Numerical Solutions of Fractional Weakly Singular Two-Dimensional Partial Volterra Integral Equations Using Euler Wavelets

Abstract

This paper presents an innovative numerical method for solving two-dimensional weakly singular Volterra integral equations, including fractional Volterra integral equations with weak singularities. Solving these equations in higher dimensions and in the presence of fractional and weak singularities is highly challenging. The proposed approach uses Euler wavelets (EWs) within an operational matrix (OM) framework combined with advanced numerical techniques, initially transforming these equations into a linear algebraic system and then solving it efficiently. This method offers very high accuracy, strong computational efficiency, and simplicity of implementation, making it suitable for a wide range of such complex problems, especially those requiring high speed and precision in the presence of intricate features.

1. Introduction

In the last decade, there has been a growing interest in the study of nonlinear FPVIEs due to their significance in physical and engineering problems. Mathematical modeling frequently involves functional equations, including integral and IDEs. An IDE combines integrals and derivatives of a function, while a PIDE arises when the unknown function or its derivatives are integrated. These equations are widely used in various applications, such as reaction–diffusion problems in cellular biology [1,2,3], water–wave systems [4], layer–liquid systems [5], plasma physics, fluid dynamics [6], and Quantum Mechanics [7]. However, solving 2DFWSPVIEs presents significant challenges, particularly in higher dimensions and in the presence of fractional orders and weak singularities. Existing numerical methods often struggle to achieve high accuracy, computational efficiency, and simplicity of implementation due to the complex interplay of fractional derivatives and singular kernels.

While strongly singular kernels with singularities of order $\mathcal{O}\left(\right|x-y{|}^{-\alpha })$ for $\alpha \ge 1$ require specialized variational principles or regularization to manage their unbounded behavior, weakly singular kernels with $\alpha <1$ arise as regularized counterparts, often representing physically tractable limits of singular interactions [8]. Strongly singular kernels, common in problems such as wave scattering or crack propagation, exhibit rapid growth that complicates numerical approximation, often necessitating singularity subtraction or Cauchy principal value integrals [8]. In contrast, weakly singular kernels, as considered here, have milder singularities that are more amenable to numerical methods like the Euler wavelet operational matrix method (EWOMM), which uses their regularity to construct sparse operational matrices without complex regularization. However, the EWOMM is less effective for strongly singular kernels, whose severe singularities demand treatments beyond its scope, such as mesh refinement near singularities or analytical preconditioning. This constraint defines the method’s applicability to weakly singular scenarios found in applications like fractional diffusion or viscoelasticity. The Euler wavelet approach developed here exploits this regularity to construct operational matrices tailored to weakly singular kernels, avoiding ad hoc singularity subtraction or asymptotic expansions while enabling high-accuracy approximations and efficient solutions of nonlinear systems involving fractional orders.

Wavelets are a set of functions generated through dilating and translating a single function called the mother wavelet. They have been a useful tool for approximating solutions to various equations [9]. The origins of wavelet theory can be traced back to the early 1960s. The wavelet algorithm offers several advantages: (1) it is a multiresolution algorithm, (2) the wavelet base is compactly supported and orthonormal, (3) increasing the degree of the mother function improves the accuracy of the approximation, (4) the coefficient matrix of the algebraic equations becomes sparse after discretization, and (5) the wavelet basis serves as an unconditional basis for $L^2\left(\mathbb{R}\right)$ [10].

In recent years, EWs have been used as effective mathematical tools for solving various integral equations [11,12]. Cetin in [13] applied EWs to solve the time-fractional Cattaneo equation with the Caputo–Fabrizio derivative. Researchers in [14] used EPs to handle stochastic Itô–Volterra integral equations. Mirzaee in [15] applied EPs to solve Fredholm IDEs. Behera et al. utilized the EWs OM technique to solve Volterra–Fredholm IDEs with weakly singular kernels [16]. Y. Wang, et al. presented two-dimensional EPs approximants for solutions of two-dimensional Volterra integral equations of fractional order [17]. Behera and Saha Ray addressed an EWs technique to tackle pantograph Volterra delay IDEs [18]. By employing EWs, ref. [19] sought to solve optimal control problems involving fractional orders.

In approximation theory, piecewise functions, especially orthogonal functions, are utilized to achieve highly accurate solutions for equations that require precision. OMs are employed to simplify complex problems and reduce computational time, making them a cost-effective and time-efficient technique for obtaining solutions. Various techniques, including CAS wavelets [20], have been employed. Syam in [21] used the Block pulse function method to solve a two-dimensional fractional integro-differential equation with a modified Atangana–Baleanu fractional derivative. Sadeghi in [22] applied the operational matrix for the Atangana–Baleanu derivative to address FDEs. The B-spline [23], Tau method [24], Laguerre series [25], Bernstein OM [26], Jacobi OM [27], Homotopy analysis method [28], Boubaker Hybrid Functions [29], Collocation method [30], Lucas polynomials [31], Wavelets method [32], and Fourier-Gegenbauer pseudospectral method [33] have been developed and applied in various studies.

