A Legendre Spectral Operational Matrix Method with Convergence Analysis for Two-Dimensional Integro-Differential Equations

Abstract

In this paper, we develop a Legendre spectral operational matrix method for the numerical solution of two-dimensional Volterra–Fredholm integro-differential equations subject to mixed boundary conditions. The proposed approach transforms the physical domain onto a reference square and approximates the unknown solution using a tensor-product Legendre polynomial expansion. Exact operational matrices for differentiation and lower-limit integration are constructed, allowing the original integro-differential problem to be reduced systematically to a finite-dimensional algebraic system for the spectral coefficients. The formulation provides a unified treatment of differential, Volterra, and Fredholm operators within a single spectral framework and avoids complicated discretizations of multidimensional integral terms. For a specialized linear form of the problem, rigorous convergence estimates are established in both $L^2$ and $L^{\infty }$ norms under suitable regularity assumptions on the coefficients and kernels. The analysis shows that the dominant convergence behavior is governed by the differential operator, while the integral terms contribute only higher-order consistency effects. Several benchmark examples involving both linear and nonlinear two-dimensional integro-differential equations are presented to demonstrate the performance of the proposed method. Numerical results exhibit rapid spectral-type error decay as the polynomial degree increases, with the numerical errors approaching machine precision for moderate truncation orders. These results confirm the accuracy, efficiency, and reliability of the proposed Legendre spectral operational matrix framework for solving a broad class of multidimensional integro-differential equations with nonlocal operators.

1. Introduction

Integro-differential equations constitute an important class of mathematical models in which differential operators are coupled with integral terms representing nonlocal interactions. Such equations arise naturally in many areas of science and engineering because they incorporate both local dynamics and accumulated effects over spatial or temporal domains. Applications of integro-differential equations include viscoelasticity, biological systems, transport theory, heat transfer, population dynamics, and environmental processes such as pollution diffusion and contaminant transport [1]. In these models, the evolution of a system at a given point depends not only on local rates of change but also on distributed interactions represented through integral operators, which capture memory effects and long-range spatial coupling. Consequently, integro-differential formulations provide a powerful mathematical framework for describing complex processes involving hereditary behavior, nonlocal interactions, and distributed sources [2,3].

A broad and important class of such models is formed by two-dimensional Volterra–Fredholm integro-differential equations, in which local differential operators are coupled with both Volterra- and Fredholm-type integral terms. The Volterra integral operator typically represents cumulative or causal effects over a partial region of the domain, whereas the Fredholm operator models global nonlocal interactions over the entire spatial region. These equations arise in many applications, including heat transfer, population dynamics, transport processes, and environmental modeling, where spatially distributed interactions must be incorporated into the mathematical description of the system. From a computational perspective, the presence of multidimensional integral operators significantly increases the complexity of the resulting problem, since the discretization of the integral terms often leads to dense algebraic systems and high computational cost. Consequently, the development of efficient and accurate numerical techniques for solving multidimensional integro-differential equations has attracted considerable attention in recent years.

A large number of numerical techniques have been developed for solving integral and integro-differential equations. Classical approaches include collocation methods, Galerkin and projection methods, quadrature-based discretizations, and spectral methods [4,5]. In the one-dimensional setting, several semi-analytical and numerical approaches, such as variational iteration methods, Adomian decomposition methods, and homotopy perturbation techniques, have been proposed for Volterra and Fredholm integro-differential equations [6,7,8]. These methods have been successfully applied to many benchmark problems and engineering applications. However, the numerical treatment of multidimensional integral and integro-differential equations is considerably more challenging due to the presence of multidimensional kernels and the dense algebraic systems generated after discretization.

To address these difficulties, several specialized numerical schemes have been introduced in the literature. For example, Galerkin and collocation methods have been widely used for two-dimensional Fredholm integral equations [9]. Wavelet-based techniques and triangular orthogonal function approaches have also been proposed for solving multidimensional integral equations efficiently [10,11,12]. In addition, meshless methods based on radial basis functions have been developed for solving two-dimensional Fredholm integral equations with high accuracy [13]. More recently, Ma et al. proposed a numerical method based on the integral mean value theorem for solving two-dimensional Fredholm integral equations, where the original problem is reduced to a system of algebraic equations together with convergence analysis [14].

Spectral methods have attracted considerable attention for the numerical solution of integral and integro-differential equations because of their high-order accuracy and excellent convergence properties for smooth problems [15,16]. These methods approximate the unknown solution by global expansions in orthogonal polynomial bases such as Legendre or Chebyshev polynomials and can therefore achieve spectral or near-spectral convergence when the solution possesses sufficient regularity. Legendre polynomial bases are widely used for problems defined on bounded intervals due to their orthogonality properties and favorable numerical stability [17]. In recent years, several researchers have applied spectral and collocation methods based on orthogonal polynomials to Volterra and Fredholm integro-differential equations and demonstrated their high accuracy and rapid convergence [18,19,20,21]. Moreover, recent studies have extended these approaches to fractional integro-differential equations using spectral techniques together with rigorous convergence analysis [22,23].

Parallel to the development of spectral approaches, operational matrix techniques have emerged as an effective computational framework for solving differential, integral, and integro-differential equations. In these methods, differentiation and integration operators are represented through precomputed operational matrices with respect to a chosen basis, thereby reducing the original problem to a system of algebraic equations for the expansion coefficients [24]. Various operational matrix formulations have been developed using block-pulse functions, triangular functions, shifted Legendre polynomials, shifted Jacobi polynomials, and Chebyshev polynomials for solving Volterra, Fredholm, and mixed integro-differential equations [25,26,27,28]. These approaches provide efficient algebraic formulations for multidimensional nonlocal problems and have been successfully applied to a variety of integral and integro-differential models.

Recent years have witnessed notable progress in the numerical analysis of Volterra–Fredholm integro-differential equations and their fractional extensions through the development of high-order approximation techniques. In particular, spectral collocation approaches based on orthogonal polynomial bases have been proposed for fractional Volterra–Fredholm integro-differential problems, where the governing equations are transformed into algebraic systems that can be solved efficiently while preserving high accuracy [29]. Moreover, pseudo-spectral and continuous Galerkin frameworks have recently been developed for Volterra integro-differential equations, providing rigorous convergence analysis and efficient multistep numerical schemes [30]. In addition, stable finite-difference algorithms have been introduced for higher-order neutral Volterra integro-differential equations, where second-order convergence and robust numerical performance are demonstrated [31]. More recently, wavelet-based spectral techniques have been employed for fractional Volterra–Fredholm integro-differential equations, where shifted Gegenbauer wavelets and polynomial-based approximations are used to construct accurate numerical solutions and compare the efficiency of different approximation strategies [32].

Recent studies have further advanced the theoretical analysis and numerical treatment of Volterra–Fredholm integro-differential equations and their fractional extensions. In particular, analytical investigations have established fundamental properties such as existence, uniqueness, and stability of solutions for fractional Volterra–Fredholm integro-differential equations within fuzzy frameworks, providing a rigorous theoretical basis for the study of nonlocal models [33]. In addition, efficient numerical techniques have been proposed for singularly perturbed Fredholm integro-differential equations arising in reaction–diffusion processes, where specially designed discretization methods are used to capture boundary-layer behavior and maintain numerical stability [34]. Furthermore, compact finite-difference schemes based on alternating direction implicit (ADI) techniques have been developed for two-dimensional fractional integro-differential equations involving Riemann–Liouville integral kernels, demonstrating improved accuracy and computational efficiency for multidimensional problems [35]. Meshless numerical procedures have also been successfully employed for solving nonlinear and fractional differential equations, providing accurate and computationally efficient approximations for complex mathematical models [36,37]. These studies demonstrate the effectiveness of meshless frameworks in the numerical treatment of fractional dynamical systems.

Overall, these developments highlight the continuing progress in both the theoretical analysis and numerical approximation of integro-differential equations arising in various applications of science and engineering. They also indicate that high-order spectral, pseudo-spectral, wavelet-based, and operational matrix techniques continue to provide powerful computational tools for solving multidimensional nonlocal problems. Despite the significant progress achieved in the development of numerical methods for integral and integro-differential equations, the efficient numerical treatment of multidimensional Volterra–Fredholm integro-differential equations remains challenging. In particular, the construction of highly accurate numerical schemes capable of effectively handling coupled differential and multidimensional integral operators while maintaining computational efficiency continues to be an active area of research. Many numerical methods have been developed for integro-differential equations; however, several difficulties persist for multidimensional Volterra–Fredholm problems. In particular, classical finite-difference, collocation, and quadrature-based methods often produce dense algebraic systems and require substantial computational effort when multidimensional integral operators are involved. Moreover, many existing spectral and operational matrix methods mainly focus on one-dimensional models or pure integral equations, while rigorous convergence analysis for coupled two-dimensional integro-differential systems remains relatively limited. The present approach is based on a unified tensor-product Legendre operational matrix formulation capable of efficiently treating differential, Volterra, and Fredholm operators within a single high-order spectral framework.

Motivated by these challenges, the present work develops a Legendre spectral operational matrix method for solving two-dimensional Volterra–Fredholm integro-differential equations. In the proposed approach, the unknown solution is approximated by a finite tensor-product expansion of Legendre polynomials. By employing operational matrices for the differentiation and integration associated with the Legendre basis, the original integro-differential equation is transformed into a finite-dimensional algebraic system for the unknown expansion coefficients. This formulation avoids complicated discretizations of multidimensional integral operators and provides an efficient computational framework for solving the problem. The main novelty of the proposed work lies in the development of a unified tensor-product Legendre spectral framework that simultaneously treats differential, Volterra, and Fredholm operators within a single operational matrix formulation. Unlike many existing approaches that mainly focus on one-dimensional or purely integral models, the present method is designed for two-dimensional integro-differential equations subject to mixed boundary conditions. In addition, rigorous convergence analysis is established in both $L^2$ and $L^{\infty }$ norms, while the numerical experiments demonstrate spectral-type accuracy, with errors approaching machine precision for moderate polynomial orders. Several numerical examples are presented to illustrate the accuracy and efficiency of the proposed method. The obtained results confirm that the Legendre spectral operational matrix approach provides highly accurate approximations and represents an effective numerical tool for solving two-dimensional integro-differential equations with nonlocal operators.

