ are arbitrary functions to be determined.

By replacing ${\phi }_n\left(y\right)=1$ and ${\phi }_n\left(y\right)=y$, respectively, in Equation (32), we obtain the following two equations that can be solved to determine ${\aleph }_k\left(x\right)$ and ${\Im }_k\left(x\right)$.

$${\int }_{k\varrho }^{(k+1)\varrho }k\left(\right|x-y\left|\right){\gamma }_n(y,1)\mathrm{d}y={\aleph }_k\left(x\right){\gamma }_n(k\varrho ,1)+{\Im }_k\left(x\right){\gamma }_n((k+1)\varrho ,1),$$

(33)

and

$${\int }_{k\varrho }^{(k+1)\varrho }k\left(\right|x-y\left|\right){\gamma }_n(y,y)\mathrm{d}y={\aleph }_k\left(x\right){\gamma }_n(k\varrho ,k\varrho )+{\Im }_k\left(x\right){\gamma }_n((k+1)\varrho ,(k+1)\varrho ).$$

(34)

Consequently, the two functions ${\aleph }_k\left(x\right)$ and ${\Im }_k\left(x\right)$ are obtained as follows:

$$\begin{array}{cc}\hfill {\aleph }_k\left(x\right)& ={\displaystyle \frac{1}{{\rm Y}}}[{\gamma }_n((k+1)\varrho ,(k+1)\varrho )A_k\left(x\right)-{\gamma }_n((k+1)\varrho ,1)B_k\left(x\right)],\hfill \\ \hfill {\Im }_k\left(x\right)& ={\displaystyle \frac{1}{{\rm Y}}}[{\gamma }_n(k\varrho ,1)B_k\left(x\right)-{\gamma }_n(k\varrho ,k\varrho )A_k\left(x\right)],\hfill \end{array}$$

(35)

where

$$A_k\left(x\right)={\int }_{k\varrho }^{(k+1)\varrho }k\left(\right|x-y\left|\right){\gamma }_n(y,1)\mathrm{d}y,B_k\left(x\right)={\int }_{k\varrho }^{(k+1)\varrho }k\left(\right|x-y\left|\right){\gamma }_n(y,y)\mathrm{d}y,$$

and

$${\rm Y}={\gamma }_n(k\varrho ,1){\gamma }_n((k+1)\varrho ,(k+1)\varrho )-{\gamma }_n(k\varrho ,k\varrho ){\gamma }_n((k+1)\varrho ,1),{\rm Y}\ne 0.$$

In view of Equations (33)–(35), the Formula (31) becomes

$${\int }_{-1}^{1}k\left(\right|x-y\left|\right){\gamma }_n(y,{\phi }_n\left(y\right))\mathrm{d}y=\sum _{k=-N}^N{U}_k\left(x\right){\gamma }_n(k\varrho ,{\phi }_n\left(k\varrho \right)),$$

where

$$\begin{array}{c}\hfill U_k\left(x\right)=\left\{\begin{array}{cc}{\aleph }_{-N}\left(x\right);\hfill & k=-N\hfill \\ {\aleph }_k\left(x\right)+{\Im }_{k-1}\left(x\right);\hfill & -N<k<N\hfill \\ {\Im }_{N-1}\left(x\right);\hfill & k=N.\hfill \end{array}\right.\end{array}$$

Thus, the integral Equation (30) takes the form

$$\begin{array}{cc}\hfill {\varpi }_m{\phi }_m\left(x\right)& =G_m\left(x\right)-\sum _{n=0}^{m-1}{\omega }_n\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{\mathsf{\Lambda }}_{mn}^{\alpha }\right){\phi }_n\left(x\right)+\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}\sum _{n=0}^m{\omega }_n{\mathsf{\Lambda }}_{mn}^{\alpha +1}{F}_{mn}\sum _{k=-N}^N{U}_k\left(x\right){\gamma }_n(k\varrho ,{\phi }_n\left(k\varrho \right)).\hfill \end{array}$$

Setting $x=\ell \varrho for-N\le \ell \le N$, we obtain:

$$\begin{array}{cc}\hfill {\varpi }_m{\phi }_m(\ell \varrho )& =G_m(\ell \varrho )-\sum _{n=0}^{m-1}{\omega }_n\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{\mathsf{\Lambda }}_{mn}^{\alpha }\right){\phi }_n(\ell \varrho )+\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}\sum _{n=0}^m{\omega }_n{\mathsf{\Lambda }}_{mn}^{\alpha +1}{F}_{mn}\sum _{k=-N}^N{U}_k(\ell \varrho ){\gamma }_n(k\varrho ,{\phi }_n\left(k\varrho \right)),\hfill \end{array}$$

with

$$G_m(\ell \varrho )={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}\sum _{n=0}^m{\omega }_n{\mathsf{\Lambda }}_{mn}^{\alpha }{g}_n(\ell \varrho )+{\mu }_0{v}_1(\ell \varrho )+{\displaystyle \frac{{\mu }_0{t}_m^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}}(v_2\ell \varrho ).$$

Using the notations defined below,

$${\phi }_{m\ell }={\phi }_m(\ell \varrho ),G_{m\ell }=G_m(\ell \varrho ),U_{k\ell }=U_k(\ell \varrho ).$$

Consequently, the system of nonlinear algebraic equations is given by:

$$\begin{array}{cc}\hfill {\varpi }_m{\phi }_{m\ell }& =G_{m\ell }-\sum _{n=0}^{m-1}{\omega }_n\left({\mu }_1+{\displaystyle \frac{{\mu }_2}{\mathsf{\Gamma }\left(\alpha \right)}}{\mathsf{\Lambda }}_{mn}^{\alpha }\right){\phi }_{n\ell }+\hfill \\ \hfill & {\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}\sum _{n=0}^m{\omega }_n{\mathsf{\Lambda }}_{mn}^{\alpha +1}{F}_{mn}\sum _{k=-N}^N{U}_{k\ell }{\gamma }_n(k\varrho ,{\phi }_n\left(k\varrho \right)),\hfill \end{array}$$

(36)

with

$$G_{m\ell }={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}\sum _{n=0}^m{\omega }_n{\mathsf{\Lambda }}_{mn}^{\alpha }{g}_{n\ell }+{\mu }_0{v}_{1\ell }+{\displaystyle \frac{{\mu }_0{t}_m^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}}\left(v_{2\ell }\right).$$

The matrix $U_{k\ell }$ can be expressed in Toeplitz form as:

$$U_{k\ell }={\mathsf{\Theta }}_{k\ell }-{\mathsf{\Xi }}_{k\ell }.$$

Here, the matrix ${\mathsf{\Theta }}_{k\ell }={\aleph }_k(\ell \varrho )+{\Im }_{k-1}(\ell \varrho ),for-N\le k,\ell \le N,$ is a Toeplitz matrix of order $2N+1$, and

$$\begin{array}{c}\hfill {\mathsf{\Xi }}_{k\ell }=\left\{\begin{array}{cc}{\Im }_{-N-1}(k\ell );\hfill & k=-N\hfill \\ 0;\hfill & -N<k<N\hfill \\ {\aleph }_N(k\ell );\hfill & k=N.\hfill \end{array}\right.\end{array}$$

The resulting nonlinear algebraic system (36) is solved using an iterative numerical scheme. In the present work, we employ standard iterative solvers available in Wolfram Mathematica, such as fixed-point iteration or Newton-type methods. The iteration is initialized using the solution of the corresponding linearized system, and convergence is achieved when the norm of the residual vector is less than a prescribed tolerance $(\epsilon ={10}^{-12})$.