Numerous previous studies have utilized two-dimensional partial Volterra integral equations and their fractional counterparts. Various OMs have been applied to solve these equations. For example, in reference [34], Boubaker polynomials were used for the numerical solution of 2DFPVIEs. Singh et al. employed the OM technique to find an approximate solution for fractional singular IDEs [35]. In [36], Triangular Functions were utilized to solve partial mixed Volterra–Fredholm integral equations. Few studies have been conducted on 2DFWSPVIE cases. For example, Behera and Saha Ray tackled weakly singular PIDEs in 2020 using a two-dimensional operational technique [37]. Khajehnasiri et al. presented two-dimensional Boubaker polynomial approximants for solving a fractional weakly singular two-dimensional partial Volterra integral equation [38]. Compact finite difference was proposed in [39] for solving two-dimensional parabolic integro-differential equations with a weakly singular kernel. Additionally, Yavuz et al. developed two-dimensional weakly singular Volterra integral equations using bivariate rational approximants [40]. Similarly, Zhang et al. [41] devised a numerical approach for Volterra integral equations with weakly singular kernels.

In our present study, we examine the 2DFWSPVIE in the folllowing form:

$$\begin{array}{c}\hfill {\upsilon }_{\kappa ,\varrho }(\kappa ,\varrho )+{\upsilon }_{\varrho }(\kappa ,\varrho )=\upsilon (\kappa ,\varrho )+g(\kappa ,\varrho )+\Theta (\kappa ,\varrho )+\Pi (\kappa ,\varrho ),\end{array}$$

(1)

subject to the initial conditions

$$\begin{array}{ccc}\hfill \upsilon (\kappa ,0)& =& {\upsilon }_0\left(\kappa \right),\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill \upsilon (0,\varrho )& =& {\upsilon }_0\left(\varrho \right),\hfill \end{array}$$

(3)

where

$$\begin{array}{ccc}\hfill \Theta (\kappa ,\varrho )& =& {\displaystyle \frac{1}{\Gamma \left({\gamma }_1\right)\Gamma \left({\gamma }_2\right)}}{\int }_0^{\kappa }{\int }_0^{\varrho }{\displaystyle \frac{H(\kappa ,\varrho ,s,y,\upsilon (s,y\left)\right)}{{(\kappa -s)}^{1-{\gamma }_1}{(\varrho -y)}^{1-{\gamma }_2}}}dyds,\hfill \\ \hfill \Pi (\kappa ,\varrho )& =& {\int }_0^{\kappa }{\int }_0^{\varrho }{\displaystyle \frac{{\rm Y}\left(\upsilon \right(s,y\left)\right)}{{(\kappa -s)}^{\alpha }}}dyds,\hfill \end{array}$$

and $0<{\gamma }_1,{\gamma }_2<1$, $\upsilon$ is an unknown function that needs to be determined; $H,{\rm Y}$, and g are known functions defined on the domain $\mathsf{\Omega }=[0,1]\times [0,1]$; ${\upsilon }_0$ is a known function defined on the interval $[0,1]$; and $\Gamma (.)$ is the Gamma function.

To address the limitations of existing approaches for solving this class of problems, our work proposes a novel method that achieves high accuracy, computational efficiency, and straightforward implementation. This method effectively handles the complexities of 2DFWSPVIEs by employing the multiresolution properties of EWs and the sparsity of OMs, offering a robust solution for intricate problems across physics, engineering, and biology. In particular, we present the EWOMM: a novel numerical method for solving this class of 2DFWSPVIEs using EWs and OM techniques. The primary contributions include (1) the development of a new OM approach based on EWs that enables efficient transformation of 2DFWSPVIEs into solvable algebraic systems; (2) the establishment of high-accuracy approximations for weakly singular kernels to address challenges associated with fractional orders and singularity; and (3) rigorous numerical validation through multiple examples that display superior accuracy and robustness across various fractional orders and initial conditions. These advancements broaden the applicability of wavelet-based methods for complex fractional integral equations in applied mathematics and related fields.

In this study, we focus on the Riemann–Liouville and Caputo derivatives due to their suitability for problems involving weakly singular kernels, such as the 2DFWSPVIEs considered here. These derivatives feature power-law kernels that naturally match the singular structure of the target equations, and their close connection to Volterra-type integrals simplifies the construction of numerical schemes within the operational matrix framework. The Caputo derivative is often preferred in physical modeling because it permits initial conditions to be specified using integer-order derivatives, consistent with classical differential equations. Both derivatives possess key mathematical properties—such as linearity and generalized product and chain rules under appropriate regularity—that facilitate their use in the analysis and numerical treatment of fractional differential and integral equations. Furthermore, their link to the Riemann–Liouville fractional integral, a foundational operator in fractional calculus, reinforces their suitability for integration-based numerical schemes.

Alternative fractional derivatives, including the two-scale fractal, He’s, and Atangana–Baleanu derivatives, can be useful in modeling complex systems (e.g., fractal-based mechanical properties of materials), but their non-singular or non-local kernels pose significant computational challenges, as discussed in Section 2. Such definitions often require fundamentally different operational matrix constructions, introduce higher computational costs due to memory effects, and may not yield improved accuracy for the weakly singular cases considered here. Our adoption of the classical fractional calculus framework, supported by decades of theoretical and computational development, ensures efficient, accurate, and tractable discretization schemes for high-dimensional problems.

This paper is organized as follows: Section 2 introduces fundamental concepts of fractional calculus. Section 3 presents EWs and their application in function approximation. Section 4 details the OMs for fractional differentiation and integration. Section 5 outlines the EWOMM. Section 6 provides illustrative examples using the EWOMM. Finally, Section 7 presents some conclusions and future directions. A list of acronyms used in this paper is provided in Appendix A.