The remainder of this paper is organized as follows. Section 2 presents the Legendre spectral operational matrix formulation for the considered two-dimensional integro-differential equations. Section 3 is devoted to the convergence analysis of the proposed method. In Section 4, several numerical examples are provided to demonstrate the accuracy and efficiency of the method. Finally, conclusions are given in Section 5.

2. Legendre Spectral Operational Matrix Method

In this section, we develop a Legendre spectral operational matrix method for the following two-dimensional Volterra–Fredholm integro-differential equation:

$$\begin{array}{cc}\hfill \mathcal{M}u(x,y) & +{\lambda }_V{\int }_a^x{\int }_a^y{\mathcal{K}}_V(x,y,\sigma ,\tau ) u(\sigma ,\tau ) d\tau d\sigma \hfill \\ \hfill & +{\lambda }_F{\int }_a^c{\int }_a^c{\mathcal{K}}_F(x,y,\sigma ,\tau ) u(\sigma ,\tau ) d\tau d\sigma \hfill \\ \hfill & =s(x,y),\hspace{1em}(x,y)\in \mathsf{\Omega },\hfill \end{array}$$

(1)

where the spatial domain is defined by

$$\mathsf{\Omega }=[a,c]\times [a,c]\subset {\mathbb{R}}^2,$$

and the differential operator $\mathcal{M}$ is given by

$$\mathcal{M}u(x,y)={\displaystyle \sum _{\mu =0}^r}{\displaystyle \sum _{\nu =0}^q}{\vartheta }_{\mu \nu }(x,y){\displaystyle \frac{{\partial }^{\mu +\nu }u(x,y)}{\partial x^{\mu }\partial y^{\nu }}}.$$

(2)

The problem is supplemented with the mixed boundary conditions

$$u(a,y)={\zeta }_1\left(y\right),\hspace{1em}{u}_x(c,y)={\zeta }_2\left(y\right),\hspace{1em}u(x,a)={\zeta }_3\left(x\right),\hspace{1em}{u}_y(x,c)={\zeta }_4\left(x\right).$$

(3)

The objective is to derive a fully algebraic numerical scheme by approximating the unknown solution through a tensor-product Legendre polynomial expansion together with operational matrices for differentiation and integration.

To map the physical domain onto the reference square, we introduce the affine transformation from $[a,c]$ to $[-1,1]$:

$$x=\mathsf{\Xi }\left(\xi \right):={\displaystyle \frac{c-a}{2}}\xi +{\displaystyle \frac{a+c}{2}},\hspace{1em}\hspace{1em}y=\mathsf{\Xi }\left(\eta \right):={\displaystyle \frac{c-a}{2}}\eta +{\displaystyle \frac{a+c}{2}},$$

(4)

and similarly,

$$\sigma =\mathsf{\Xi }\left(\rho \right),\hspace{1em}\hspace{1em}\tau =\mathsf{\Xi }\left(\theta \right).$$

Define

$$U(\xi ,\eta ):=u\left(\mathsf{\Xi }\right(\xi ),\mathsf{\Xi }(\eta \left)\right).$$

Then, by the chain rule,

$${\displaystyle \frac{{\partial }^{\mu +\nu }u(x,y)}{\partial x^{\mu }\partial y^{\nu }}}={\left({\displaystyle \frac{2}{c-a}}\right)}^{\mu +\nu }{\displaystyle \frac{{\partial }^{\mu +\nu }U(\xi ,\eta )}{\partial {\xi }^{\mu }\partial {\eta }^{\nu }}}.$$

(5)

Therefore, problem (1) is transformed onto the reference square

$$\widehat{\mathsf{\Omega }}={[-1,1]}^2$$

as

$$\begin{array}{cc}\hfill \widehat{\mathcal{M}}U(\xi ,\eta ) & +{\lambda }_V{\left({\displaystyle \frac{c-a}{2}}\right)}^2{\int }_{-1}^{\xi }{\int }_{-1}^{\eta }{\widehat{\mathcal{K}}}_V(\xi ,\eta ,\rho ,\theta )U(\rho ,\theta ) d\theta d\rho \hfill \\ \hfill & +{\lambda }_F{\left({\displaystyle \frac{c-a}{2}}\right)}^2{\int }_{-1}^1{\int }_{-1}^1{\widehat{\mathcal{K}}}_F(\xi ,\eta ,\rho ,\theta )U(\rho ,\theta ) d\theta d\rho \hfill \\ \hfill & =\widehat{s}(\xi ,\eta ),\hspace{1em}(\xi ,\eta )\in \widehat{\mathsf{\Omega }}.\hfill \end{array}$$

(6)

Here,

$$\widehat{\mathcal{M}}U(\xi ,\eta )={\displaystyle \sum _{\mu =0}^r}{\displaystyle \sum _{\nu =0}^q}{\widehat{\vartheta }}_{\mu \nu }(\xi ,\eta ){\left({\displaystyle \frac{2}{c-a}}\right)}^{\mu +\nu }{\displaystyle \frac{{\partial }^{\mu +\nu }U(\xi ,\eta )}{\partial {\xi }^{\mu }\partial {\eta }^{\nu }}},$$

(7)

where

$${\widehat{\vartheta }}_{\mu \nu }(\xi ,\eta )={\vartheta }_{\mu \nu }(\mathsf{\Xi }\left(\xi \right),\mathsf{\Xi }\left(\eta \right)),\hspace{1em}\hspace{1em}\widehat{s}(\xi ,\eta )=s(\mathsf{\Xi }\left(\xi \right),\mathsf{\Xi }\left(\eta \right)),$$

and

$$\begin{array}{c}{\widehat{\mathcal{K}}}_V(\xi ,\eta ,\rho ,\theta )={\mathcal{K}}_V\left(\mathsf{\Xi }\left(\xi \right),\mathsf{\Xi }\left(\eta \right),\mathsf{\Xi }\left(\rho \right),\mathsf{\Xi }\left(\theta \right)\right),\\ {\widehat{\mathcal{K}}}_F(\xi ,\eta ,\rho ,\theta )={\mathcal{K}}_F\left(\mathsf{\Xi }\left(\xi \right),\mathsf{\Xi }\left(\eta \right),\mathsf{\Xi }\left(\rho \right),\mathsf{\Xi }\left(\theta \right)\right).\end{array}$$

We now construct a spectral approximation of the transformed solution on $\widehat{\mathsf{\Omega }}$ using a tensor-product Legendre polynomial basis together with operational matrices for differentiation and integration.

2.1. Legendre Basis and Tensor Expansion

Let ${\left\{L_j\left(t\right)\right\}}_{j\ge 0}$ denote the classical Legendre polynomials on the interval $[-1,1]$, defined by Rodrigues’ formula

$$L_j\left(t\right)={\displaystyle \frac{1}{2^{j}j!}}{\displaystyle \frac{d^j}{d{t}^j}}\left[{(t^2-1)}^j\right].$$

(8)

These polynomials satisfy the orthogonality relation

$${\int }_{-1}^1{L}_i\left(t\right)L_j\left(t\right) dt={\displaystyle \frac{2}{2j+1}}{\delta }_{ij}.$$

(9)

Define the one-dimensional Legendre basis vector by

$$\mathcal{L}\left(t\right):=\left[\begin{array}{c}{L}_0\left(t\right)\\ L_1\left(t\right)\\ ⋮\\ L_N\left(t\right)\end{array}\right].$$

(10)

The transformed solution is approximated by the tensor-product expansion

$$U_N(\xi ,\eta )={\displaystyle \sum _{i=0}^N}{\displaystyle \sum _{j=0}^N}{z}_{ij}{L}_i\left(\xi \right)L_j\left(\eta \right)=\mathcal{L}{\left(\xi \right)}^T\mathbf{Z}\mathcal{L}\left(\eta \right),$$

(11)

where

$$\mathbf{Z}={\left[z_{ij}\right]}_{i,j=0}^N\in {\mathbb{R}}^{(N+1)\times (N+1)}$$

(12)

is the unknown coefficient matrix.

2.2. Derivative Operational Matrix

We now derive the exact derivative representation in the truncated Legendre basis.

Lemma 1.

For each integer $n\ge 1$, the derivative of the Legendre polynomial $L_n\left(t\right)$ is given by

$${\displaystyle \frac{d}{dt}}{L}_n\left(t\right)=\sum _{\begin{array}{c}0\le k\le n-1\\ n-k \mathrm{odd}\end{array}}(2k+1)L_k\left(t\right).$$

(13)

Proof. 

A standard identity for Legendre polynomials is

$$(2n+1)L_n\left(t\right)={\displaystyle \frac{d}{dt}}\left[L_{n+1}\left(t\right)-L_{n-1}\left(t\right)\right],\hspace{1em}\hspace{1em}n\ge 1.$$

(14)

Using (14) recursively gives

$$L_n^{\prime }\left(t\right)=(2n-1)L_{n-1}\left(t\right)+(2n-5)L_{n-3}\left(t\right)+\cdots ,$$

where the expansion terminates at $L_0$ or $L_1$, depending on the parity of n. Therefore, (13) follows immediately. □

Consequently, there exists a derivative operational matrix

$$\mathbf{G}\in {\mathbb{R}}^{(N+1)\times (N+1)}$$

such that

$${\displaystyle \frac{d}{dt}}\mathcal{L}\left(t\right)=\mathbf{G}\mathcal{L}\left(t\right),$$

(15)

where the entries of $\mathbf{G}$ are given by

$${\left(\mathbf{G}\right)}_{k,n}=\left\{\begin{array}{cc}2k+1, \hfill & 0\le k\le n-1,\hspace{1em}n-k \mathrm{odd},\hfill \\ 0, \hfill & \mathrm{otherwise},\hfill \end{array}\right.\hspace{1em}0\le k,n\le N.$$

(16)

Proposition 1.