The computational complexity of the proposed numerical scheme depends primarily on the temporal and spatial discretization parameters. Let M denote the number of time discretization points and $2N+1$ the number of spatial nodes in the interval $[-1,1]$. The discretization of the Volterra integral term requires summations over previous time levels, resulting in $O\left(M\right)$ operations per time step and $O\left(M^2\right)$ operations overall. For each time level, the spatial integral with the singular kernel is approximated using $O\left(N\right)$ operations for each spatial node, leading to an overall cost of approximately $O\left(M^{2}N\right)$ for assembling the algebraic system. The resulting nonlinear system is solved iteratively using standard nonlinear solvers. Each iteration involves matrix–vector operations with the Toeplitz-type matrix, which require $O\left(N^2\right)$ operations. Therefore, if K iterations are required for convergence, the total computational complexity of the algorithm can be estimated as $O(M^{2}N+K{N}^2)$. The Toeplitz structure of the coefficient matrix also reduces memory requirements and enables efficient numerical implementation compared with general dense-matrix formulations.

6. The Convergence of the System of Nonlinear Algebraic Equations

In this section, we analyze the convergence of the kernel $k\left(\right|x-y\left|\right)$ and, consequently, the convergence of the system of nonlinear algebraic Equations (36).

Lemma 1. 

Let $k_n\left(\right|x-y\left|\right)\in L_2([-1,1]\times [-1,1])$ be a sequence of continuous kernels that satisfy the condition

$$\underset{n\to \infty }{lim}{\left\{{\int }_{-1}^1{\int }_{-1}^1|k_n\left(\right|x-y\left|\right)-k\left(\right|x-{y\left|\right)|}^2\mathrm{d}x\mathrm{d}y\right\}}^{{\displaystyle \frac{1}{2}}}=0.$$

(37)

Then there exists a positive integer $n_0$ such that

$${\left\{{\int }_{-1}^1{\int }_{-1}^1|k_n\left(\right|x-{y\left|\right)|}^2\mathrm{d}x\mathrm{d}y\right\}}^{{\displaystyle \frac{1}{2}}}\le C<\infty ,\forall n>n_0.$$

(38)

Proof. 

Using the triangle inequality, we have

$${\int }_{-1}^1{\int }_{-1}^1|k_n\left(\right|x-{y\left|\right)|}^2\mathrm{d}x\mathrm{d}y\le {\int }_{-1}^1{\int }_{-1}^1\left[\right|k_n\left(\right|x-y\left|\right)-k\left(\right|x-y\left|\right)|+\left|k\right(|x-{y\left|\right)\left|\right]}^2\mathrm{d}x\mathrm{d}y.$$

Taking square roots and applying standard inequalities yields

$$\begin{array}{cc}\hfill {\left\{{\int }_{-1}^1{\int }_{-1}^1|k_n\left(\right|x-{y\left|\right)|}^2\mathrm{d}x\mathrm{d}y\right\}}^{{\displaystyle \frac{1}{2}}}& \le {\left\{{\int }_{-1}^1{\int }_{-1}^1|k(|x-{y\left|\right)|}^2\mathrm{d}x\mathrm{d}y\right\}}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +{\left\{{\int }_{-1}^1{\int }_{-1}^1|k_n\left(\right|x-y\left|\right)-k\left(\right|x-{y\left|\right)|}^2\mathrm{d}x\mathrm{d}y\right\}}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

There is a positive integer $n_0$ such that the above inequality can be modified in the following form by using condition (37) and condition (i):

$${\left\{{\int }_{-1}^1{\int }_{-1}^1|k_n\left(\right|x-{y\left|\right)|}^2\mathrm{d}x\mathrm{d}y\right\}}^{{\displaystyle \frac{1}{2}}}\le \epsilon +C,\forall n>n_0.$$

Since $\epsilon$ is arbitrary, inequality (38) is obtained. □

Lemma 2. 

If the kernel of Equation (30) satisfies the following conditions:

(i*) 

${\left\{{\int }_{k\varrho }^{(k+1)\varrho }{\int }_{k\varrho }^{(k+1)\varrho }\left|k\right(|x-{y\left|\right)|}^2\mathrm{d}x\mathrm{d}y\right\}}^{{\displaystyle \frac{1}{2}}}\le C$.

(ii*) 

${lim}_{\overline{x}\to x}\parallel k\left(\right|\overline{x}-y\left|\right)-k\left(\right|x-{y\left|\right)\parallel }_{L_2}=0;\overline{x},x\in [-1,1]$.

Then,

$$\begin{array}{cc}\hfill \mathsf{1}.& \underset{N}{sup}\sum _{k=-N}^N\left|U_{k\ell }\right|\mathrm{exists}.\hfill \\ \hfill \mathsf{2}.& \underset{\overline{k}\to k}{lim}\underset{N}{sup}\sum _{\ell =-N}^N\parallel U_{\overline{k}\ell }-U_{k\ell }\parallel =0.\hfill \end{array}$$

(39)

Proof. 

To prove the first part of Lemma 2, from Formula (35), we have

$$\begin{array}{cc}\hfill |{\aleph }_k\left(x\right)|\le & {\displaystyle \frac{1}{|{\rm Y}|}}\left[|{\gamma }_n((k+1)\varrho ,(k+1)\varrho )|{\int }_{k\varrho }^{(k+1)\varrho }\left|k\right(|x-y\left|\right)\left|\right|{\gamma }_n(y,1)|\mathrm{d}y\right.\hfill \\ \hfill & +\left.|{\gamma }_n((k+1)\varrho ,1)|{\int }_{k\varrho }^{(k+1)\varrho }\left|k\right(|x-y\left|\right)\left|\right|{\gamma }_n(y,y)|\mathrm{d}y\right].\hfill \end{array}$$

When the Cauchy–Schwarz inequality is applied and the sum from $k=-N$ to $k=N$ is taken, the inequality above provides

$$\begin{array}{cc}\hfill \sum _{k=-N}^N\left|{\aleph }_k\left(x\right)\right|\le & {\displaystyle \frac{1}{|{\rm Y}|}}\parallel k\left(\right|x-y\left|\right)\parallel \left[\sum _{k=-N}^N|{\gamma }_n((k+1)\varrho ,(k+1)\varrho )|\parallel {\gamma }_n(y,1)\parallel \right.\hfill \\ \hfill & +\left.\sum _{k=-N}^N|{\gamma }_n((k+1)\varrho ,1)|\parallel {\gamma }_n(y,y)\parallel \right].\hfill \end{array}$$

In view of the condition $\left(i^{*}\right)$, and the continuity of the function $\gamma$ in the interval $[-1,1]$, there exists a small constant $H_1$, such that

$$\sum _{k=-N}^N\left|{\aleph }_k\left(x\right)\right|\le H_1;\forall N.$$

Since each term of ${\sum }_{k=-N}^N{\aleph }_k\left(x\right)$ is bounded above, for $x=\ell \varrho ,-N\le \ell \le N$, we deduce

$$\underset{N}{sup}\sum _{k=-N}^N\left|{\aleph }_k\left(x\right)\right|\le H_1.$$

(40)

Also, from the Formula (35) for the function ${\Im }_k\left(x\right)$, as previously shown, there exists a small constant $H_2$, such that

$$\underset{N}{sup}\sum _{k=-N}^N\left|{\Im }_k\left(x\right)\right|\le H_2.$$

(41)

Using (40) and (41), we have

$$\begin{array}{cc}\hfill \underset{N}{sup}\sum _{k=-N}^N\left|U_{k\ell }\right|& \le \underset{N}{sup}\sum _{k=-N}^N|{\aleph }_k\left(x\right)|+\underset{N}{sup}\sum _{k=-N}^N|{\Im }_k\left(x\right)|\hfill \\ \hfill & \le H_1+H_2\hfill \\ \hfill & \le H;(H=H_1+H_2).\hfill \end{array}$$

Hence, ${sup}_N{\sum }_{k=-N}^N\left|U_{k\ell }\right|$ exists.