2. Preliminaries

The fundamental definitions and characteristics of the fractional derivative and integral have been outlined below [42].

Definition 1.

One may define the Riemann–Liouville fractional integral operator $I^{{\gamma }_1}$ of order ${\gamma }_1\ge 0$ of a function $\upsilon \in L^1\left([0,1]\right)$ as

$$\begin{array}{ccc}\hfill \left(I^{{\gamma }_1}\right)\upsilon \left(\kappa \right)& =& \left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma \left({\gamma }_1\right)}}{\int }_0^{\kappa }{\displaystyle \frac{\upsilon \left(s\right)}{{(\kappa -s)}^{1-{\gamma }_1}}}ds,\hfill & {\gamma }_1>0,\hfill \\ \upsilon \left(\kappa \right),\hfill & {\gamma }_1=0,\hfill \end{array}\right.\hfill \end{array}$$

For $v\left(\kappa \right)={\kappa }^{{\gamma }_2}:{\gamma }_2>-1$, the above definition gives

$$\begin{array}{c}\hfill I^{{\gamma }_1}{\kappa }^{{\gamma }_2}={\displaystyle \frac{\Gamma ({\gamma }_2+1)}{\Gamma ({\gamma }_1+{\gamma }_2+1)}}{\kappa }^{{\gamma }_1+{\gamma }_2}.\end{array}$$

Definition 2.

The Caputo fractional derivative ${}^{c}D^{\gamma 1}$ of order ${\gamma }_1$ is defined as

$$\begin{array}{c}\hfill {(}^c{D}^{{\gamma }_1}\upsilon )\left(\kappa \right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (n-{\gamma }_1)}}{\int }_0^{\kappa }{\displaystyle \frac{{\upsilon }^{\left(n\right)}\left(s\right)}{{(\kappa -s)}^{{\gamma }_1+1-n}}}ds,\hfill & n-1<{\gamma }_1<n,n\in \mathbb{N},\hfill \\ {\displaystyle \frac{d^n\upsilon \left(\kappa \right)}{d{\kappa }^n}},\hfill & {\gamma }_1=n,\hfill \end{array}\right.\end{array}$$

where $\upsilon \in C^n\left([0,1]\right)$ is integrable with respect to the singular weight function, and $D={\displaystyle \frac{d}{d\kappa }}$.

Definition 3.

The Caputo partial fractional derivative of $\upsilon (\kappa ,\varrho )$ with respect to κ of order ${\gamma }_1>0$ is defined as

$$\begin{array}{c}\hfill {(}^c{D}_{\kappa }^{{\gamma }_1}\upsilon )(\kappa ,\varrho )={\displaystyle \frac{{\partial }^{{\gamma }_1}\upsilon (\kappa ,\varrho )}{\partial {\kappa }^{{\gamma }_1}}}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (n-{\gamma }_1)}}{\int }_0^{\kappa }{\displaystyle \frac{{\partial }^n\upsilon (s,\varrho )}{\partial s^n}}{\displaystyle \frac{ds}{{(\kappa -s)}^{{\gamma }_1+1-n}}},\hfill & n-1<{\gamma }_1<n,n\in \mathbb{N},\hfill \\ {\displaystyle \frac{{\partial }^n\upsilon (\kappa ,\varrho )}{\partial {\kappa }^n}},\hfill & {\gamma }_1=n,\hfill \end{array}\right.\end{array}$$

Definition 4

([43]). Let $({\gamma }_1,{\gamma }_2)\in (0,\infty )\times (0,\infty ),\gamma =(0,0),\mathsf{\Omega }:=[0,a]\times [0,b],$ and $\upsilon \in L^1\left(\mathsf{\Omega }\right)$. The left-sided mixed Riemann–Liouville integral of order $({\gamma }_1,{\gamma }_2)$ of υ is defined by

$$\left(I_{\gamma }^{({\gamma }_1,{\gamma }_2)}\upsilon \right)(\kappa ,\varrho )={\displaystyle \frac{1}{\Gamma \left({\gamma }_1\right)\Gamma \left({\gamma }_2\right)}}{\int }_0^{\kappa }{\int }_0^{\varrho }{(\kappa -s)}^{({\gamma }_1-1)}{(\varrho -y)}^{({\gamma }_2-1)}\upsilon (s,y)dyds.$$

(4)

In particular,

1.

$\left(I_{\gamma }^{({\gamma }_1,{\gamma }_2)}\upsilon \right)(\kappa ,\varrho ):=\upsilon (\kappa ,\varrho )$;

2.

$\left(I_{\gamma }^{\left(\sigma \right)}\upsilon \right)(\kappa ,\varrho )={\int }_0^{\kappa }{\int }_0^{\varrho }\upsilon (s,y)dyds,(\kappa ,\varrho )\in \mathsf{\Omega },\sigma =(1,1)$;

3.

$\left(I_{\gamma }^{({\gamma }_1,{\gamma }_2)}\upsilon \right)(\kappa ,0)=\left(I_{\gamma }^{({\gamma }_1,{\gamma }_2)}\right)(0,\varrho )=0,\upsilon \in [0,a],\varrho \in [0,b]$;

4.