The partial derivatives of $U_N$ satisfy

$${\displaystyle \frac{\partial U_N}{\partial \xi }}=\mathcal{L}{\left(\xi \right)}^T{\mathbf{G}}^T\mathbf{Z}\mathcal{L}\left(\eta \right),\hspace{1em}{\displaystyle \frac{\partial U_N}{\partial \eta }}=\mathcal{L}{\left(\xi \right)}^T\mathbf{Z}\mathbf{G}\mathcal{L}\left(\eta \right).$$

(17)

More generally, for nonnegative integers μ and ν,

$${\displaystyle \frac{{\partial }^{\mu +\nu }{U}_N}{\partial {\xi }^{\mu }\partial {\eta }^{\nu }}}=\mathcal{L}{\left(\xi \right)}^T{\left({\mathbf{G}}^{\mu }\right)}^T\mathbf{Z}{\mathbf{G}}^{\nu }\mathcal{L}\left(\eta \right).$$

(18)

Proof. 

From (11),

$$U_N(\xi ,\eta )=\mathcal{L}{\left(\xi \right)}^T\mathbf{Z}\mathcal{L}\left(\eta \right).$$

Differentiating with respect to $\xi$ and using (15), we obtain

$${\displaystyle \frac{\partial U_N}{\partial \xi }}={\left({\displaystyle \frac{d}{d\xi }}\mathcal{L}\left(\xi \right)\right)}^T\mathbf{Z}\mathcal{L}\left(\eta \right)={\left(\mathbf{G}\mathcal{L}\left(\xi \right)\right)}^T\mathbf{Z}\mathcal{L}\left(\eta \right).$$

Hence,

$${\displaystyle \frac{\partial U_N}{\partial \xi }}=\mathcal{L}{\left(\xi \right)}^T{\mathbf{G}}^T\mathbf{Z}\mathcal{L}\left(\eta \right).$$

The formula for $\partial U_N/\partial \eta$ follows analogously. Repeated differentiation with respect to $\xi$ and $\eta$ yields (18). □

2.3. Operational Matrix for the Volterra Integral

We next derive the operational integration matrix corresponding to the lower-limit integral operator

$${\int }_{-1}^{\xi }(·) d\rho .$$

Lemma 2.

For each integer $n\ge 1$,

$${\int }_{-1}^{\xi }{L}_n\left(\rho \right) d\rho ={\displaystyle \frac{L_{n+1}\left(\xi \right)-L_{n-1}\left(\xi \right)}{2n+1}},\hspace{1em}n\ge 1,$$

(19)

and

$${\int }_{-1}^{\xi }{L}_0\left(\rho \right) d\rho =1+\xi =L_0\left(\xi \right)+L_1\left(\xi \right).$$

(20)

Proof. 

Using the identity

$$(2n+1)L_n\left(t\right)={\displaystyle \frac{d}{dt}}\left[L_{n+1}\left(t\right)-L_{n-1}\left(t\right)\right],\hspace{1em}n\ge 1,$$

(21)

and integrating both sides from $-1$ to $\xi$, we obtain

$${\int }_{-1}^{\xi }{L}_n\left(\rho \right) d\rho ={\displaystyle \frac{L_{n+1}\left(\xi \right)-L_{n-1}\left(\xi \right)}{2n+1}}.$$

This proves (19). The Formula (20) follows directly from

$$L_0\left(\rho \right)=1.$$

□

Consequently, there exists an operational integration matrix

$$\mathbf{J}\in {\mathbb{R}}^{(N+1)\times (N+1)}$$

such that

$${\int }_{-1}^{\xi }\mathcal{L}\left(\rho \right) d\rho =\mathbf{J}\mathcal{L}\left(\xi \right).$$

(22)

The columns of $\mathbf{J}$ are determined from (19) and (20). In particular,

$$\mathbf{J}{e}_0=e_0+e_1,$$

(23)

and for $n\ge 1$,

$$\mathbf{J}{e}_n={\displaystyle \frac{1}{2n+1}}{e}_{n+1}-{\displaystyle \frac{1}{2n+1}}{e}_{n-1},$$

(24)

where $e_n$ denotes the canonical basis vector. Terms outside the index set $\{0,1,\dots ,N\}$ are omitted in the truncated representation.

Proposition 2.

Let

$$V_N(\xi ,\eta )=\mathcal{L}{\left(\xi \right)}^T\mathbf{M}\mathcal{L}\left(\eta \right).$$

Then,

$${\int }_{-1}^{\xi }{\int }_{-1}^{\eta }{V}_N(\rho ,\theta ) d\theta d\rho =\mathcal{L}{\left(\xi \right)}^T{\mathbf{J}}^T\mathbf{M}\mathbf{J}\mathcal{L}\left(\eta \right).$$

(25)

Proof. 

Using Fubini’s theorem together with (22), we obtain

$${\int }_{-1}^{\xi }{\int }_{-1}^{\eta }\mathcal{L}{\left(\rho \right)}^T\mathbf{M}\mathcal{L}\left(\theta \right) d\theta d\rho$$

$$={\left({\int }_{-1}^{\xi }\mathcal{L}\left(\rho \right) d\rho \right)}^T\mathbf{M}\left({\int }_{-1}^{\eta }\mathcal{L}\left(\theta \right) d\theta \right).$$

Applying (22) again yields

$$={\left(\mathbf{J}\mathcal{L}\left(\xi \right)\right)}^T\mathbf{M}\left(\mathbf{J}\mathcal{L}\left(\eta \right)\right),$$

which gives

$$=\mathcal{L}{\left(\xi \right)}^T{\mathbf{J}}^T\mathbf{M}\mathbf{J}\mathcal{L}\left(\eta \right).$$

Hence, (25) follows. □

2.4. Kernel Approximation

To construct the operational matrix formulation, the kernels are approximated by tensor-product Legendre expansions over the four-dimensional domain ${[-1,1]}^4$. For the Volterra kernel, we write

$${\widehat{\mathcal{K}}}_V(\xi ,\eta ,\rho ,\theta )\approx {\displaystyle \sum _{i,j,p,q=0}^N}{\kappa }_{ijpq}^{\left(V\right)}{L}_i\left(\xi \right)L_j\left(\eta \right)L_p\left(\rho \right)L_q\left(\theta \right).$$

(26)

Similarly, the Fredholm kernel is approximated by

$${\widehat{\mathcal{K}}}_F(\xi ,\eta ,\rho ,\theta )\approx {\displaystyle \sum _{i,j,p,q=0}^N}{\kappa }_{ijpq}^{\left(F\right)}{L}_i\left(\xi \right)L_j\left(\eta \right)L_p\left(\rho \right)L_q\left(\theta \right).$$

(27)

For sufficiently smooth kernels, the tensor-product Legendre expansions in (26) and (27) converge rapidly in both $L^2$ and $L^{\infty }$ norms. In particular, the truncation error decreases spectrally with respect to the polynomial order N. Consequently, the kernel approximation contributes only higher-order consistency errors to the overall numerical scheme. The coefficients ${\kappa }_{ijpq}^{\left(V\right)} \mathrm{and} {\kappa }_{ijpq}^{\left(F\right)}$ may be computed using Legendre projection or high-order Gauss-Legendre quadrature over the four-dimensional domain. Using the approximate solution $U_N$, we obtain

$$\begin{array}{cc}\hfill & {\widehat{\mathcal{K}}}_V(\xi ,\eta ,\rho ,\theta )U_N(\rho ,\theta )\hfill \\ \hfill & \approx {\displaystyle \sum _{i,j,p,q=0}^N}{\displaystyle \sum _{\alpha ,\beta =0}^N}{\kappa }_{ijpq}^{\left(V\right)}{z}_{\alpha \beta }{L}_i\left(\xi \right)L_j\left(\eta \right)L_p\left(\rho \right)L_q\left(\theta \right)L_{\alpha }\left(\rho \right)L_{\beta }\left(\theta \right).\hfill \end{array}$$

(28)

To preserve the algebraic structure explicitly, we use the product expansion

$$L_p\left(t\right)L_{\alpha }\left(t\right)={\displaystyle \sum _{m=0}^N}{\pi }_m^{(p,\alpha )}{L}_m\left(t\right).$$

(29)

Hence, the Volterra integrand can be rewritten in tensor-product basis form. By (25), its integral representation can be expressed through a matrix operator acting on $\mathbf{Z}$. Consequently, there exists a matrix operator ${\mathbf{Q}}_V\left(\mathbf{Z}\right)$ such that

$${\int }_{-1}^{\xi }{\int }_{-1}^{\eta }{\widehat{\mathcal{K}}}_V(\xi ,\eta ,\rho ,\theta )U_N(\rho ,\theta ) d\theta d\rho \approx \mathcal{L}{\left(\xi \right)}^T{\mathbf{Q}}_V\left(\mathbf{Z}\right)\mathcal{L}\left(\eta \right).$$

(30)

Similarly, there exists a matrix operator ${\mathbf{Q}}_F\left(\mathbf{Z}\right)$ such that

$${\int }_{-1}^1{\int }_{-1}^1{\widehat{\mathcal{K}}}_F(\xi ,\eta ,\rho ,\theta )U_N(\rho ,\theta ) d\theta d\rho \approx \mathcal{L}{\left(\xi \right)}^T{\mathbf{Q}}_F\left(\mathbf{Z}\right)\mathcal{L}\left(\eta \right).$$

(31)

Remark 1.