To prove the second part of Lemma 2, from Formula (35), we obtain for $\overline{x},x\in [-1,1]$:

$$\begin{array}{cc}\hfill |{\aleph }_k\left(\overline{x}\right)|\le & {\displaystyle \frac{1}{|{\rm Y}|}}\left[|{\gamma }_n((k+1)\varrho ,(k+1)\varrho )|{\int }_{k\varrho }^{(k+1)\varrho }\left|k\right(|\overline{x}-y\left|\right)\left|\right|{\gamma }_n(y,1)|\mathrm{d}y\right.\hfill \\ \hfill & +\left.|{\gamma }_n((k+1)\varrho ,1)|{\int }_{k\varrho }^{(k+1)\varrho }\left|k\right(|\overline{x}-y\left|\right)\left|\right|{\gamma }_n(y,y)|\mathrm{d}y\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill |{\aleph }_k\left(x\right)|\le & {\displaystyle \frac{1}{|{\rm Y}|}}\left[|{\gamma }_n((k+1)\varrho ,(k+1)\varrho )|{\int }_{k\varrho }^{(k+1)\varrho }\left|k\right(|x-y\left|\right)\left|\right|{\gamma }_n(y,1)|\mathrm{d}y\right.\hfill \\ \hfill & +\left.|{\gamma }_n((k+1)\varrho ,1)|{\int }_{k\varrho }^{(k+1)\varrho }\left|k\right(|x-y\left|\right)\left|\right|{\gamma }_n(y,y)|\mathrm{d}y\right].\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill |{\aleph }_k\left(\overline{x}\right)-{\aleph }_k\left(x\right)|\le & {\displaystyle \frac{1}{|{\rm Y}|}}\left[|{\gamma }_n((k+1)\varrho ,(k+1)\varrho )|{\int }_{k\varrho }^{(k+1)\varrho }\left|k\right(|\overline{x}-y\left|\right)-k\left(\right|x-y\left|\right)\left|\right|{\gamma }_n(y,1)|\mathrm{d}y\right.\hfill \\ \hfill & +\left.|{\gamma }_n((k+1)\varrho ,1)|{\int }_{k\varrho }^{(k+1)\varrho }\left|k\right(|\overline{x}-y\left|\right)-k\left(\right|x-y\left|\right)\left|\right|{\gamma }_n(y,y)|\mathrm{d}y\right].\hfill \end{array}$$

When the Cauchy–Schwarz inequality is applied and the sum from $k=-N$ to $k=N$ is taken, taking into account the continuity of the function $\gamma$, the inequality mentioned before can be modified as

$$\begin{array}{cc}\hfill \underset{N}{sup}\sum _{k=-N}^N|{\aleph }_k\left(\overline{x}\right)-{\aleph }_k\left(x\right)|\le & {\displaystyle \frac{1}{|{\rm Y}|}}\parallel k\left(\right|\overline{x}-y\left|\right)-k\left(\right|x-y\left|\right)\parallel \left[\underset{N}{sup}\sum _{k=-N}^N|{\gamma }_n((k+1)\varrho ,(k+1)\varrho )|\parallel {\gamma }_n(y,1)\parallel \right.\hfill \\ \hfill & +\left.\underset{N}{sup}\sum _{k=-N}^N|{\gamma }_n((k+1)\varrho ,1)|\parallel {\gamma }_n(y,y)\parallel \right].\hfill \end{array}$$

Putting $x=\ell \varrho ,\overline{x}=\overline{\ell }\varrho$ and $\ell ,\overline{\ell }\in [-N,N]$, and applying the condition $\left(i{i}^{*}\right)$, we get as ${lim}_{\overline{x}\to x}$

$$\underset{\overline{\ell }\to \ell }{lim}\underset{N}{sup}\sum _{k=-N}^N|{\aleph }_k\left(\overline{\ell }\varrho \right)-{\aleph }_k(\ell \varrho )|=0.$$

(42)

Similarly, in view of Formula (35), we can prove

$$\underset{\overline{\ell }\to \ell }{lim}\underset{N}{sup}\sum _{k=-N}^N|{\Im }_k\left(\overline{\ell }\varrho \right)-{\Im }_k(\ell \varrho )|=0.$$

(43)

Finally, with the aid of (42) and (43), we have

$$\underset{\overline{\ell }\to \ell }{lim}\underset{N}{sup}\sum _{k=-N}^N|U_{k\overline{\ell }}-U_{k\ell }|=0.$$

□

Now, we prove the convergence of the system of nonlinear algebraic Equations (36) in the Banach space ${\ell }^{\infty }$.

Theorem 2. 

If the Equation (39) is satisfied, then the system of nonlinear algebraic Equations (36) is convergent.

Proof. 

The proof follows from Lemmas 1 and 2 together with standard results on the convergence of nonlinear systems in Banach spaces ${\ell }^{\infty }$. □

7. Numerical Results

In this section, we are dealing with the problems that are of interest to researchers yet cannot be solved in an analytical way. The problems are characterized by logarithmic, Cauchy kernels, and other singular kernels [27]. The absolute errors were computed for different values of x, t, and $\alpha$, and it was demonstrated that the method in this paper is accurate.

Example 1. 

Consider the following frPI-DE with a symmetric singular kernel $ln|y-x|$:

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{{\partial }^{\alpha +1}}{\partial t^{\alpha +1}}}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}+{\displaystyle \frac{1}{2}}\right)\phi (x,t)& =g(x,t)+0.001{\int }_0^t{t}^2{\tau }^2\mathrm{d}\tau {\int }_{-1}^{1}ln|y-x|{\phi }^2(y,\tau )\mathrm{d}y,\hfill \end{array}$$

(44)

where the function $g(x,t)$ is chosen such that the exact solution is given by $\phi =x^2{t}^2$.

Under the initial conditions,

$$\phi (x,0)=v_1\left(x\right)=0,{\displaystyle \frac{{\partial }^{\alpha }\phi (x,0)}{\partial t^{\alpha }}}=v_2\left(x\right)=0.$$

(45)

Integrating Equation (44), we obtain

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}\phi (x,t)-{\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}\phi (x,0)\right)& +{\displaystyle \frac{1}{4}}{I}^{1-\alpha }\phi (x,t)+{\displaystyle \frac{1}{2}}{\int }_0^t\phi (x,\tau )\mathrm{d}\tau ={\int }_0^{t}g(x,\tau )\mathrm{d}\tau \hfill \\ \hfill & +0.001{\int }_0^t{\int }_{-1}^1(t-\tau )t^2{\tau }^{2}ln|y-x|{\phi }^2(y,\tau )\mathrm{d}y\mathrm{d}\tau .\hfill \end{array}$$

Using condition (45), we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}\phi (x,t)& +{\displaystyle \frac{1}{4}}{I}^{1-\alpha }\phi (x,t)+{\displaystyle \frac{1}{2}}{\int }_0^t\phi (x,\tau )\mathrm{d}\tau ={\int }_0^{t}g(x,\tau )\mathrm{d}\tau \hfill \\ \hfill & +0.001{\int }_0^t{\int }_{-1}^1(t-\tau )t^2{\tau }^{2}ln|y-x|{\phi }^2(y,\tau )\mathrm{d}y\mathrm{d}\tau .\hfill \end{array}$$

By applying Cauchy’s formula for repeated integration with the fundamental Caputo-fractional integral formula, we get

$$\begin{array}{cc}\hfill & [\phi (x,t)-\phi (x,0)]+{\int }_0^t\left({\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{2\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right)\phi (x,\tau )\mathrm{d}\tau =\hfill \\ \hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha }g(x,\tau )\mathrm{d}\tau +{\displaystyle \frac{0.001}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\int }_{-1}^1{(t-\tau )}^{\alpha +1}{t}^2{\tau }^{2}ln|y-x|{\phi }^2(y,\tau )\mathrm{d}y\mathrm{d}\tau .\hfill \end{array}$$

Using condition (45), we obtain the NV-FIE:

$$\begin{array}{cc}\hfill \phi (x,t)& +{\int }_0^t\left({\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{2\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right)\phi (x,\tau )\mathrm{d}\tau ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha }g(x,\tau )\mathrm{d}\tau +\hfill \\ \hfill & {\displaystyle \frac{0.001}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\int }_{-1}^1{(t-\tau )}^{\alpha +1}{t}^2{\tau }^{2}ln|y-x|{\phi }^2(y,\tau )\mathrm{d}y\mathrm{d}\tau .\hfill \end{array}$$

(46)

The modified Toeplitz matrix method (TMM) is applied with $N=5$ to approximate the solution of Equation (46). The exact and approximate solutions are computed at various positions $x\in [-1,1]$ and time values $t\in [0,0.7]$.