$I_{\gamma }^{{\gamma }_1,{\gamma }_2}{\kappa }^{\lambda }{\varrho }^{\omega }={\displaystyle \frac{\Gamma (1+\lambda )\Gamma (1+\omega )}{\Gamma (1+\lambda +{\gamma }_1)\Gamma (1+\omega +{\gamma }_2)}}{\kappa }^{\lambda +{\gamma }_1}{\varrho }^{\omega +{\gamma }_2},(\kappa ,\varrho )\in \mathsf{\Omega },\lambda ,\omega \in (-1,\infty )$.

To provide a comprehensive theoretical foundation, we introduce alternative fractional derivative definitions that are relevant to complex system modeling, particularly in applications like fractal-based mechanical properties of materials [44,45].

Definition 5.

(i) Two-Scale Fractal Derivative: For a function $\upsilon \left(\kappa \right)$ and a fractal dimension $\alpha \in (0,1)$, the two-scale fractal derivative is defined as [44]

$$\begin{array}{c}\hfill D^{\alpha }\upsilon \left(\kappa \right)=\underset{\Delta \kappa \to 0}{lim}{\displaystyle \frac{\upsilon (\kappa +\Delta \kappa )-\upsilon \left(\kappa \right)}{{(\Delta \kappa )}^{\alpha }}},\end{array}$$

where the scaling is adapted to fractal structures, making it suitable for modeling materials with irregular geometries, such as recycled aggregate concretes.

(ii) He’s Fractional Derivative: He’s fractional derivative, also known as the fractal derivative, is defined similarly but emphasizes local scaling properties [44]:

$$\begin{array}{c}\hfill D^{\alpha }\upsilon \left(\kappa \right)={\displaystyle \frac{d\upsilon }{d{\kappa }^{\alpha }}}\approx {\displaystyle \frac{\upsilon (\kappa +\Delta \kappa )-\upsilon \left(\kappa \right)}{{(\Delta \kappa )}^{\alpha }}},\end{array}$$

for small $\Delta \kappa$, focusing on fractal media with non-integer dimensions.

(iii) Atangana–Baleanu Fractional Derivative: For $\upsilon \in H^1(0,1)$ and $0<\alpha <1$, the Atangana–Baleanu fractional derivative in the Caputo sense is defined as [45]

$$\begin{array}{c}\hfill {}^{ABC}D^{\alpha }\upsilon \left(\kappa \right)={\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{\int }_0^{\kappa }{\upsilon }^{\prime }\left(s\right)E_{\alpha }\left(-{\displaystyle \frac{\alpha }{1-\alpha }}{(\kappa -s)}^{\alpha }\right)ds,\end{array}$$

where $B\left(\alpha \right)>0$ is a normalization function with $B\left(0\right)=B\left(1\right)=1$, and $E_{\alpha }$ is the Mittag–Leffler function. This derivative uses a non-singular kernel, making it suitable for modeling memory-dependent processes like anomalous diffusion.

It is noteworthy to mention that the Riemann–Liouville and Caputo derivatives, defined above, use power-law kernels that naturally align with the weakly singular structure of 2DFWSPVIEs, enabling efficient operational matrix constructions. In contrast, the two-scale fractal and He’s fractional derivatives are designed for fractal media, requiring specialized numerical schemes due to their local scaling properties, which are less compatible with the integral-based framework of this study [44,46]. The Atangana–Baleanu derivative, while effective for non-local systems, introduces computational complexity due to its Mittag–Leffler kernel, which demands higher computational resources for accurate discretization compared to the power-law kernels of Riemann–Liouville and Caputo derivatives. For the 2DFWSPVIEs considered here, the classical derivatives offer a balance of theoretical simplicity and numerical efficiency, making them preferable for the EWOMM framework. Alternative derivatives may be explored in future work to address problems with fractal or non-local characteristics.

3. EWs

This section defines EWs and their properties, which are used as a basis for approximating functions in the context of solving 2DFWSPVIE.

The EWs ${\tilde{\varphi }}_{n,m}\left(\kappa \right)=\tilde{\kappa }(k,n,m,\kappa )$ have four parameters: $n=1,2,\dots ,2^{k-1}$, where k is a positive integer, m is the degree of the EPs, and $\kappa$ is the normalized time. The EWs are defined on the interval $[0,1)$ for $m=0,1,\dots ,M-1$, where $M>0$ is a fixed integer, as follows:

$$\begin{array}{ccc}\hfill {\tilde{\varphi }}_{n,m}\left(\kappa \right)& =& \left\{\begin{array}{cc}{2}^{{\displaystyle \frac{k-1}{2}}}{\tilde{P}}_m(2^{k-1}\kappa -n+1),\hfill & {\displaystyle \frac{n-1}{2^{k-1}}}\le \kappa <{\displaystyle \frac{n}{2^{k-1}}},\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\hfill \end{array}$$

with

$$\begin{array}{ccc}\hfill {\tilde{P}}_m\left(\kappa \right)& =& \left\{\begin{array}{cc}1,\hfill & m=0,\hfill \\ {\displaystyle \frac{1}{\sqrt{\frac{2{(-1)}^{m-1}{(m!)}^2}{\left(2m\right)!}{P}_{2m+1}\left(0\right)}}}{P}_m\left(\kappa \right),\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