If the kernels are linear in u, then both ${\mathbf{Q}}_V$ and ${\mathbf{Q}}_F$ are linear operators in $\mathbf{Z}$, and may be written as

$${\mathbf{Q}}_V\left(\mathbf{Z}\right)={\displaystyle \sum _{\alpha ,\beta =0}^N}{z}_{\alpha \beta }{\mathbf{V}}^{(\alpha ,\beta )},$$

$${\mathbf{Q}}_F\left(\mathbf{Z}\right)={\displaystyle \sum _{\alpha ,\beta =0}^N}{z}_{\alpha \beta }{\mathbf{F}}^{(\alpha ,\beta )}.$$

For nonlinear kernels, the same procedure leads to nonlinear algebraic terms.

2.5. Spectral Representation of the Differential Operator

Using (18), the differential operator (7) applied to the approximate solution $U_N$ can be written as

$$\widehat{\mathcal{M}}{U}_N(\xi ,\eta )={\displaystyle \sum _{\mu =0}^r}{\displaystyle \sum _{\nu =0}^q}{\widehat{\vartheta }}_{\mu \nu }(\xi ,\eta ){\left({\displaystyle \frac{2}{c-a}}\right)}^{\mu +\nu }\mathcal{L}{\left(\xi \right)}^T{\left({\mathbf{G}}^{\mu }\right)}^T\mathbf{Z}{\mathbf{G}}^{\nu }\mathcal{L}\left(\eta \right).$$

(32)

To obtain a closed spectral representation, the coefficient functions are approximated using the same tensor-product Legendre basis:

$${\widehat{\vartheta }}_{\mu \nu }(\xi ,\eta )\approx \mathcal{L}{\left(\xi \right)}^T{\mathbf{C}}^{(\mu ,\nu )}\mathcal{L}\left(\eta \right),\hspace{1em}\widehat{s}(\xi ,\eta )\approx \mathcal{L}{\left(\xi \right)}^T\mathbf{H}\mathcal{L}\left(\eta \right).$$

(33)

Using the product re-expansion of Legendre polynomials described previously, each term in (32) can be expressed in tensor-product matrix form. Consequently, each contribution generates a matrix operator ${\mathbf{D}}^{(\mu ,\nu )}\left(\mathbf{Z}\right)$. Summing over all differential terms yields

$$\widehat{\mathcal{M}}{U}_N(\xi ,\eta )\approx \mathcal{L}{\left(\xi \right)}^T{\mathbf{D}}_{\mathrm{tot}}\left(\mathbf{Z}\right)\mathcal{L}\left(\eta \right),$$

(34)

where

$${\mathbf{D}}_{\mathrm{tot}}\left(\mathbf{Z}\right):={\displaystyle \sum _{\mu =0}^r}{\displaystyle \sum _{\nu =0}^q}{\left({\displaystyle \frac{2}{c-a}}\right)}^{\mu +\nu }{\mathbf{D}}^{(\mu ,\nu )}\left(\mathbf{Z}\right).$$

(35)

Substituting (30), (31), (33), and (34) into (6) gives

$$\mathcal{L}{\left(\xi \right)}^T\left[{\mathbf{D}}_{\mathrm{tot}}\left(\mathbf{Z}\right)+{\lambda }_V{\left({\displaystyle \frac{c-a}{2}}\right)}^2{\mathbf{Q}}_V\left(\mathbf{Z}\right)+{\lambda }_F{\left({\displaystyle \frac{c-a}{2}}\right)}^2{\mathbf{Q}}_F\left(\mathbf{Z}\right)-\mathbf{H}\right]\mathcal{L}\left(\eta \right)=0.$$

(36)

Since the Legendre basis functions are linearly independent, the coefficient matrix must vanish. Therefore,

$${\mathbf{D}}_{\mathrm{tot}}\left(\mathbf{Z}\right)+{\lambda }_V{\left({\displaystyle \frac{c-a}{2}}\right)}^2{\mathbf{Q}}_V\left(\mathbf{Z}\right)+{\lambda }_F{\left({\displaystyle \frac{c-a}{2}}\right)}^2{\mathbf{Q}}_F\left(\mathbf{Z}\right)=\mathbf{H}.$$

(37)

Equation (37) represents a system of ${(N+1)}^2$ algebraic equations for the unknown coefficient matrix $\mathbf{Z}$.

The boundary conditions (3) are transformed onto the reference square as

$$\begin{array}{cccc}\hfill U(-1,\eta )& ={\widehat{\zeta }}_1\left(\eta \right), \hfill & \hfill U_{\xi }(1,\eta )& ={\displaystyle \frac{c-a}{2}}{\widehat{\zeta }}_2\left(\eta \right),\hfill \\ \hfill U(\xi ,-1)& ={\widehat{\zeta }}_3\left(\xi \right), \hfill & \hfill U_{\eta }(\xi ,1)& ={\displaystyle \frac{c-a}{2}}{\widehat{\zeta }}_4\left(\xi \right),\hfill \end{array}$$

(38)

where

$${\widehat{\zeta }}_1\left(\eta \right)={\zeta }_1\left(\mathsf{\Xi }\left(\eta \right)\right),\hspace{1em}{\widehat{\zeta }}_2\left(\eta \right)={\zeta }_2\left(\mathsf{\Xi }\left(\eta \right)\right),\hspace{1em}{\widehat{\zeta }}_3\left(\xi \right)={\zeta }_3\left(\mathsf{\Xi }\left(\xi \right)\right),\hspace{1em}{\widehat{\zeta }}_4\left(\xi \right)={\zeta }_4\left(\mathsf{\Xi }\left(\xi \right)\right).$$

Using (11) and (17), we obtain

$$U_N(-1,\eta )=\mathcal{L}{(-1)}^T\mathbf{Z}\mathcal{L}\left(\eta \right),$$

(39)

$${\displaystyle \frac{\partial U_N}{\partial \xi }}(1,\eta )=\mathcal{L}{\left(1\right)}^T{\mathbf{G}}^T\mathbf{Z}\mathcal{L}\left(\eta \right),$$

(40)

$$U_N(\xi ,-1)=\mathcal{L}{\left(\xi \right)}^T\mathbf{Z}\mathcal{L}(-1),$$

(41)

$${\displaystyle \frac{\partial U_N}{\partial \eta }}(\xi ,1)=\mathcal{L}{\left(\xi \right)}^T\mathbf{Z}\mathbf{G}\mathcal{L}\left(1\right).$$

(42)

The boundary data are expanded in the one-dimensional Legendre basis as

$${\widehat{\zeta }}_1\left(\eta \right)\approx {\mathbf{b}}_1^T\mathcal{L}\left(\eta \right),\hspace{1em}{\widehat{\zeta }}_2\left(\eta \right)\approx {\mathbf{b}}_2^T\mathcal{L}\left(\eta \right),$$

(43)

$${\widehat{\zeta }}_3\left(\xi \right)\approx {\mathbf{b}}_3^T\mathcal{L}\left(\xi \right),\hspace{1em}{\widehat{\zeta }}_4\left(\xi \right)\approx {\mathbf{b}}_4^T\mathcal{L}\left(\xi \right).$$

(44)

Thus, the boundary equations become

$$\mathcal{L}{(-1)}^T\mathbf{Z}={\mathbf{b}}_1^T,$$

(45)

$$\mathcal{L}{\left(1\right)}^T{\mathbf{G}}^T\mathbf{Z}={\displaystyle \frac{c-a}{2}}{\mathbf{b}}_2^T,$$

(46)

$$\mathbf{Z}\mathcal{L}(-1)={\mathbf{b}}_3,$$

(47)

$$\mathbf{Z}\mathbf{G}\mathcal{L}\left(1\right)={\displaystyle \frac{c-a}{2}}{\mathbf{b}}_4.$$

(48)

For numerical implementation, it is convenient to vectorize the matrix Equation (37). Let

$$\mathbf{z}:=vec\left(\mathbf{Z}\right)\in {\mathbb{R}}^{{(N+1)}^2}.$$

If the operators ${\mathbf{D}}_{\mathrm{tot}}$, ${\mathbf{Q}}_V$, and ${\mathbf{Q}}_F$ are linear in $\mathbf{Z}$, then there exist matrices

$$\mathbb{D},\hspace{1em}\mathbb{V},\hspace{1em}\mathbb{F}\in {\mathbb{R}}^{{(N+1)}^2\times {(N+1)}^2}$$

such that

$$vec\left({\mathbf{D}}_{\mathrm{tot}}\left(\mathbf{Z}\right)\right)=\mathbb{D}\mathbf{z},\hspace{1em}vec\left({\mathbf{Q}}_V\left(\mathbf{Z}\right)\right)=\mathbb{V}\mathbf{z},\hspace{1em}vec\left({\mathbf{Q}}_F\left(\mathbf{Z}\right)\right)=\mathbb{F}\mathbf{z}.$$

(49)

The interior system then becomes

$$\left[\mathbb{D}+{\lambda }_V{\left({\displaystyle \frac{c-a}{2}}\right)}^2\mathbb{V}+{\lambda }_F{\left({\displaystyle \frac{c-a}{2}}\right)}^2\mathbb{F}\right]\mathbf{z}=\mathbf{h},\hspace{1em}\mathbf{h}:=vec\left(\mathbf{H}\right).$$

(50)

The boundary Equations (45)–(48) are incorporated into (50) using standard techniques.