To check the effectiveness of the proposed method, the absolute error between the exact and approximate solutions is calculated at the same points. To further demonstrate the performance of the numerical scheme, several values of the fractional order α were considered, specifically $\alpha =0.003,0.05,$ and 0.7. These values were selected to represent different regimes of fractional dynamics within the interval $0<\alpha <1$. The small values $\alpha =0.003$ and $\alpha =0.05$ correspond to cases where the fractional derivative is close to the classical integer-order derivative, allowing us to observe the behavior of the numerical method when the memory effect is relatively weak. In contrast, the value $\alpha =0.7$ represents a stronger fractional behavior, which reflects more pronounced nonlocal memory effects in the model.

Table 1. Absolute errors for Example 1 using TMM with $N=5$ and $t=0.1$ for different values of $\alpha$.

x	Errors ($\mathit{\alpha }$ = 0.003)	Errors ($\mathit{\alpha }$ = 0.05)	Errors ($\mathit{\alpha }$ = 0.7)
−1.0	$6.10213\times {10}^{-13}$	$5.26801\times {10}^{-13}$	$2.46317\times {10}^{-13}$
−0.8	$3.52401\times {10}^{-13}$	$1.98753\times {10}^{-13}$	$1.53807\times {10}^{-13}$
−0.6	$1.58234\times {10}^{-13}$	$1.08324\times {10}^{-13}$	$1.86372\times {10}^{-14}$
−0.4	$4.75283\times {10}^{-14}$	$4.38215\times {10}^{-14}$	$1.58327\times {10}^{-14}$
−0.2	$3.52487\times {10}^{-14}$	$2.31412\times {10}^{-14}$	$1.35876\times {10}^{-14}$
0.0	$2.54717\times {10}^{-14}$	$1.13547\times {10}^{-14}$	$1.05732\times {10}^{-14}$
0.2	$3.20574\times {10}^{-14}$	$2.23604\times {10}^{-14}$	$1.53648\times {10}^{-14}$
0.4	$5.39801\times {10}^{-14}$	$4.71085\times {10}^{-14}$	$1.93854\times {10}^{-14}$
0.6	$5.53687\times {10}^{-14}$	$4.92387\times {10}^{-14}$	$2.36542\times {10}^{-14}$
0.8	$2.03658\times {10}^{-13}$	$1.82657\times {10}^{-13}$	$1.46827\times {10}^{-13}$
1.0	$4.12365\times {10}^{-13}$	$3.96542\times {10}^{-13}$	$1.73692\times {10}^{-13}$

, Table 2, Table 3 and Table 4 present the absolute errors for different values of t, while Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 illustrate the exact solution, the numerical solution obtained using the modified Toeplitz matrix method (TMM), and the corresponding absolute error for different values of α.

Figure 1. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.1$ and $\alpha =0.003$ with $N=5$.

Figure 1. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.1$ and $\alpha =0.003$ with $N=5$.

Figure 2. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.1$ and $\alpha =0.05$ with $N=5$.

Figure 2. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.1$ and $\alpha =0.05$ with $N=5$.

Figure 3. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.1$ and $\alpha =0.7$ with $N=5$.

Figure 3. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.1$ and $\alpha =0.7$ with $N=5$.

Table 2. Absolute errors for Example 1 using TMM with $N=5$ and $t=0.3$ for different values of $\alpha$.

x	Errors ($\mathit{\alpha }$ = 0.003)	Errors ($\mathit{\alpha }$ = 0.05)	Errors ($\mathit{\alpha }$ = 0.7)
$-1.0$	$8.63971\times {10}^{-13}$	$5.96384\times {10}^{-13}$	$4.32187\times {10}^{-13}$
$-0.8$	$3.98247\times {10}^{-13}$	$3.68127\times {10}^{-13}$	$2.35468\times {10}^{-13}$
$-0.6$	$3.62845\times {10}^{-13}$	$2.68347\times {10}^{-13}$	$4.58261\times {10}^{-14}$
$-0.4$	$6.38724\times {10}^{-14}$	$4.93128\times {10}^{-14}$	$2.46831\times {10}^{-14}$
$-0.2$	$4.38974\times {10}^{-14}$	$2.66831\times {10}^{-14}$	$2.35186\times {10}^{-14}$
$0.0$	$2.96538\times {10}^{-14}$	$1.74101\times {10}^{-14}$	$1.53698\times {10}^{-14}$
$0.2$	$5.38617\times {10}^{-14}$	$2.56387\times {10}^{-14}$	$2.03584\times {10}^{-14}$
$0.4$	$7.43628\times {10}^{-14}$	$5.23984\times {10}^{-14}$	$3.56218\times {10}^{-14}$
$0.6$	$7.96381\times {10}^{-14}$	$7.61395\times {10}^{-14}$	$4.76298\times {10}^{-14}$
$0.8$	$4.62801\times {10}^{-13}$	$3.90576\times {10}^{-13}$	$3.62184\times {10}^{-13}$
$1.0$	$5.53918\times {10}^{-13}$	$5.36182\times {10}^{-13}$	$5.31781\times {10}^{-13}$

Table 2. Absolute errors for Example 1 using TMM with $N=5$ and $t=0.3$ for different values of $\alpha$.

x	Errors ($\mathit{\alpha }$ = 0.003)	Errors ($\mathit{\alpha }$ = 0.05)	Errors ($\mathit{\alpha }$ = 0.7)
$-1.0$	$8.63971\times {10}^{-13}$	$5.96384\times {10}^{-13}$	$4.32187\times {10}^{-13}$
$-0.8$	$3.98247\times {10}^{-13}$	$3.68127\times {10}^{-13}$	$2.35468\times {10}^{-13}$
$-0.6$	$3.62845\times {10}^{-13}$	$2.68347\times {10}^{-13}$	$4.58261\times {10}^{-14}$
$-0.4$	$6.38724\times {10}^{-14}$	$4.93128\times {10}^{-14}$	$2.46831\times {10}^{-14}$
$-0.2$	$4.38974\times {10}^{-14}$	$2.66831\times {10}^{-14}$	$2.35186\times {10}^{-14}$
$0.0$	$2.96538\times {10}^{-14}$	$1.74101\times {10}^{-14}$	$1.53698\times {10}^{-14}$
$0.2$	$5.38617\times {10}^{-14}$	$2.56387\times {10}^{-14}$	$2.03584\times {10}^{-14}$
$0.4$	$7.43628\times {10}^{-14}$	$5.23984\times {10}^{-14}$	$3.56218\times {10}^{-14}$
$0.6$	$7.96381\times {10}^{-14}$	$7.61395\times {10}^{-14}$	$4.76298\times {10}^{-14}$
$0.8$	$4.62801\times {10}^{-13}$	$3.90576\times {10}^{-13}$	$3.62184\times {10}^{-13}$
$1.0$	$5.53918\times {10}^{-13}$	$5.36182\times {10}^{-13}$	$5.31781\times {10}^{-13}$

Figure 4. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.3$ and $\alpha =0.003$ with $N=5$.

Figure 4. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.3$ and $\alpha =0.003$ with $N=5$.

Figure 5. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.3$ and $\alpha =0.05$ with $N=5$.

Figure 5. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.3$ and $\alpha =0.05$ with $N=5$.

Figure 6. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.3$ and $\alpha =0.7$ with $N=5$.

Figure 6. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.3$ and $\alpha =0.7$ with $N=5$.