The normalization coefficient is given by ${\displaystyle \frac{1}{\sqrt{\frac{2{(-1)}^{m-1}{(m!)}^2}{\left(2m\right)!}{P}_{2m+1}\left(0\right)}}}$. The functions $P_m\left(\kappa \right)$ denote the familiar EPs of order m, which can be expressed through the following generating functions [47]:

$${\displaystyle \frac{2e^{\kappa x}}{e^x+1}}=\sum _{m=0}^{\infty }{P}_m\left(\kappa \right){\displaystyle \frac{x^m}{m!}}.$$

(5)

The EPs of the first type for $m=0,\dots ,N$ can be generated using the following equation:

$$\sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)P_l\left(\kappa \right)+P_m\left(\kappa \right)=2{\kappa }^m,$$

(6)

where $\left(\begin{array}{c}m\\ l\end{array}\right)$ is a binomial coefficient. The initial EPs are as follows:

$$P_0\left(\kappa \right)=1,P_1\left(\kappa \right)=\kappa -{\displaystyle \frac{1}{2}},P_2\left(\kappa \right)={\kappa }^2-\kappa ,P_3\left(\kappa \right)={\kappa }^3-{\displaystyle \frac{3}{2}}{\kappa }^2+{\displaystyle \frac{1}{4}},\dots .$$

At $\kappa =0$, the initial Euler numbers are as follows:

$$P_0\left(0\right)=1,P_1\left(0\right)=-{\displaystyle \frac{1}{2}},P_3\left(0\right)={\displaystyle \frac{1}{4}},P_5\left(0\right)=-{\displaystyle \frac{1}{2}}.$$

(7)

These polynomials satisfy the formula below:

$${\int }_0^1{P}_m\left(\kappa \right)P_n\left(\kappa \right)d\kappa ={(-1)}^{n-1}{\displaystyle \frac{m!(n+1)!}{(m+n+1)!}}{P}_{m+n+1}\left(0\right),m,n\ge 1;$$

(8)

moreover, they form a complete basis over the interval $[0,1]$.

3.1. Function Approximation

Consider a function $\upsilon \left(\kappa \right)\in L^2[0,1]$, which can be defined in terms of the EWs as follows:

$$\upsilon \left(\kappa \right)\approx \sum _{n=1}^{2^{k-1}}\sum _{m=0}^{M-1}{\tilde{c}}_{n,m}{\tilde{\varphi }}_{n,m}\left(\kappa \right)={\tilde{C}}^T\Phi \left(\kappa \right),$$

(9)

where $\tilde{C}$ and $\Phi \left(\kappa \right)$ are vectors, each of dimension $(2^{k-1}M\times 1)$, and are given by

$$\tilde{C}={[{\tilde{c}}_{1,0},{\tilde{c}}_{1,1},\dots ,{\tilde{c}}_{1,M-1},...,{\tilde{c}}_{2^{k-1},0},{\tilde{c}}_{2^{k-1},1},...,{\tilde{c}}_{2^{k-1},M-1}]}^T,$$

(10)

and

$$\Phi \left(\kappa \right)={[{\tilde{\varphi }}_{1,0}\left(\kappa \right),{\tilde{\varphi }}_{1,1}\left(\kappa \right),\dots ,{\tilde{\varphi }}_{1,M-1}\left(\kappa \right),...,{\tilde{\varphi }}_{2^{k-1},0}\left(\kappa \right),{\tilde{\varphi }}_{2^{k-1},1}\left(\kappa \right),...,{\tilde{\varphi }}_{2^{k-1},M-1}\left(\kappa \right)]}^T.$$

(11)

The coefficient vector $\tilde{C}$ can be derived from Equation (9) in the following manner:

$${\tilde{C}}^T=\left({\int }_0^1\Phi \left(\kappa \right)\upsilon \left(\kappa \right)d\kappa \right){\tilde{D}}^{-1},$$

(12)

where $\tilde{D}$ is a square matrix of size $2^{k-1}M$ and is defined as

$$\tilde{D}={\int }_0^1\Phi \left(\kappa \right){\Phi }^T\left(\kappa \right)d\kappa .$$

(13)

Similarly, the kernel function $k(\kappa ,\varrho )$ can be approximated in the following way:

$$k(\kappa ,\varrho )\approx {\Phi }^T\left(\kappa \right)\tilde{\omega }\Phi \left(\varrho \right),$$

(14)

where $\tilde{\omega }$ is a square matrix of size $2^{k-1}M$ and is defined as

$$\tilde{\omega }={\tilde{D}}^{-1}\left({\int }_0^1\Phi \left(\varrho \right)\left({\int }_0^{1}k(\kappa ,\varrho ){\Phi }^T\left(\kappa \right)d\kappa \right)d\varrho \right){\tilde{D}}^{-1}.$$

(15)

3.2. 2D-EWs

The 2D-EWs can be represented as a multiplication of 1D-EWs in the following way:

$$\begin{array}{ccc}\hfill {\tilde{\varphi }}_{n,m,n_1,m_1}(\kappa ,\varrho )& =& \left\{\begin{array}{cc}{\tilde{\varphi }}_{n,m}\left(\kappa \right){\tilde{\varphi }}_{n_1,m_1}\left(\varrho \right),\hfill & {\displaystyle \frac{n-1}{2^{k-1}}}\le \kappa <{\displaystyle \frac{n}{2^{k-1}}},{\displaystyle \frac{m_1-1}{2^{h-1}}}\le \varrho <{\displaystyle \frac{m_1}{2^{h-1}}},\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\hfill \end{array}$$