2.6. Specialized Linear Form for Implementation

For practical implementation and for the subsequent convergence analysis, we now specialize the general two-dimensional Volterra–Fredholm integro-differential model to the linear case. The restriction to the linear form is introduced mainly to facilitate the rigorous convergence analysis of the proposed method. Nevertheless, the operational matrix framework remains applicable to nonlinear problems, where the resulting algebraic system can be solved iteratively, although a complete nonlinear convergence analysis is beyond the scope of the present study. Let $\mathsf{\Omega }=[a,c]\times [a,c]$. We consider

$$Lu(x,y)+K_{1}u(x,y)+K_{2}u(x,y)=g(x,y),\hspace{1em}(x,y)\in \mathsf{\Omega },$$

(51)

where

$$\begin{array}{cc}\hfill Lu(x,y) & =a_1(x,y){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+a_2(x,y){\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}+a_3(x,y){\displaystyle \frac{{\partial }^{2}u}{\partial x\partial y}}\hfill \\ \hfill & +b_1(x,y){\displaystyle \frac{\partial u}{\partial x}}+b_2(x,y){\displaystyle \frac{\partial u}{\partial y}}+c_1(x,y)u(x,y),\hfill \end{array}$$

(52)

$$K_{1}u(x,y)={\lambda }_V{\int }_a^x{\int }_a^y{\tilde{k}}_1(x,y;s,t) u(s,t) dt ds,$$

(53)

$$K_{2}u(x,y)={\lambda }_F{\int }_a^c{\int }_a^c{\tilde{k}}_2(x,y;s,t) u(s,t) dt ds.$$

(54)

The mixed boundary conditions are

$$u(a,y)=f_1\left(y\right),\hspace{1em}{u}_x(c,y)=f_1^{\ast }\left(y\right),\hspace{1em}u(x,a)=f_2\left(x\right),\hspace{1em}{u}_y(x,c)=f_2^{\ast }\left(x\right).$$

(55)

Under the affine transformation

$$\begin{array}{c}x=\mathsf{\Xi }\left(\xi \right):={\displaystyle \frac{c-a}{2}}\xi +{\displaystyle \frac{a+c}{2}},\hspace{1em}y=\mathsf{\Xi }\left(\eta \right):={\displaystyle \frac{c-a}{2}}\eta +{\displaystyle \frac{a+c}{2}},\hfill \\ s=\mathsf{\Xi }\left(\rho \right),\hspace{1em}t=\mathsf{\Xi }\left(\theta \right),\hfill \end{array}$$

(56)

and with

$$U(\xi ,\eta ):=u(\mathsf{\Xi }\left(\xi \right),\mathsf{\Xi }\left(\eta \right)),\hspace{1em}(\xi ,\eta )\in \widehat{\mathsf{\Omega }}:={[-1,1]}^2,$$

(57)

problem (51) becomes

$$\mathcal{A}U=F\hspace{1em}\mathrm{in} \widehat{\mathsf{\Omega }},$$

(58)

where

$$\begin{array}{cc}\hfill \mathcal{A}U=& {\alpha }_1(\xi ,\eta )U_{\xi \xi }+{\alpha }_2(\xi ,\eta )U_{\eta \eta }+{\alpha }_3(\xi ,\eta )U_{\xi \eta }+{\beta }_1(\xi ,\eta )U_{\xi }+{\beta }_2(\xi ,\eta )U_{\eta }+\gamma (\xi ,\eta )U\hfill \\ \hfill & \hspace{1em} +{\lambda }_V\left(\mathcal{V}U\right)(\xi ,\eta )+{\lambda }_F\left(\mathcal{F}U\right)(\xi ,\eta ),\hfill \end{array}$$

(59)

with

$${\alpha }_j(\xi ,\eta )={\left({\displaystyle \frac{2}{c-a}}\right)}^2{a}_j(\mathsf{\Xi }\left(\xi \right),\mathsf{\Xi }\left(\eta \right)),\hspace{1em}j=1,2,3,$$

(60)

$${\beta }_{\mathit{\ell }}(\xi ,\eta )={\displaystyle \frac{2}{c-a}}{b}_{\ell }(\mathsf{\Xi }\left(\xi \right),\mathsf{\Xi }\left(\eta \right)),\hspace{1em}\ell =1,2,\hspace{1em}\gamma (\xi ,\eta )=c_1(\mathsf{\Xi }\left(\xi \right),\mathsf{\Xi }\left(\eta \right)),$$

(61)

$$F(\xi ,\eta )=g\left(\mathsf{\Xi }\right(\xi ),\mathsf{\Xi }(\eta \left)\right),$$

(62)

$$\left(\mathcal{V}U\right)(\xi ,\eta )={\left({\displaystyle \frac{c-a}{2}}\right)}^2{\int }_{-1}^{\xi }{\int }_{-1}^{\eta }{\widehat{k}}_1(\xi ,\eta ;\rho ,\theta )U(\rho ,\theta ) d\theta d\rho ,$$

(63)

$$\left(\mathcal{F}U\right)(\xi ,\eta )={\left({\displaystyle \frac{c-a}{2}}\right)}^2{\int }_{-1}^1{\int }_{-1}^1{\widehat{k}}_2(\xi ,\eta ;\rho ,\theta )U(\rho ,\theta ) d\theta d\rho ,$$

(64)

where

$${\widehat{k}}_i(\xi ,\eta ;\rho ,\theta )={\tilde{k}}_i(\mathsf{\Xi }\left(\xi \right),\mathsf{\Xi }\left(\eta \right);\mathsf{\Xi }\left(\rho \right),\mathsf{\Xi }\left(\theta \right)),\hspace{1em}i=1,2.$$

(65)

The transformed boundary conditions are

$$\begin{array}{cc}U(-1,\eta ) =\widehat{f_1}\left(\eta \right), \hfill & \hfill U_{\xi }(1,\eta ) ={\displaystyle \frac{c-a}{2}}\widehat{f_1^{\ast }}\left(\eta \right),\\ U(\xi ,-1) =\widehat{f_2}\left(\xi \right), \hfill & \hfill U_{\eta }(\xi ,1) ={\displaystyle \frac{c-a}{2}}\widehat{f_2^{\ast }}\left(\xi \right),\end{array}$$

(66)

where

$$\begin{array}{cc}{\widehat{f}}_1\left(\eta \right)=f_1\left(\mathsf{\Xi }\left(\eta \right)\right), & \widehat{f_1^{\ast }}\left(\eta \right)=f_1^{\ast }\left(\mathsf{\Xi }\left(\eta \right)\right),\\ {\widehat{f}}_2\left(\xi \right)=f_2\left(\mathsf{\Xi }\left(\xi \right)\right), & \widehat{f_2^{\ast }}\left(\xi \right)=f_2^{\ast }\left(\mathsf{\Xi }\left(\xi \right)\right).\end{array}$$

Using the tensor-product Legendre approximation

$$U_N(\xi ,\eta )=\mathcal{L}{\left(\xi \right)}^T\mathbf{Z}\mathcal{L}\left(\eta \right),$$

together with the derivative and integration matrices $\mathbf{G}$ and $\mathbf{J}$, the differential part of the specialized operator may be represented in matrix form by

$${\mathbf{D}}_{\mathrm{sp}}\left(\mathbf{Z}\right):={\mathbf{A}}_1{\left({\mathbf{G}}^2\right)}^T\mathbf{Z}+{\mathbf{A}}_2\mathbf{Z}{\mathbf{G}}^2+{\mathbf{A}}_3{\mathbf{G}}^T\mathbf{Z}\mathbf{G}+{\mathbf{B}}_1{\mathbf{G}}^T\mathbf{Z}+{\mathbf{B}}_2\mathbf{Z}\mathbf{G}+\mathbf{C}\mathbf{Z},$$

(67)

where ${\mathbf{A}}_j$, ${\mathbf{B}}_{\ell }$, and $\mathbf{C}$ denote the Legendre coefficient matrices associated with ${\alpha }_j$, ${\beta }_{\ell }$, and $\gamma$, respectively.

Likewise, the Volterra and Fredholm terms admit the matrix representations

$$\left(\mathcal{V}{U}_N\right)(\xi ,\eta )=\mathcal{L}{\left(\xi \right)}^T{\mathbf{Q}}_V\left(\mathbf{Z}\right)\mathcal{L}\left(\eta \right),\hspace{1em}\left(\mathcal{F}{U}_N\right)(\xi ,\eta )=\mathcal{L}{\left(\xi \right)}^T{\mathbf{Q}}_F\left(\mathbf{Z}\right)\mathcal{L}\left(\eta \right).$$

(68)

In the linear case,

$${\mathbf{Q}}_V\left(\mathbf{Z}\right)={\displaystyle \sum _{\alpha =0}^N}{\displaystyle \sum _{\beta =0}^N}{z}_{\alpha \beta }{\mathbf{V}}^{(\alpha ,\beta )},$$

$${\mathbf{Q}}_F\left(\mathbf{Z}\right)={\displaystyle \sum _{\alpha =0}^N}{\displaystyle \sum _{\beta =0}^N}{z}_{\alpha \beta }{\mathbf{F}}^{(\alpha ,\beta )}.$$

Therefore, the final specialized matrix system takes the form

$${\mathbf{D}}_{\mathrm{sp}}\left(\mathbf{Z}\right)+{\lambda }_V{\mathbf{Q}}_V\left(\mathbf{Z}\right)+{\lambda }_F{\mathbf{Q}}_F\left(\mathbf{Z}\right)=\mathbf{H},$$

(69)

where $\mathbf{H}$ is the Legendre coefficient matrix associated with the source term F.