Table 3. Absolute errors for Example 1 using TMM with $N=5$ and $t=0.5$ for different values of $\alpha$.

x	Errors ($\mathit{\alpha }$ = 0.003)	Errors ($\mathit{\alpha }$ = 0.05)	Errors ($\mathit{\alpha }$ = 0.7)
$-1.0$	$9.63258\times {10}^{-13}$	$6.82374\times {10}^{-13}$	$6.32987\times {10}^{-13}$
$-0.8$	$6.32847\times {10}^{-13}$	$6.31587\times {10}^{-13}$	$4.32054\times {10}^{-13}$
$-0.6$	$5.43021\times {10}^{-13}$	$4.32864\times {10}^{-13}$	$7.62316\times {10}^{-14}$
$-0.4$	$2.36584\times {10}^{-13}$	$6.31864\times {10}^{-14}$	$5.32184\times {10}^{-14}$
$-0.2$	$6.32187\times {10}^{-14}$	$5.36218\times {10}^{-14}$	$4.30568\times {10}^{-14}$
$0.0$	$4.23548\times {10}^{-14}$	$3.65482\times {10}^{-14}$	$3.25682\times {10}^{-14}$
$0.2$	$6.38241\times {10}^{-14}$	$4.36021\times {10}^{-14}$	$4.32168\times {10}^{-14}$
$0.4$	$9.34587\times {10}^{-14}$	$5.93167\times {10}^{-14}$	$5.46931\times {10}^{-14}$
$0.6$	$1.68234\times {10}^{-13}$	$8.52346\times {10}^{-14}$	$6.31592\times {10}^{-14}$
$0.8$	$6.32913\times {10}^{-13}$	$4.83168\times {10}^{-13}$	$4.78216\times {10}^{-13}$
$1.0$	$2.30576\times {10}^{-12}$	$7.46315\times {10}^{-13}$	$6.87413\times {10}^{-13}$

Table 3. Absolute errors for Example 1 using TMM with $N=5$ and $t=0.5$ for different values of $\alpha$.

x	Errors ($\mathit{\alpha }$ = 0.003)	Errors ($\mathit{\alpha }$ = 0.05)	Errors ($\mathit{\alpha }$ = 0.7)
$-1.0$	$9.63258\times {10}^{-13}$	$6.82374\times {10}^{-13}$	$6.32987\times {10}^{-13}$
$-0.8$	$6.32847\times {10}^{-13}$	$6.31587\times {10}^{-13}$	$4.32054\times {10}^{-13}$
$-0.6$	$5.43021\times {10}^{-13}$	$4.32864\times {10}^{-13}$	$7.62316\times {10}^{-14}$
$-0.4$	$2.36584\times {10}^{-13}$	$6.31864\times {10}^{-14}$	$5.32184\times {10}^{-14}$
$-0.2$	$6.32187\times {10}^{-14}$	$5.36218\times {10}^{-14}$	$4.30568\times {10}^{-14}$
$0.0$	$4.23548\times {10}^{-14}$	$3.65482\times {10}^{-14}$	$3.25682\times {10}^{-14}$
$0.2$	$6.38241\times {10}^{-14}$	$4.36021\times {10}^{-14}$	$4.32168\times {10}^{-14}$
$0.4$	$9.34587\times {10}^{-14}$	$5.93167\times {10}^{-14}$	$5.46931\times {10}^{-14}$
$0.6$	$1.68234\times {10}^{-13}$	$8.52346\times {10}^{-14}$	$6.31592\times {10}^{-14}$
$0.8$	$6.32913\times {10}^{-13}$	$4.83168\times {10}^{-13}$	$4.78216\times {10}^{-13}$
$1.0$	$2.30576\times {10}^{-12}$	$7.46315\times {10}^{-13}$	$6.87413\times {10}^{-13}$

Figure 7. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.5$ and $\alpha =0.003$ with $N=5$.

Figure 7. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.5$ and $\alpha =0.003$ with $N=5$.

Figure 8. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.5$ and $\alpha =0.05$ with $N=5$.

Figure 8. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.5$ and $\alpha =0.05$ with $N=5$.

Figure 9. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.5$ and $\alpha =0.7$ with $N=5$.

Figure 9. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.5$ and $\alpha =0.7$ with $N=5$.

Table 4. Absolute errors for Example 1 using TMM with $N=5$ and $t=0.7$ for different values of $\alpha$.

x	Errors ($\mathit{\alpha }$ = 0.003)	Errors ($\mathit{\alpha }$ = 0.05)	Errors ($\mathit{\alpha }$ = 0.7)
$-1.0$	$2.64713\times {10}^{-12}$	$9.32547\times {10}^{-13}$	$5.53682\times {10}^{-13}$
$-0.8$	$8.62374\times {10}^{-13}$	$7.53168\times {10}^{-13}$	$5.23874\times {10}^{-13}$
$-0.6$	$6.84731\times {10}^{-13}$	$6.32841\times {10}^{-13}$	$3.82364\times {10}^{-13}$
$-0.4$	$3.25476\times {10}^{-13}$	$2.62857\times {10}^{-13}$	$2.56384\times {10}^{-13}$
$-0.2$	$1.54682\times {10}^{-13}$	$2.65848\times {10}^{-13}$	$1.52378\times {10}^{-13}$
$0.0$	$6.32187\times {10}^{-14}$	$4.53687\times {10}^{-14}$	$5.23478\times {10}^{-14}$
$0.2$	$9.31576\times {10}^{-14}$	$6.38472\times {10}^{-14}$	$5.93215\times {10}^{-14}$
$0.4$	$3.54761\times {10}^{-13}$	$3.25874\times {10}^{-13}$	$2.96541\times {10}^{-13}$
$0.6$	$4.58632\times {10}^{-13}$	$4.53128\times {10}^{-13}$	$3.62584\times {10}^{-13}$
$0.8$	$6.25478\times {10}^{-12}$	$5.23841\times {10}^{-12}$	$5.03642\times {10}^{-12}$
$1.0$	$7.57613\times {10}^{-12}$	$6.32185\times {10}^{-12}$	$5.56381\times {10}^{-12}$

Table 4. Absolute errors for Example 1 using TMM with $N=5$ and $t=0.7$ for different values of $\alpha$.

x	Errors ($\mathit{\alpha }$ = 0.003)	Errors ($\mathit{\alpha }$ = 0.05)	Errors ($\mathit{\alpha }$ = 0.7)
$-1.0$	$2.64713\times {10}^{-12}$	$9.32547\times {10}^{-13}$	$5.53682\times {10}^{-13}$
$-0.8$	$8.62374\times {10}^{-13}$	$7.53168\times {10}^{-13}$	$5.23874\times {10}^{-13}$
$-0.6$	$6.84731\times {10}^{-13}$	$6.32841\times {10}^{-13}$	$3.82364\times {10}^{-13}$
$-0.4$	$3.25476\times {10}^{-13}$	$2.62857\times {10}^{-13}$	$2.56384\times {10}^{-13}$
$-0.2$	$1.54682\times {10}^{-13}$	$2.65848\times {10}^{-13}$	$1.52378\times {10}^{-13}$
$0.0$	$6.32187\times {10}^{-14}$	$4.53687\times {10}^{-14}$	$5.23478\times {10}^{-14}$
$0.2$	$9.31576\times {10}^{-14}$	$6.38472\times {10}^{-14}$	$5.93215\times {10}^{-14}$
$0.4$	$3.54761\times {10}^{-13}$	$3.25874\times {10}^{-13}$	$2.96541\times {10}^{-13}$
$0.6$	$4.58632\times {10}^{-13}$	$4.53128\times {10}^{-13}$	$3.62584\times {10}^{-13}$
$0.8$	$6.25478\times {10}^{-12}$	$5.23841\times {10}^{-12}$	$5.03642\times {10}^{-12}$
$1.0$	$7.57613\times {10}^{-12}$	$6.32185\times {10}^{-12}$	$5.56381\times {10}^{-12}$

Figure 10. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.7$ and $\alpha =0.003$ with $N=5$.

Figure 10. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.7$ and $\alpha =0.003$ with $N=5$.

Figure 11. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.7$ and $\alpha =0.05$ with $N=5$.

Figure 11. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.7$ and $\alpha =0.05$ with $N=5$.

Figure 12. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.7$ and $\alpha =0.7$ with $N=5$.

Figure 12. Comparison of the exact solution, the numerical solution obtained using TMM, and the corresponding absolute error for Example 1 at $t=0.7$ and $\alpha =0.7$ with $N=5$.