(16)

where h is the dilation level for $\varrho$, $n_1=1,2,\dots ,2^{h-1}$, and $m_1=0,1,\dots ,M_1-1$. It is easy to verify that ${\tilde{\varphi }}_{n,m}\left(\kappa \right){\tilde{\varphi }}_{n_1,m_1}\left(\varrho \right)$ form an orthonormal set over $[0,1)\times [0,1)$. Similarly, any two-variable function $\upsilon (\kappa ,\varrho )$ defined over $[0,1)\times [0,1)$ can be represented using a basis of 2D-EWs as follows:

$$\upsilon (\kappa ,\varrho )\approx \sum _{n=1}^{2^{k-1}}\sum _{m=0}^{M-1}\sum _{n_1=1}^{2^{h-1}}\sum _{m_1=0}^{M_1-1}{c}_{n,m,n_1,m_1}{\tilde{\varphi }}_{n,m}\left(\kappa \right){\tilde{\varphi }}_{n_1,m_1}\left(\varrho \right)=V^T\Phi (\kappa ,\varrho ),$$

(17)

where V is a $2^{k-1}M\times 2^{h-1}{M}_1$ coefficient matrix. $\Phi (\kappa ,\varrho )$ is given by

$$\begin{array}{ccc}\hfill \Phi (\kappa ,\varrho )& =& [{\tilde{\varphi }}_{1,0,1,0}(\kappa ,\varrho ),{\tilde{\varphi }}_{1,0,1,1}(\kappa ,\varrho ),\dots ,{\tilde{\varphi }}_{1,0,2^{h-1},M_1-1}(\kappa ,\varrho ),{\tilde{\varphi }}_{1,1,1,0}(\kappa ,\varrho ),\dots ,{\tilde{\varphi }}_{1,1,2^{h-1},M_1-1}(\kappa ,\varrho ),\hfill \\ & & \dots ,{\tilde{\varphi }}_{2^{k-1},M-1,1,0}(\kappa ,\varrho ),\dots ,{\tilde{\varphi }}_{2^{k-1},M-1,2^{h-1},M_1-1}(\kappa ,\varrho ){]}^T.\hfill \end{array}$$

(18)

We can express $\Phi (\kappa ,\varrho )$ in the following form:

$$\Phi (\kappa ,\varrho )=\Phi \left(\kappa \right)\otimes \Phi \left(\varrho \right),$$

(19)

where ⊗ is the Kronecker product. Using the approximating formula,

$$\begin{array}{ccc}\hfill {\left[v(\kappa ,\varrho )\right]}^2& \approx & \Phi (\kappa ,\varrho )V{\Phi }^T(\kappa ,\varrho )=C_2^T\Phi (\kappa ,\varrho ),\hfill \end{array}$$

and we can calculate an approximation to ${\left[v(\kappa ,\varrho )\right]}^p$ as follows:

$$\begin{array}{ccc}\hfill {\left[v(\kappa ,\varrho )\right]}^p& \approx & [\Phi (\kappa ,\varrho )V]\left[{\Phi }^T(\kappa ,\varrho )C_{p-1}\right]=C_p^T\Phi (\kappa ,\varrho ),\hfill \end{array}$$

where $C_p$ is the coefficient vector for ${\left[\upsilon (\kappa ,\varrho )\right]}^p$, which is computed iteratively.

4. OMs

Here, we will derive the OMs of EWs by using the following identity:

$${\int }_0^{\kappa }\Phi \left(t\right)dt={\tau }_{\kappa }\Phi \left(\kappa \right),$$

(20)

where ${\tau }_{\kappa }$ is a square integration matrix of size $2^{k-1}M\times 2^{k-1}M$, and $\Phi \left(\kappa \right)$ is the EWs vector for $0<\alpha <1$ as defined by Equation (11). The OMI based on 2D-EWs with respect to the variable $\kappa$ can be obtained from Equation (20) as follows:

$$\begin{array}{ccc}\hfill {\int }_0^{\kappa }\Phi (x,\varrho )dx& =& {\int }_0^{\kappa }(\Phi \left(x\right)\otimes \Phi \left(\varrho \right))dx=({\int }_0^{\kappa }\Phi \left(x\right)dx)\otimes \Phi \left(\varrho \right)\hfill \\ & =& [{\tau }_{\kappa }\Phi \left(\kappa \right)]\otimes [I\Phi \left(\varrho \right)]=({\tau }_{\kappa }\otimes I)[\Phi \left(\kappa \right)\otimes \Phi \left(\varrho \right)]={\widehat{\tau }}_{\kappa }\Phi (\kappa ,\varrho ),\hfill \end{array}$$

(21)

where I is the identity matrix, and ${\widehat{\tau }}_{\kappa }={\tau }_{\kappa }\otimes I$ is a square matrix of size $2^{k-1}{2}^{h-1}M{M}_1\times 2^{k-1}{2}^{h-1}M{M}_1$. In a similar way, we find that

$$\begin{array}{ccc}\hfill {\int }_0^{\varrho }\Phi (\kappa ,y)dy& =& {\int }_0^{\varrho }(\Phi \left(\kappa \right)\otimes \Phi \left(y\right))dy=\Phi \left(\kappa \right)\otimes ({\int }_0^{\varrho }\Phi \left(y\right)dy)\hfill \\ & =& [I\Phi \left(\kappa \right)]\otimes [{\tau }_{\varrho }\Phi \left(\varrho \right)]=(I\otimes {\tau }_{\varrho })[\Phi \left(\kappa \right)\otimes \Phi \left(\varrho \right)]={\widehat{\tau }}_{\varrho }\Phi (\kappa ,\varrho ),\hfill \end{array}$$