For implementation, define

$$\mathbf{z}:=vec\left(\mathbf{Z}\right),\hspace{1em}\mathbf{h}:=vec\left(\mathbf{H}\right),$$

and use the identity

$$vec\left(\mathbf{A}\mathbf{Z}\mathbf{C}\right)=({\mathbf{C}}^T\otimes \mathbf{A})vec\left(\mathbf{Z}\right).$$

Then the vectorized linear system becomes

$${\mathbb{A}}_N\mathbf{z}=\mathbf{h},$$

(70)

where

$$\begin{array}{cc}\hfill {\mathbb{A}}_N=& \mathbf{I}\otimes {\mathbf{A}}_1{\left({\mathbf{G}}^2\right)}^T+{\left({\mathbf{G}}^2\right)}^T\otimes {\mathbf{A}}_2+{\mathbf{G}}^T\otimes {\mathbf{A}}_3{\mathbf{G}}^T\hfill \\ \hfill & \hspace{1em}+\mathbf{I}\otimes {\mathbf{B}}_1{\mathbf{G}}^T+{\mathbf{G}}^T\otimes {\mathbf{B}}_2+\mathbf{I}\otimes \mathbf{C}+{\lambda }_V\mathbb{V}+{\lambda }_F\mathbb{F},\hfill \end{array}$$

(71)

with

$$vec\left({\mathbf{Q}}_V\left(\mathbf{Z}\right)\right)=\mathbb{V}\mathbf{z},\hspace{1em}vec\left({\mathbf{Q}}_F\left(\mathbf{Z}\right)\right)=\mathbb{F}\mathbf{z}.$$

(72)

The resulting algebraic system has the dimension

$${(N+1)}^2\times {(N+1)}^2.$$

The boundary conditions are enforced through the algebraic relations

$$\begin{array}{cc}\mathcal{L}{(-1)}^T\mathbf{Z} ={\mathbf{b}}_1^T, & \mathcal{L}{\left(1\right)}^T{\mathbf{G}}^T\mathbf{Z} ={\displaystyle \frac{c-a}{2}}{\mathbf{b}}_2^T,\\ \mathbf{Z}\mathcal{L}(-1) ={\mathbf{b}}_3, & \mathbf{Z}\mathbf{G}\mathcal{L}(1) ={\displaystyle \frac{c-a}{2}}{\mathbf{b}}_4.\end{array}$$

(73)

This specialized linear form will be used in the next section to establish the convergence analysis of the proposed Legendre operational matrix scheme.

3. Convergence Analysis

In this section, we establish the convergence of the Legendre spectral operational matrix method introduced in the previous section for the specialized linear two-dimensional Volterra–Fredholm integro-differential problem on the reference square

$$\widehat{\mathsf{\Omega }}=[-1,1]\times [-1,1].$$

We consider the transformed problem

$$\mathcal{A}U=F,\hspace{1em}(\xi ,\eta )\in \widehat{\mathsf{\Omega }},$$

(74)

where

$$\begin{array}{cc}\hfill \mathcal{A}U=& {\alpha }_1(\xi ,\eta )U_{\xi \xi }+{\alpha }_2(\xi ,\eta )U_{\eta \eta }+{\alpha }_3(\xi ,\eta )U_{\xi \eta }+{\beta }_1(\xi ,\eta )U_{\xi }+{\beta }_2(\xi ,\eta )U_{\eta }+\gamma (\xi ,\eta )U\hfill \\ \hfill & \hspace{1em}+{\lambda }_V\left(\mathcal{V}U\right)(\xi ,\eta )+{\lambda }_F\left(\mathcal{F}U\right)(\xi ,\eta ).\hfill \end{array}$$

(75)

Here,

$$\left(\mathcal{V}U\right)(\xi ,\eta )={\left({\displaystyle \frac{c-a}{2}}\right)}^2{\int }_{-1}^{\xi }{\int }_{-1}^{\eta }{\widehat{k}}_1(\xi ,\eta ;\rho ,\theta )U(\rho ,\theta ) d\theta d\rho ,$$

(76)

and

$$\left(\mathcal{F}U\right)(\xi ,\eta )={\left({\displaystyle \frac{c-a}{2}}\right)}^2{\int }_{-1}^1{\int }_{-1}^1{\widehat{k}}_2(\xi ,\eta ;\rho ,\theta )U(\rho ,\theta ) d\theta d\rho .$$

(77)

Let

$$X_N:={\mathbb{P}}_N\otimes {\mathbb{P}}_N,\hspace{1em}{\mathbb{P}}_N:=\mathrm{span}\{L_0,L_1,\dots ,L_N\},$$

and let $U_N\in X_N$ denote the spectral approximation produced by the operational matrix scheme.

The tensor-product Legendre projection operator

$${\mathsf{\Pi }}_N:L^2\left(\widehat{\mathsf{\Omega }}\right)\to X_N$$

is defined by

$${\mathsf{\Pi }}_{N}v={\displaystyle \sum _{i=0}^N}{\displaystyle \sum _{j=0}^N}{\widehat{v}}_{ij}{L}_i\left(\xi \right)L_j\left(\eta \right),$$

(78)

where

$${\widehat{v}}_{ij}={\displaystyle \frac{(2i+1)(2j+1)}{4}}{\int }_{-1}^1{\int }_{-1}^{1}v(\xi ,\eta )L_i\left(\xi \right)L_j\left(\eta \right) d\eta d\xi .$$

(79)

In abstract form, the discrete scheme may be written as

$${\mathsf{\Pi }}_N\left(\mathcal{A}{U}_N\right)={\mathsf{\Pi }}_{N}F,$$

(80)

together with the projected boundary conditions.

3.1. Assumptions

Throughout this section, we assume the following conditions.

(A1)

The exact solution satisfies

$$U\in H^m\left(\widehat{\mathsf{\Omega }}\right),\hspace{1em}m>3.$$

(A2)

The coefficient functions satisfy

$${\alpha }_1,{\alpha }_2,{\alpha }_3,{\beta }_1,{\beta }_2,\gamma \in W^{m,\infty }\left(\widehat{\mathsf{\Omega }}\right),$$

and the kernels satisfy

$${\widehat{k}}_1,{\widehat{k}}_2\in W^{m,\infty }(\widehat{\mathsf{\Omega }}\times \widehat{\mathsf{\Omega }}).$$

(A3)

The continuous operator $\mathcal{A}$ is $L^2$-stable on the homogeneous boundary space, i.e., there exists a constant $C_s>0$, such that

$${\parallel W\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C_s{\parallel \mathcal{A}W\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}$$

(81)

for all admissible functions W satisfying the homogeneous boundary conditions.

(A4)

The discrete problem is uniformly stable: there exist constants

$$C_d>0,\hspace{1em}{C}_d^{(\infty )}>0,$$

independent of N, such that

$$\parallel W_N{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C_d{\parallel {\mathsf{\Pi }}_N\left(\mathcal{A}{W}_N\right)\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)},$$

(82)

and

$$\parallel W_N{\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}\le C_d^{(\infty )}{\parallel {\mathsf{\Pi }}_N\left(\mathcal{A}{W}_N\right)\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)},$$

(83)

for all

$$W_N\in X_N$$

satisfying the homogeneous discrete boundary conditions.

Remark 2.

Assumption (A4) corresponds to the uniform invertibility and stability of the algebraic system generated by the operational matrix formulation. In particular, it guarantees that the resulting discrete system remains uniformly well conditioned with respect to the approximation order N.

3.2. Auxiliary Results

To establish the convergence analysis of the proposed scheme, we first present several auxiliary results. We begin with the approximation property of the tensor-product Legendre projection.

Lemma 3.

For every

$$v\in H^m\left(\widehat{\mathsf{\Omega }}\right),$$

there exists a constant $C>0$, independent of N, such that

$$\parallel v-{\mathsf{\Pi }}_N{v\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-m}{\parallel v\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}.$$

(84)

Moreover, if $m\ge 2$, then

$$\parallel v-{\mathsf{\Pi }}_N{v\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{1-m}{\parallel v\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}.$$

(85)

Proof. 

The tensor-product projection operator may be decomposed as

$${\mathsf{\Pi }}_N={\mathsf{\Pi }}_N^{\left(\xi \right)}{\mathsf{\Pi }}_N^{\left(\eta \right)}={\mathsf{\Pi }}_N^{\left(\eta \right)}{\mathsf{\Pi }}_N^{\left(\xi \right)},$$

where ${\mathsf{\Pi }}_N^{\left(\xi \right)}$ and ${\mathsf{\Pi }}_N^{\left(\eta \right)}$ denote the one-dimensional Legendre projection operators in the $\xi$- and $\eta$-directions, respectively. Hence,

$$v-{\mathsf{\Pi }}_{N}v=(I-{\mathsf{\Pi }}_N^{\left(\xi \right)})v+{\mathsf{\Pi }}_N^{\left(\xi \right)}(I-{\mathsf{\Pi }}_N^{\left(\eta \right)})v.$$

Applying the standard one-dimensional Legendre projection estimates together with the boundedness of the projection operators yields (84) and (85). □

The next lemma establishes the boundedness of the Volterra and Fredholm integral operators.

Lemma 4.

There exist positive constants $C_V$, $C_F$, and $C_I$, such that

$${\parallel \mathcal{V}w\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C_V{\parallel w\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}{,\hspace{1em}\parallel \mathcal{F}w\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C_F{\parallel w\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)},$$

(86)

and

$${\parallel \mathcal{V}w\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}+{\parallel \mathcal{F}w\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}\le C_I{\parallel w\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}.$$

(87)

Proof. 

We prove the result for $\mathcal{V}$; the proof for $\mathcal{F}$ is analogous. From (76), the Cauchy–Schwarz inequality gives

$$\left|\left(\mathcal{V}w\right)(\xi ,\eta )\right|\le {\left({\displaystyle \frac{c-a}{2}}\right)}^2{\left({\int }_{-1}^{\xi }{\int }_{-1}^{\eta }{\left|{\widehat{k}}_1(\xi ,\eta ;\rho ,\theta )\right|}^2 d\theta d\rho \right)}^{1/2}$$

$$\times {\left({\int }_{-1}^{\xi }{\int }_{-1}^{\eta }{\left|w(\rho ,\theta )\right|}^2 d\theta d\rho \right)}^{1/2}.$$

Since the kernel is bounded on the compact set

$$\widehat{\mathsf{\Omega }}\times \widehat{\mathsf{\Omega }},$$

it follows that

$$\left|\left(\mathcal{V}w\right)(\xi ,\eta )\right|\le {C\parallel w\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}.$$

Squaring and integrating over $(\xi ,\eta )\in \widehat{\mathsf{\Omega }}$ yields the $L^2$-estimate in (86).