In this example, the numerical results reported in Table 1, Table 2, Table 3 and Table 4 demonstrate that the proposed method yields highly accurate approximations for all tested values of the spatial variable x, time variable t, and fractional $\alpha$. In particular, the absolute errors are consistently of the order ${10}^{-14}$–${10}^{-12}$, even though the spatial discretization parameter is fixed at the relatively small value $N=5$. These small error values indicate that the method is converging rapidly and that it is working well with the logarithmic singularity.

From the tables, we find that the smallest errors occur near the center of the spatial domain, while slightly larger errors appear near $x=\pm 1$. This behavior is expected and can be explained by the fact that the logarithmic kernel is more influential near the boundaries, where numerical quadrature and interpolation errors tend to be more pronounced. Nevertheless, even at the boundaries, the errors remain extremely small, confirming the robustness of the scheme.

Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 confirm the data provided in the tables. In all cases, the approximate solutions obtained by the TMM are almost indistinguishable from the exact solutions, with error curves that are smooth, symmetric, and several orders of magnitude smaller than the solution itself. These visual comparisons provide qualitative confirmation of the high accuracy of the proposed approach.

Example 2. 

Consider the following frPI-DE with a nonsymmetric kernel:

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{{\partial }^{\alpha +1}}{\partial t^{\alpha +1}}}+{\displaystyle \frac{1}{3}}{\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}+{\displaystyle \frac{1}{5}}\right)\phi (x,t)& =g(x,t)+0.003{\int }_0^{t}t\tau \mathrm{d}\tau {\int }_{\mathsf{\Omega }}{\displaystyle \frac{1}{(x-y)}}{\phi }^3(y,\tau )\mathrm{d}y,\hfill \end{array}$$

(47)

where the function $g(x,t)$ is chosen such that the exact solution is given by $\phi (x,t)=x{e}^t$.

Under the initial conditions,

$$\phi (x,0)=v_1\left(x\right)=x,{\displaystyle \frac{{\partial }^{\alpha }x}{\partial t^{\alpha }}}=v_2\left(x\right).$$

Analogously to Example 1, we obtain the following NV-FIE:

$$\begin{array}{cc}\hfill \phi (x,t)& +{\int }_0^t\left({\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{5\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right)\phi (x,\tau )\mathrm{d}\tau =G(x,t)+\hfill \\ \hfill & {\displaystyle \frac{0.003}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\int }_{-1}^1{(t-\tau )}^{\alpha +1}{\displaystyle \frac{t\tau }{(x-y)}}{\phi }^3(y,\tau )\mathrm{d}y\mathrm{d}\tau ,\hfill \end{array}$$

(48)

where

$$G(x,t)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha }g(x,\tau )\mathrm{d}\tau +x+{\displaystyle \frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}}{v}_2\left(x\right).$$

Table 5, Table 6, Table 7 and Table 8 illustrate the behavior of the exact solution of Equation (48) over the domain $x\in [-1,1]$ and the time $t\in [0,0.9]$, evaluated at selected points of position and time for different values of α. To check the effectiveness of the proposed method, the approximate solution of Equation (48) is computed at the same points using TMM with $N=10,$ and the corresponding absolute errors are reported.

Table 5. Exact solution, approximate solution, and absolute error for Example 2 computed using TMM with $N=10$ and $\alpha =0.005$ at different values of x and t.

(x, t)	Exact Solution	Approximate Solution	Errors
$(0.0,0.0)$	0	0	0
$(0.1,0.1)$	$0.110517092$	$0.110517091$	$1.10517\times {10}^{-9}$
$(0.2,0.2)$	$0.244280552$	$0.244280549$	$2.44281\times {10}^{-9}$
$(0.3,0.3)$	$0.404957642$	$0.404957638$	$4.04958\times {10}^{-9}$
$(0.4,0.4)$	$0.596729879$	$0.596729873$	$5.96730\times {10}^{-9}$
$(0.5,0.5)$	$0.824360635$	$0.824360627$	$8.24361\times {10}^{-9}$
$(0.6,0.6)$	$1.093271280$	$1.093271269$	$1.09327\times {10}^{-8}$
$(0.7,0.7)$	$1.409626895$	$1.409626881$	$1.40963\times {10}^{-8}$
$(0.8,0.8)$	$1.780432743$	$1.780432725$	$1.78043\times {10}^{-8}$
$(0.9,0.9)$	$2.213642800$	$2.213642778$	$2.21364\times {10}^{-8}$

Table 5. Exact solution, approximate solution, and absolute error for Example 2 computed using TMM with $N=10$ and $\alpha =0.005$ at different values of x and t.

(x, t)	Exact Solution	Approximate Solution	Errors
$(0.0,0.0)$	0	0	0
$(0.1,0.1)$	$0.110517092$	$0.110517091$	$1.10517\times {10}^{-9}$
$(0.2,0.2)$	$0.244280552$	$0.244280549$	$2.44281\times {10}^{-9}$
$(0.3,0.3)$	$0.404957642$	$0.404957638$	$4.04958\times {10}^{-9}$
$(0.4,0.4)$	$0.596729879$	$0.596729873$	$5.96730\times {10}^{-9}$
$(0.5,0.5)$	$0.824360635$	$0.824360627$	$8.24361\times {10}^{-9}$
$(0.6,0.6)$	$1.093271280$	$1.093271269$	$1.09327\times {10}^{-8}$
$(0.7,0.7)$	$1.409626895$	$1.409626881$	$1.40963\times {10}^{-8}$
$(0.8,0.8)$	$1.780432743$	$1.780432725$	$1.78043\times {10}^{-8}$
$(0.9,0.9)$	$2.213642800$	$2.213642778$	$2.21364\times {10}^{-8}$

Figure 13. Comparison of the exact solution and the numerical solution obtained using TMM for Example 2 with $N=10$ and fractional order $\alpha =0.005$ at different values of x and t.

Figure 13. Comparison of the exact solution and the numerical solution obtained using TMM for Example 2 with $N=10$ and fractional order $\alpha =0.005$ at different values of x and t.

Table 6. Exact solution, approximate solution, and absolute error for Example 2 computed using TMM with $N=10$ and $\alpha =0.04$ at different values of x and t.

(x, t)	Exact Solution	Approximate Solution	Errors
$(0.0,0.0)$	0	0	0
$(0.1,0.1)$	$0.110517092$	$0.110517092$	$2.21034\times {10}^{-10}$
$(0.2,0.2)$	$0.244280552$	$0.244280551$	$4.88561\times {10}^{-10}$
$(0.3,0.3)$	$0.404957642$	$0.404957641$	$8.09915\times {10}^{-10}$
$(0.4,0.4)$	$0.596729879$	$0.596729878$	$1.19346\times {10}^{-9}$
$(0.5,0.5)$	$0.824360635$	$0.824360634$	$1.64872\times {10}^{-9}$
$(0.6,0.6)$	$1.093271280$	$1.093271278$	$2.18654\times {10}^{-9}$
$(0.7,0.7)$	$1.409626895$	$1.409626892$	$2.81925\times {10}^{-9}$
$(0.8,0.8)$	$1.780432743$	$1.780432739$	$3.56087\times {10}^{-9}$
$(0.9,0.9)$	$2.213642800$	$2.213642796$	$4.42729\times {10}^{-9}$

Table 6. Exact solution, approximate solution, and absolute error for Example 2 computed using TMM with $N=10$ and $\alpha =0.04$ at different values of x and t.