(22)

where ${\widehat{\tau }}_{\varrho }=I\otimes {\tau }_{\varrho }$ is a square matrix of size $2^{k-1}{2}^{h-1}M{M}_1\times 2^{k-1}{2}^{h-1}M{M}_1$. Combining the above results yields

$$\begin{array}{ccc}\hfill {\int }_0^{\kappa }{\int }_0^{\varrho }\Phi (x,y)dxdy& =& {\int }_0^{\kappa }{\int }_0^{\varrho }(\Phi \left(x\right)\otimes \Phi \left(y\right))dxdy=({\int }_0^{\kappa }\Phi \left(x\right)dx)\otimes ({\int }_0^{\varrho }\Phi \left(y\right)dy)\hfill \\ & =& [{\tau }_{\kappa }\Phi \left(\kappa \right)]\otimes [{\tau }_{\varrho }\Phi \left(\varrho \right)]=({\tau }_{\kappa }\otimes {\tau }_{\varrho })[\Phi \left(\kappa \right)\otimes \Phi \left(\varrho \right)]\hfill \\ & =& {\widehat{\tau }}_{\kappa \varrho }\Phi (\kappa ,\varrho ),\hfill \end{array}$$

(23)

where ${\widehat{\tau }}_{\kappa \varrho }={\tau }_{\kappa }\otimes {\tau }_{\varrho }$. For integrals with respect to $\kappa$ involving a singular kernel, we can express them as follows:

$${\int }_0^{\kappa }{\displaystyle \frac{\Phi \left(t\right)}{{(\kappa -t)}^{\alpha }}}dt={\rho }_{\kappa }\Phi \left(\kappa \right),$$

(24)

where ${\rho }_{\kappa }$ is the $(2^{k-1}M\times 2^{k-1}M)$-dimensional OMI with singularity given by

$${\rho }_{\kappa }=\left({\int }_0^1\left({\int }_0^{\kappa }{\displaystyle \frac{\Phi \left(t\right)}{{(\kappa -t)}^{\alpha }}}dt\right){\Phi }^T\left(t\right)dt\right){\tilde{D}}^{-1}.$$

(25)

Similarly, for $\Phi (\kappa ,y)$, we obtain

$$\begin{array}{ccc}\hfill {\int }_0^{\kappa }{\displaystyle \frac{\Phi (s,y)}{{(\kappa -s)}^{\alpha }}}ds& =& {\int }_0^{\kappa }{\displaystyle \frac{\Phi \left(s\right)\otimes \Phi \left(y\right)}{{(\kappa -s)}^{\alpha }}}ds={\int }_0^{\kappa }{\displaystyle \frac{\Phi \left(s\right)}{{(\kappa -s)}^{\alpha }}}ds\otimes \Phi \left(y\right)\hfill \\ & =& [{\rho }_{\kappa }\Phi \left(\kappa \right)]\otimes [I\Phi \left(y\right)]=({\rho }_{\kappa }\otimes I)[\Phi \left(\kappa \right)\otimes \Phi \left(y\right)]={\tilde{\rho }}_{\kappa }\Phi (\kappa ,y),\hfill \end{array}$$

(26)

where ${\tilde{\rho }}_{\kappa }={\rho }_{\kappa }\otimes I$ is a square OM of size $2^{k-1}{2}^{h-1}M{M}_1\times 2^{k-1}{2}^{h-1}M{M}_1$. Furthermore, we define the OM for the product of two 2D-EWs vectors, $\tilde{R}$, to be the $2^{k-1}{2}^{h-1}M{M}_1\times 2^{k-1}{2}^{h-1}M{M}_1$ matrix that satisfies the following relation:

$$\Phi (\kappa ,\varrho ){\Phi }^T(\kappa ,\varrho )R=\tilde{R}\Phi (\kappa ,\varrho ).$$

(27)

Also,

$$\Phi (\kappa ,\varrho )D{\Phi }^T(\kappa ,\varrho )=\widehat{D}\Phi (\kappa ,\varrho ),$$

(28)

where $\widehat{D}$ is an $M{M}_1$-vector with elements equal to the diagonal entries of the matrix D.

The following derivations focus on constructing OMs for differentiation using 2D-EWs, which are essential for handling the partial derivatives in the 2DFWSPVIE. Let $\Phi (\kappa ,\varrho )$ be the one-dimensional vector of EWs as defined by Equation (18). Its associated derivative matrix, $D_{\kappa }$, can be derived as follows:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial }{\partial \kappa }}\Phi (\kappa ,\varrho )& =& {\displaystyle \frac{\partial }{\partial \kappa }}\Phi \left(\kappa \right)\otimes \Phi \left(\varrho \right)\hfill \\ & =& [D\Phi (\kappa \left)\right]\otimes [I\Phi (\varrho \left)\right]\hfill \\ & =& (D\otimes I)[\Phi \left(\kappa \right)\otimes \Phi \left(\varrho \right)]=D_{\kappa }\Phi (\kappa ,\varrho ),\hfill \end{array}$$

(29)

where $D_{\kappa }=D\otimes I$. Similarly, we can readily write

$${\displaystyle \frac{\partial }{\partial \varrho }}\Phi (\kappa ,\varrho )=D_{\varrho }\Phi (\kappa ,\varrho ),$$

(30)

where $D_{\varrho }=I\otimes D$.