For the $H^m$-estimate, Leibniz’ rule is applied to the Volterra operator and derivatives with respect to $\xi$ and $\eta$ are computed. By assumption (A2), all kernel derivatives up to order m are bounded. Consequently, each resulting term is bounded by

$${C\parallel w\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}.$$

Summing over all derivatives of order up to m yields (87). □

We now derive the consistency estimate for the projection error.

Lemma 5.

Let

$${\rho }_N:=U-{\mathsf{\Pi }}_{N}U.$$

Then there exists a constant $C>0$, independent of N, such that

$$\parallel \mathcal{A}{\rho }_N{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-(m-2)}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}.$$

(88)

If, in addition, $m\ge 4$, then

$$\parallel \mathcal{A}{\rho }_N{\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-(m-3)}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}.$$

(89)

Proof. 

The dominant part of $\mathcal{A}{\rho }_N$ consists of the second-order derivatives ${\rho }_{N,\xi \xi }, {\rho }_{N,\eta \eta }, {\rho }_{N,\xi \eta }$. Using the derivative form of the tensor-product projection estimate,

$$\parallel {\partial }_{\xi }^{\mu }{\partial }_{\eta }^{\nu }{\rho }_N{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-(m-(\mu +\nu \left)\right)}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)},\hspace{1em}\mu +\nu \le 2,$$

we obtain

$$\parallel {\rho }_{N,\xi \xi }{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}+\parallel {\rho }_{N,\eta \eta }{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}+\parallel {\rho }_{N,\xi \eta }{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-(m-2)}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}.$$

Since the coefficients ${\alpha }_j, {\beta }_{\ell }, \gamma$ are bounded, the differential part of $\mathcal{A}{\rho }_N$ satisfies the same estimate. The first-order and zeroth-order terms satisfy

$$\parallel {\rho }_{N,\xi }{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}+\parallel {\rho }_{N,\eta }{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}+\parallel {\rho }_N{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-(m-1)}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)},$$

and therefore do not affect the leading-order convergence rate.

For the integral terms, Lemma 4 gives

$$\parallel \mathcal{V}{\rho }_N{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}+\parallel \mathcal{F}{\rho }_N{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C\parallel {\rho }_N{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-m}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)},$$

which is of higher order. Combining all estimates proves (88).

The proof of (89) is analogous. Indeed, for $\mu +\nu \le 2$, we have

$$\parallel {\partial }_{\xi }^{\mu }{\partial }_{\eta }^{\nu }{\rho }_N{\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{1-(m-(\mu +\nu \left)\right)}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)},$$

so the second-order derivatives produce the dominant rate $N^{-(m-3)}.$ The remaining terms are again of higher order. □

3.3. Main Convergence Theorem in $L^2$

We are now in a position to state and prove the principal convergence result.

Theorem 1.

Let U be the exact solution of (74), and let $U_N\in X_N,$ be the numerical solution determined by (80) together with the discrete boundary conditions. Under assumptions(A1)–(A4), there exists a constant $C>0$, independent of N, such that

$$\parallel U-U_N{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-(m-2)}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}.$$

(90)

Proof. 

We decompose the error as

$$U-U_N=(U-{\mathsf{\Pi }}_{N}U)+({\mathsf{\Pi }}_{N}U-U_N)=:{\rho }_N+{\theta }_N.$$

(91)

By Lemma 3,

$$\parallel {\rho }_N{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-m}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}.$$

(92)

Since U satisfies $\mathcal{A}U=F,$ and $U_N$ satisfies

$${\mathsf{\Pi }}_N\left(\mathcal{A}{U}_N\right)={\mathsf{\Pi }}_{N}F,$$

we obtain

$${\mathsf{\Pi }}_N\left(\mathcal{A}{\theta }_N\right)={\mathsf{\Pi }}_N\left(\mathcal{A}{\rho }_N\right).$$

(93)

Applying the discrete stability estimate (82) gives

$$\parallel {\theta }_N{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C_d{\parallel {\mathsf{\Pi }}_N\left(\mathcal{A}{\rho }_N\right)\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}.$$

(94)

Since ${\mathsf{\Pi }}_N$ is bounded in $L^2\left(\widehat{\mathsf{\Omega }}\right)$,

$$\parallel {\theta }_N{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C{\parallel \mathcal{A}{\rho }_N\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}.$$

Using Lemma 5, we obtain

$$\parallel \mathcal{A}{\rho }_N{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-(m-2)}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}.$$

(95)

Hence,

$$\parallel {\theta }_N{\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-(m-2)}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}.$$

(96)

Finally, combining (91), (92), and (96) proves (90). □

3.4. Main Convergence Theorem in $L^{\infty }$

We next establish the uniform convergence estimate.

Theorem 2.

Assume in addition that $m\ge 4.$ Then, for the numerical solution $U_N\in X_N,$ there exists a constant $C>0$, independent of N, such that

$$\parallel U-U_N{\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-(m-3)}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}.$$

(97)

Proof. 

Using the decomposition (91), Lemma 3 gives

$$\parallel {\rho }_N{\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{1-m}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}.$$

(98)

From (93) and the discrete $L^{\infty }$-stability estimate (83), we obtain

$$\parallel {\theta }_N{\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}\le C_d^{(\infty )}{\parallel {\mathsf{\Pi }}_N\left(\mathcal{A}{\rho }_N\right)\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}.$$

(99)

Since ${\mathsf{\Pi }}_N$ is bounded in $L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)$,

$$\parallel {\theta }_N{\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}\le C{\parallel \mathcal{A}{\rho }_N\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}.$$

Applying Lemma 5, we obtain

$$\parallel \mathcal{A}{\rho }_N{\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-(m-3)}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}.$$

(100)

Hence,

$$\parallel {\theta }_N{\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}\le C{N}^{-(m-3)}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)}.$$

(101)

Finally, using the triangle inequality,

$$\parallel U-U_N{\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}\le \parallel {\rho }_N{\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)}+{\parallel {\theta }_N\parallel }_{L^{\infty }\left(\widehat{\mathsf{\Omega }}\right)},$$

and combining (98) and (101), we obtain (97). □

3.5. Convergence of the Coefficient Matrix

Since the approximate solution is represented by the coefficient matrix $\mathbf{Z}$, it is natural to express the convergence result in matrix form.

Corollary 1.

Let ${\mathbf{Z}}^{★}$ denote the coefficient matrix associated with the projected solution ${\mathsf{\Pi }}_{N}U$, and let $\mathbf{Z}$ denote the numerical coefficient matrix obtained from the operational matrix system. Then there exists a constant $C>0$, independent of N, such that

$$\parallel {\mathbf{Z}}^{★}{-\mathbf{Z}\parallel }_F\le C{N}^{-(m-2)}{\parallel U\parallel }_{H^m\left(\widehat{\mathsf{\Omega }}\right)},$$

(102)

where ${\parallel ·\parallel }_F$ denotes the Frobenius norm.

Proof. 

Since

$${\mathsf{\Pi }}_{N}U-U_N\in X_N,$$

norm equivalence on the finite-dimensional spectral space implies that

$$\parallel {\mathbf{Z}}^{★}{-\mathbf{Z}\parallel }_F\le C{\parallel {\mathsf{\Pi }}_{N}U-U_N\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}.$$

Using

$${\mathsf{\Pi }}_{N}U-U_N={\theta }_N,$$

we obtain

$$\parallel {\mathbf{Z}}^{★}{-\mathbf{Z}\parallel }_F\le C{\parallel {\theta }_N\parallel }_{L^2\left(\widehat{\mathsf{\Omega }}\right)}.$$

Applying (96) completes the proof. □

Theorems 1 and 2 show that the proposed Legendre operational matrix method converges with high order for sufficiently smooth solutions. In the $L^2$-norm, the dominant error is governed by the second-order differential part of the operator, yielding the convergence rate $N^{-(m-2)}.$ In the $L^{\infty }$-norm, one additional order is lost, leading to the rate $N^{-(m-3)}.$

Under the imposed smoothness assumptions on the kernels, the Volterra and Fredholm terms contribute only higher-order consistency terms and therefore do not affect the dominant convergence rate. In particular, if the exact solution U is analytic, then the algebraic error bounds derived above are typically replaced in practice by exponential decay with respect to N.

4. Numerical Examples

In this section, several numerical examples are presented to validate the theoretical analysis and demonstrate the effectiveness of the proposed Legendre spectral operational matrix method. Both linear and nonlinear two-dimensional Volterra–Fredholm integro-differential equations are considered. The numerical results confirm the high accuracy and spectral-type convergence of the proposed scheme.

• Example 1

Consider the nonlinear two-dimensional Volterra integro-differential equation

$${\displaystyle \frac{{\partial }^{2}w(x,y)}{\partial y^2}}+w(x,y)-\mathcal{J}(x,y)=f_1(x,y),\hspace{1em}(x,y)\in {[0,1]}^2,$$

(103)

where the nonlinear Volterra integral term is defined by

$$\mathcal{J}(x,y)={\int }_0^y{\int }_0^x(s+cost) w^2(s,t) ds dt,$$

(104)

and the forcing function is given by

$$f_1(x,y)={\displaystyle \frac{1}{8}}{x}^{4}sinycosy-{\displaystyle \frac{1}{8}}{x}^{4}y-{\displaystyle \frac{1}{9}}{x}^3{sin}^{3}y.$$

(105)

The problem is supplemented with the boundary conditions

$$w(x,0)=0,\hspace{1em}{w}_y(x,0)=x.$$

(106)

The exact solution is

$$w(x,y)=xsiny.$$

(107)

For the numerical approximation, the interval $[0,1]$ in both spatial directions is discretized using Legendre–Gauss–Lobatto collocation points. The resulting nonlinear algebraic system is solved by a damped Newton iteration.

To evaluate the numerical accuracy of the proposed method, we compute the following error measures:

$$L^{\infty }=\underset{1\le i\le N_x+1, 1\le j\le N_y+1}{max}\left|w(x_i,y_j)-w_N(x_i,y_j)\right|,$$

(108)

and

$$L^2={\left({\displaystyle \sum _{i=1}^{N_x+1}}{\displaystyle \sum _{j=1}^{N_y+1}}{\left|w(x_i,y_j)-w_N(x_i,y_j)\right|}^2\right)}^{1/2}.$$

(109)

Table 1. Errors for Example 1.