(x, t)	Exact Solution	Approximate Solution	Errors
$(0.0,0.0)$	0	0	0
$(0.1,0.1)$	$0.110517092$	$0.110517092$	$2.21034\times {10}^{-10}$
$(0.2,0.2)$	$0.244280552$	$0.244280551$	$4.88561\times {10}^{-10}$
$(0.3,0.3)$	$0.404957642$	$0.404957641$	$8.09915\times {10}^{-10}$
$(0.4,0.4)$	$0.596729879$	$0.596729878$	$1.19346\times {10}^{-9}$
$(0.5,0.5)$	$0.824360635$	$0.824360634$	$1.64872\times {10}^{-9}$
$(0.6,0.6)$	$1.093271280$	$1.093271278$	$2.18654\times {10}^{-9}$
$(0.7,0.7)$	$1.409626895$	$1.409626892$	$2.81925\times {10}^{-9}$
$(0.8,0.8)$	$1.780432743$	$1.780432739$	$3.56087\times {10}^{-9}$
$(0.9,0.9)$	$2.213642800$	$2.213642796$	$4.42729\times {10}^{-9}$

Figure 14. Comparison of the exact solution and the numerical solution obtained using TMM for Example 2 with $N=10$ and fractional order $\alpha =0.04$ at different values of x and t.

Figure 14. Comparison of the exact solution and the numerical solution obtained using TMM for Example 2 with $N=10$ and fractional order $\alpha =0.04$ at different values of x and t.

Table 7. Exact solution, approximate solution, and absolute error for Example 2 computed using TMM with $N=10$ and $\alpha =0.2$ at different values of x and t.

(x, t)	Exact Solution	Approximate Solution	Errors
$(0.0,0.0)$	0	0	0
$(0.1,0.1)$	$0.110517092$	$0.110517092$	$1.43672\times {10}^{-10}$
$(0.2,0.2)$	$0.244280552$	$0.244280551$	$3.17565\times {10}^{-10}$
$(0.3,0.3)$	$0.404957642$	$0.404957642$	$5.26445\times {10}^{-10}$
$(0.4,0.4)$	$0.596729879$	$0.596729878$	$7.75749\times {10}^{-10}$
$(0.5,0.5)$	$0.824360635$	$0.824360634$	$1.07167\times {10}^{-9}$
$(0.6,0.6)$	$1.093271280$	$1.093271279$	$1.42125\times {10}^{-9}$
$(0.7,0.7)$	$1.409626895$	$1.409626893$	$1.83252\times {10}^{-9}$
$(0.8,0.8)$	$1.780432743$	$1.780432740$	$2.31456\times {10}^{-9}$
$(0.9,0.9)$	$2.213642800$	$2.213642797$	$2.87774\times {10}^{-9}$

Table 7. Exact solution, approximate solution, and absolute error for Example 2 computed using TMM with $N=10$ and $\alpha =0.2$ at different values of x and t.

(x, t)	Exact Solution	Approximate Solution	Errors
$(0.0,0.0)$	0	0	0
$(0.1,0.1)$	$0.110517092$	$0.110517092$	$1.43672\times {10}^{-10}$
$(0.2,0.2)$	$0.244280552$	$0.244280551$	$3.17565\times {10}^{-10}$
$(0.3,0.3)$	$0.404957642$	$0.404957642$	$5.26445\times {10}^{-10}$
$(0.4,0.4)$	$0.596729879$	$0.596729878$	$7.75749\times {10}^{-10}$
$(0.5,0.5)$	$0.824360635$	$0.824360634$	$1.07167\times {10}^{-9}$
$(0.6,0.6)$	$1.093271280$	$1.093271279$	$1.42125\times {10}^{-9}$
$(0.7,0.7)$	$1.409626895$	$1.409626893$	$1.83252\times {10}^{-9}$
$(0.8,0.8)$	$1.780432743$	$1.780432740$	$2.31456\times {10}^{-9}$
$(0.9,0.9)$	$2.213642800$	$2.213642797$	$2.87774\times {10}^{-9}$

Figure 15. Comparison of the exact solution and the numerical solution obtained using TMM for Example 2 with $N=10$ and fractional order $\alpha =0.2$ at different values of x and t.

Figure 15. Comparison of the exact solution and the numerical solution obtained using TMM for Example 2 with $N=10$ and fractional order $\alpha =0.2$ at different values of x and t.

Table 8. Exact solution, approximate solution, and absolute error for Example 2 computed using TMM with $N=10$ and $\alpha =0.9$ at different values of x and t.

(x, t)	Exact Solution	Approximate Solution	Errors
$(0.0,0.0)$	0	0	0
$(0.1,0.1)$	$0.110517092$	$0.110517092$	$1.3262\times {10}^{-11}$
$(0.2,0.2)$	$0.244280552$	$0.244280552$	$2.93137\times {10}^{-11}$
$(0.3,0.3)$	$0.404957642$	$0.404957642$	$4.85949\times {10}^{-11}$
$(0.4,0.4)$	$0.596729879$	$0.596729879$	$7.16076\times {10}^{-11}$
$(0.5,0.5)$	$0.824360635$	$0.824360635$	$9.89233\times {10}^{-11}$
$(0.6,0.6)$	$1.093271280$	$1.093271280$	$1.31192\times {10}^{-10}$
$(0.7,0.7)$	$1.409626895$	$1.409626895$	$1.69155\times {10}^{-10}$
$(0.8,0.8)$	$1.780432743$	$1.780432743$	$2.13652\times {10}^{-10}$
$(0.9,0.9)$	$2.213642800$	$2.213642800$	$2.65637\times {10}^{-10}$

Table 8. Exact solution, approximate solution, and absolute error for Example 2 computed using TMM with $N=10$ and $\alpha =0.9$ at different values of x and t.

(x, t)	Exact Solution	Approximate Solution	Errors
$(0.0,0.0)$	0	0	0
$(0.1,0.1)$	$0.110517092$	$0.110517092$	$1.3262\times {10}^{-11}$
$(0.2,0.2)$	$0.244280552$	$0.244280552$	$2.93137\times {10}^{-11}$
$(0.3,0.3)$	$0.404957642$	$0.404957642$	$4.85949\times {10}^{-11}$
$(0.4,0.4)$	$0.596729879$	$0.596729879$	$7.16076\times {10}^{-11}$
$(0.5,0.5)$	$0.824360635$	$0.824360635$	$9.89233\times {10}^{-11}$
$(0.6,0.6)$	$1.093271280$	$1.093271280$	$1.31192\times {10}^{-10}$
$(0.7,0.7)$	$1.409626895$	$1.409626895$	$1.69155\times {10}^{-10}$
$(0.8,0.8)$	$1.780432743$	$1.780432743$	$2.13652\times {10}^{-10}$
$(0.9,0.9)$	$2.213642800$	$2.213642800$	$2.65637\times {10}^{-10}$

Figure 16. Comparison of the exact solution and the numerical solution obtained using TMM for Example 2 with $N=10$ and fractional order $\alpha =0.9$ at different values of x and t.

Figure 16. Comparison of the exact solution and the numerical solution obtained using TMM for Example 2 with $N=10$ and fractional order $\alpha =0.9$ at different values of x and t.

This example is devoted to assessing the performance of the proposed modified Toeplitz Matrix Method (TMM) when applied to a fractional partial integro-differential equation involving a Cauchy-type singular kernel. This example is of particular importance, as Cauchy kernels are well known for inducing numerical instability and loss of accuracy in classical discretization schemes. The corresponding tables and figures show that the proposed method is robust, accurate, and stable.

The absolute errors at the chosen spatial points and time levels show that the modified TMM produces a numerical solution that closely agrees with the exact solution over the whole spatial domain. For all tested values of the fractional order $\alpha$, the magnitude of the error remains remarkably small, typically ranging between ${10}^{-9}$ and ${10}^{-8}$. The degree of accuracy is obtained with a relatively coarse Toeplitz grid, which indicates the efficiency of the computation method and substantiates its adequacy for practical use in the case of fractional operators and singular kernels.

In Figure 13, Figure 14, Figure 15 and Figure 16, it is seen that the numerical solution is identical to the exact solution. The error plots are all smooth and well-behaved, with no oscillations. Such behavior is indicative of convergence and confirms that the proposed method preserves the qualitative features of the exact solution.

Example 3. 

Consider the following frPI-DE with a weakly singular kernel, $1/\sqrt{x-y}$:

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{{\partial }^{\alpha +1}}{\partial t^{\alpha +1}}}+0.25{\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}+0.5\right)\phi (x,t)& =g(x,t)+0.55{\int }_0^t{t}^2\tau \mathrm{d}\tau {\int }_{-1}^1{\displaystyle \frac{1}{\sqrt{x-y}}}{\phi }^2(y,\tau )\mathrm{d}y,\hfill \end{array}$$

(49)

where the function $g(x,t)$ is chosen such that the exact solution is given by $\phi =t^2(1+xt)$.

Under the initial conditions,

$$\phi (x,0)=0,{\displaystyle \frac{{\partial }^{\alpha }\phi (x,0)}{\partial t^{\alpha }}}=0.$$

Analogously to Example 1, we obtain the following NV-FIE:

$$\begin{array}{cc}\hfill \phi (x,t)& +{\int }_0^t\left(0.25+{\displaystyle \frac{0.5}{\mathsf{\Gamma }\left(\alpha \right)}}{(t-\tau )}^{\alpha }\right)\phi (x,\tau )\mathrm{d}\tau ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha }g(x,\tau )\mathrm{d}\tau +\hfill \\ \hfill & {\displaystyle \frac{0.55}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{\int }_{-1}^1{(t-\tau )}^{\alpha +1}{t}^2\tau {\displaystyle \frac{1}{\sqrt{x-y}}}{\phi }^2(y,\tau )\mathrm{d}y\mathrm{d}\tau .\hfill \end{array}$$

(50)

In this example, we compare the approximate solutions of Equation (50) obtained via the proposed method TMM and the Shifted Chebyshev Polynomials (SCP) method.

This comparison is conducted to evaluate the accuracy and efficiency of the proposed method. To achieve this objective, the absolute error for both methods is computed at various values of $x\in [-1,0.8]$ and over different time $t\in [0,0.7]$, while also varying the number of nodes $\left(N\right)$. The results are presented in

Table 9. Comparison of numerical errors for Example 3 using TMM and SCP at selected x and t for $\alpha =0.02,N=8$ and $\alpha =0.5,N=15$.

x	t	Errors of TMM		Errors of SCP
		$\mathit{\alpha }$ = 0.02, N = 8	$\mathit{\alpha }$ = 0.5, N = 15	$\alpha$ = 0.02, N = 8	$\alpha$ = 0.5, N = 15
−1.0	0.3	$1.45876\times {10}^{-10}$	$7.36985\times {10}^{-10}$	$1.36587\times {10}^{-9}$	$8.65847\times {10}^{-10}$
−0.9		$8.52347\times {10}^{-11}$	$2.84756\times {10}^{-10}$	$3.65847\times {10}^{-10}$	$3.13584\times {10}^{-10}$
−0.7		$5.26871\times {10}^{-11}$	$3.58749\times {10}^{-11}$	$8.36110\times {10}^{-11}$	$5.25487\times {10}^{-11}$
−0.6		$8.32654\times {10}^{-12}$	$5.23687\times {10}^{-12}$	$6.85243\times {10}^{-11}$	$3.25874\times {10}^{-11}$
−0.2	0.5	$5.62684\times {10}^{-12}$	$5.33874\times {10}^{-12}$	$8.62348\times {10}^{-12}$	$3.12848\times {10}^{-11}$
−0.1		$4.58768\times {10}^{-12}$	$3.38756\times {10}^{-12}$	$5.32686\times {10}^{-12}$	$4.54875\times {10}^{-12}$
0.0		$3.36587\times {10}^{-12}$	$2.58764\times {10}^{-12}$	$5.02368\times {10}^{-12}$	$4.32904\times {10}^{-12}$
0.1		$5.32896\times {10}^{-12}$	$4.36871\times {10}^{-12}$	$2.57694\times {10}^{-11}$	$2.28745\times {10}^{-11}$
0.3	0.7	$3.56877\times {10}^{-11}$	$2.36587\times {10}^{-11}$	$6.32584\times {10}^{-11}$	$5.92179\times {10}^{-11}$
0.4		$4.32874\times {10}^{-10}$	$4.32504\times {10}^{-11}$	$6.02341\times {10}^{-10}$	$1.32587\times {10}^{-10}$
0.5		$7.35987\times {10}^{-10}$	$6.32580\times {10}^{-10}$	$8.10470\times {10}^{-10}$	$7.32567\times {10}^{-10}$
0.8		$7.96328\times {10}^{-10}$	$7.36584\times {10}^{-10}$	$3.58746\times {10}^{-9}$	$9.32658\times {10}^{-10}$

, which provides a quantitative comparison of the performance of the two methods, highlighting the accuracy and effectiveness of the proposed approach in comparison with the conventional method.

Table 9 presents a comparative analysis of the numerical errors obtained using TMM and SCP under different parameter settings, specifically for $\alpha =0.02,N=8$ and $\alpha =0.5,$ $N=15$, across various spatial points x and time levels t.

The results clearly indicate that both methods yield highly accurate approximations, with error magnitudes predominantly ranging between ${10}^{-12}$ and ${10}^{-9}$. This demonstrates the excellent numerical performance of both TMM and SCP in solving the considered problem. A detailed comparison reveals that the TMM method generally produces slightly smaller errors than the SCP method, particularly for smaller values of $\alpha$. This suggests that TMM exhibits superior numerical accuracy and better convergence behavior under finer parameter settings.

Overall, the results presented in Table 9 confirm that both TMM and SCP are highly efficient and reliable numerical techniques. Nevertheless, TMM demonstrates a slight advantage in terms of accuracy, making it a preferable choice for solving similar problems requiring high-precision numerical solutions.

Remark on the limiting values of the fractional order $\alpha$:

The fractional order $\alpha$ considered in this work satisfies $0<\alpha <1$. When $\alpha$ approaches the limiting values of this interval, the governing equation gradually transitions toward classical operators. In particular, as $\alpha \to 1,$ the Caputo fractional derivative approaches the first-order classical derivative, while as $\alpha \to 0,$ the operator behaves similarly to an integral-type memory term. In the proposed numerical scheme, the discretization weights involve the Gamma function coefficients $\mathsf{\Gamma }\left(\alpha \right)$ and $\mathsf{\Gamma }(\alpha +1)$, which remain well defined for values of $\alpha$ close to these limits. Consequently, the modified Toeplitz matrix method remains numerically stable for fractional orders approaching both endpoints of the interval. The numerical experiments presented in this section demonstrate that the algorithm maintains accuracy and stability for different values of $\alpha$, confirming the robustness of the proposed approach across the admissible fractional range.

8. General Conclusions and Future Work

We can deduce the following from the study discussed above:

• There are numerous approaches to solving continuous integral equations, whether they be linear or nonlinear. These techniques could be explicit numerical techniques or semi-analytical techniques.
• There are two approaches to solving singular integral equations: There is the Nyström multiplication method [28], which takes a long time to analyze and transforms the integral system into a nonlinear system. The Toeplitz matrix method is another approach.
• The Toeplitz matrix approach is characterized by its simplicity in converting singular integrals into non-singular integrals that can be easily calculated. It is further characterized by the quick transformation of the integral system into an algebraic system that may be investigated.
• In the numerical examples, the parameter $\lambda$ is chosen to satisfy the sufficient condition (8), which guarantees the existence and uniqueness of the solution. Although this condition is not necessary, it ensures the convergence of the proposed numerical scheme. For larger values of $\lambda$, convergence may still be observed numerically; however, a rigorous theoretical justification is beyond the scope of the present work and will be investigated in future studies.
• The proposed method differs from conventional Toeplitz-based discretization in several ways. It transforms the original fractional partial integro-differential equation into a nonlinear Volterra–Fredholm integral equation, enabling a unified treatment of temporal fractional operators and spatial singular integrals. The resulting discretization yields a structured nonlinear system with a Toeplitz-type matrix, ensuring computational efficiency.

Concerning future work, the proposed methods will be extended to address high-dimensional modeling problems. In particular, further investigations will focus on developing efficient numerical schemes capable of handling the increased computational complexity associated with high-dimensional systems, as well as analyzing their accuracy, stability, and convergence properties. These extensions are expected to broaden the applicability of the proposed approach to more realistic and complex models.