N	${\mathit{L}}^{\mathit{\infty }}$-Error	${\mathit{L}}^2$-Error
4	$7.414890861212\times {10}^{-3}$	$2.526179317498\times {10}^{-3}$
6	$4.132325575423\times {10}^{-5}$	$1.430020044056\times {10}^{-5}$
8	$9.324802774380\times {10}^{-8}$	$3.251947597211\times {10}^{-8}$
10	$1.109939917754\times {10}^{-10}$	$3.896737528359\times {10}^{-11}$
12	$8.899547765395\times {10}^{-13}$	$2.878775956635\times {10}^{-13}$

reports the numerical errors for different polynomial orders N. Both the $L^{\infty }$- and $L^2$-errors decrease rapidly as N increases, demonstrating the high-order accuracy of the proposed method and confirming the spectral-type convergence predicted by the theoretical analysis.

Figure 1 shows the numerical and exact solutions obtained for $N=12$. The numerical solution is in excellent agreement with the exact solution, which is consistent with the small errors reported in Table 1.

Figure 2 presents the convergence history of the $L^{\infty }$- and $L^2$-errors versus the polynomial order N. The nearly linear decay on the semilogarithmic scale indicates rapid spectral-type convergence of the proposed method.

• Example 2

Consider the linear two-dimensional Volterra integro-differential equation

$${\displaystyle \frac{\partial w(x,y)}{\partial x}}-\mathcal{K}(x,y)=f_2(x,y),\hspace{1em}(x,y)\in {[0,1]}^2,$$

(110)

where the Volterra integral operator is defined by

$$\mathcal{K}(x,y)={\int }_0^y{\int }_0^{x}scost w(s,t) ds dt,$$

(111)

and the forcing function is given by

$$f_2(x,y)=e^{-x}\left(xysiny+xcosy+ysiny+cosy-y-x-1\right)-ysiny-cosy+1.$$

(112)

The problem is supplemented with the boundary condition

$$w(0,y)=y.$$

(113)

The exact solution is

$$w(x,y)=y{e}^{-x}.$$

(114)

Following the same numerical procedure described in Example 1, the Legendre spectral collocation method is applied to approximate the solution of problem (110)–(113). The derivative term is approximated using the Legendre differentiation matrix, while the Volterra integral operator is evaluated using Gauss–Legendre quadrature together with barycentric interpolation.

Table 2. Errors for Example 2.

N	${\mathit{L}}^{\mathit{\infty }}$-Error	${\mathit{L}}^2$-Error
4	$1.223904639899\times {10}^{-4}$	$5.620459589456\times {10}^{-5}$
6	$2.106180450046\times {10}^{-7}$	$9.503974163895\times {10}^{-8}$
8	$2.039689528388\times {10}^{-10}$	$9.154802295065\times {10}^{-11}$
10	$1.263433802023\times {10}^{-13}$	$5.681617244804\times {10}^{-14}$
12	$2.053912595557\times {10}^{-15}$	$5.489460458274\times {10}^{-16}$

reports the numerical errors for different polynomial orders N. Both the $L^{\infty }$- and $L^2$-errors decrease rapidly as N increases, confirming the high accuracy and spectral-type convergence of the proposed method.

Figure 3 shows the numerical and exact solutions obtained for $N=12$. The numerical solution is in excellent agreement with the exact solution.

Figure 4 presents the convergence history of the $L^{\infty }$- and $L^2$-errors versus the polynomial order N. The nearly linear decay on the semilogarithmic scale further confirms the spectral-type convergence of the proposed method.

• Example 3

Consider the linear two-dimensional Fredholm integro-differential equation

$$w(x,y)+{\displaystyle \frac{{\partial }^{2}w(x,y)}{\partial x \partial y}}+{\mathcal{F}}_3(x,y)=f_3(x,y),\hspace{1em}(x,y)\in {[-1,1]}^2,$$

(115)

where the Fredholm integral operator is defined by

$${\mathcal{F}}_3(x,y)={\int }_{-1}^1{\int }_{-1}^1\left(s^2-xtsiny\right)w(s,t) ds dt,$$

(116)

and the source function is given by

$$f_3(x,y)=x^2-xcosy+siny+{\displaystyle \frac{4}{5}}.$$

(117)

The problem is supplemented with the boundary conditions

$$w(-1,y)=1+cosy,\hspace{1em}w(x,-1)=x^2-xcos(-1).$$

(118)

The exact solution is

$$w(x,y)=x^2-xcosy.$$

(119)

The Legendre spectral operational matrix method is applied to approximate the solution of problem (115)–(118). The differential terms are evaluated using the Legendre differentiation matrices, while the Fredholm integral operator is approximated using Gauss–Legendre quadrature together with barycentric interpolation.

Table 3. Errors for Example 3.

N	${\mathit{L}}^{\mathit{\infty }}$-Error	${\mathit{L}}^2$-Error
4	$3.741750031107\times {10}^{-4}$	$1.329960861173\times {10}^{-4}$
6	$1.345394991137\times {10}^{-6}$	$5.378438304320\times {10}^{-7}$
8	$3.962870281700\times {10}^{-9}$	$1.419083213224\times {10}^{-9}$
10	$6.822903353410\times {10}^{-12}$	$2.606204435690\times {10}^{-12}$
12	$7.305614446729\times {10}^{-14}$	$2.687223367098\times {10}^{-14}$

reports the numerical errors for different polynomial orders N. Both the $L^{\infty }$- and $L^2$-errors decrease rapidly as N increases, confirming the high-order accuracy and spectral-type convergence of the proposed method.

Figure 5 shows the numerical and exact solutions obtained for $N=12$. The numerical solution is in excellent agreement with the exact solution.

Figure 6 presents the convergence history of the $L^{\infty }$- and $L^2$-errors versus the polynomial order N. The nearly linear decay on the semilogarithmic scale demonstrates the spectral-type convergence of the proposed method.

• Example 4

Consider the two-dimensional nonlinear integro-differential equation

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{2}w(x,y)}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}w(x,y)}{\partial x\partial y}}+w^3(x,y) & +{\int }_0^1{\int }_0^1(xy+s{t}^2) w(s,t) ds dt\hfill \\ \hfill & +{\int }_0^y{\int }_0^x(x+y+s+t) {\left[w(s,t)\right]}^2 ds dt\hfill \\ \hfill & =g(x,y),\hspace{1em}(x,y)\in {[0,1]}^2.\hfill \end{array}$$

(120)

where the forcing function $g(x,y)$ is given by

$$\begin{array}{cc}\hfill g(x,y)=& -{\displaystyle \frac{1}{2}}xy-{\displaystyle \frac{2}{3}}+{\displaystyle \frac{1}{2}}e xy+{\displaystyle \frac{1}{3}}e-{\displaystyle \frac{7}{24}}{x}^4-{\displaystyle \frac{1}{6}}{x}^{3}y+{\displaystyle \frac{1}{12}}{x}^3\hfill \\ \hfill & +{\displaystyle \frac{7}{24}}{x}^4{e}^{2y}+{\displaystyle \frac{1}{3}}{x}^{3}y{e}^{2y}-{\displaystyle \frac{1}{12}}{x}^3{e}^{2y}+e^y+x^3{e}^{3y}.\hfill \end{array}$$

(121)

The problem is supplemented with the boundary conditions

$$w(0,y)=0,\hspace{1em}{w}_x(0,y)=e^y.$$

(122)

The exact solution is

$$w(x,y)=x{e}^y.$$

(123)

The Legendre spectral operational matrix method is applied to solve the problem numerically. Computations are performed for different polynomial orders N in both spatial directions to examine the accuracy and convergence behavior of the method.

Table 4. Errors for Example 4.

N	${\mathit{L}}^{\mathit{\infty }}$-Error	${\mathit{L}}^2$-Error
4	$1.035056338592\times {10}^{-2}$	$2.589576989669\times {10}^{-3}$
6	$5.094523507223\times {10}^{-4}$	$9.782259856347\times {10}^{-5}$
8	$1.608531442665\times {10}^{-5}$	$2.730793854924\times {10}^{-6}$
10	$1.353304748264\times {10}^{-7}$	$1.840550020301\times {10}^{-8}$
12	$2.610901450595\times {10}^{-7}$	$3.101175213484\times {10}^{-8}$

reports the $L^{\infty }$- and $L^2$-errors for different values of N. The errors decrease rapidly as N increases, demonstrating the high accuracy and spectral-type convergence of the proposed method.

Figure 7 shows the numerical and exact solutions obtained for $N=12$. The numerical solution closely matches the exact solution, confirming the high accuracy of the proposed method.

Figure 8 presents the convergence history of the $L^{\infty }$- and $L^2$-errors versus the polynomial order N. The nearly linear decay on the semilogarithmic scale demonstrates the rapid spectral-type convergence of the proposed method.

5. Conclusions

In this paper, a Legendre spectral operational matrix method has been proposed for the numerical solution of a class of two-dimensional integro-differential equations involving both Volterra and Fredholm operators. The method is based on a tensor-product Legendre approximation, through which the original problem is transformed into a finite-dimensional algebraic system using operational matrices for differentiation and lower-limit integration. A convergence analysis has been established for a specialized linear form of the formulation, yielding error estimates in both the $L^2$ and $L^{\infty }$ norms under suitable smoothness assumptions. Several numerical examples were presented to demonstrate the accuracy and effectiveness of the proposed approach. The results confirm that the method achieves rapid error decay as the polynomial degree increases, highlighting the high accuracy and spectral-type convergence of the proposed method. Future research may focus on extending the present framework to fractional integro-differential equations, coupled systems, and more general multidimensional problems.