5. The EWOMM

To solve the nonlinear 2DFPVIE (1), we need to approximate the functions ${\upsilon }_{\kappa ,\varrho }(\kappa ,\varrho )$, ${\upsilon }_{\varrho }(\kappa ,\varrho )$, $\upsilon (\kappa ,\varrho )$, $g(\kappa ,\varrho )$, $\upsilon (\kappa ,0)$, and $\upsilon (0,\varrho )$ in terms of 2D-EWs as follows:

$${\left[\upsilon (\kappa ,\varrho )\right]}^P\approx {\Phi }^T(\kappa ,\varrho )C_p,$$

(31)

$${\upsilon }_{\kappa ,\varrho }(\kappa ,\varrho )\approx C^T\Phi (\kappa ,\varrho ),$$

(32)

$$g(\kappa ,\varrho )\approx G^T\Phi (\kappa ,\varrho ),$$

(33)

$${\upsilon }_{\varrho }(0,\varrho )\approx Z^T\Phi (0,\varrho ),$$

(34)

$$\upsilon (\kappa ,0)\approx U^T\Phi (\kappa ,0),$$

(35)

$$\upsilon (0,\varrho )\approx V^T\Phi (0,\varrho ),$$

(36)

$$\upsilon (0,0)\approx D^T\Phi (0,0).$$

(37)

By integrating both sides of Equation (32) with respect to $\kappa$ and using Equation (34), we have

$$\begin{array}{ccc}\hfill {\upsilon }_{\varrho }(\kappa ,\varrho )& \approx & {\upsilon }_{\varrho }(0,\varrho )+C^T{\int }_0^{\varrho }\Phi (t,\varrho )dt\hfill \\ & =& (Z^T+C^T{\widehat{\tau }}_{\varrho })\Phi (\kappa ,\varrho )=Q^T\Phi (\kappa ,\varrho ),\hfill \end{array}$$

(38)

where $Q^T=Z^T+C^T{\widehat{\tau }}_{\varrho }$. Similarly, by integrating both sides of Equation (38) with respect to $\varrho$ and using Equations (23) and (35)–(37), we have

$$\begin{array}{ccc}\hfill \upsilon (\kappa ,\varrho )& \approx & \upsilon (\kappa ,0)+\upsilon (0,\varrho )-\upsilon (0,0)+C^T{\int }_0^{\kappa }{\int }_0^{\varrho }\Phi (t,s)dsdt\hfill \\ & \approx & U^T\Phi (\kappa ,\varrho )+V^T\Phi (\kappa ,\varrho )-D^T\Phi (\kappa ,\varrho )+C^T{\widehat{\tau }}_{\kappa ,\varrho }\Phi (\kappa ,\varrho )\hfill \\ & =& R^T\Phi (\kappa ,\varrho ),\hfill \end{array}$$

(39)

where $R^T=U^T+V^T-D^T+C^T{\widehat{\tau }}_{\kappa \varrho }$. By differentiating both sides of Equation (38) with respect to $\varrho$, we get

$${\upsilon }_{\varrho \varrho }(\kappa ,\varrho )\approx Q^T{\Phi }_{\varrho }(\kappa ,\varrho )=Q^T{D}_{\varrho }\Phi (\kappa ,\varrho ).$$

(40)

Now, we calculate the integral part as follows:

$$\begin{array}{ccc}& & {\int }_0^{\kappa }{\int }_0^{\varrho }{\displaystyle \frac{k(\kappa ,\varrho ,s,y){\left[\upsilon (s,y)\right]}^p}{{(\kappa -s)}^{1-{\gamma }_1}{(\varrho -y)}^{1-{\gamma }_2}}}dyds\hfill \\ & \approx & {\int }_0^{\kappa }{\int }_0^{\varrho }{(\kappa -s)}^{{\gamma }_1-1}{(\varrho -y)}^{{\gamma }_2-1}{\Phi }^T(\kappa ,\varrho )K\Phi (s,y){\Phi }^T(s,y)C_{p}dyds\hfill \\ & =& {\Phi }^T(\kappa ,\varrho )K{\int }_0^{\kappa }{\int }_0^{\varrho }{(\kappa -s)}^{{\gamma }_1-1}{(\varrho -y)}^{{\gamma }_2-1}\Phi (s,y){\Phi }^T(s,y)C_{p}dyds\hfill \\ & =& {\Phi }^T(\kappa ,\varrho )K\widehat{C_p}{\int }_0^{\kappa }{\int }_0^{\varrho }{(\kappa -s)}^{{\gamma }_1-1}{(\varrho -y)}^{{\gamma }_2-1}\Phi (s,y)dyds\hfill \\ & =& \Gamma \left({\gamma }_1\right)\Gamma \left({\gamma }_2\right){\Phi }^T(\kappa ,\varrho )K\widehat{C_p}\left[I^{{\gamma }_1,{\gamma }_2}\Phi (\kappa ,\varrho )\right]\hfill \\ & =& \Gamma \left({\gamma }_1\right)\Gamma \left({\gamma }_2\right){\Phi }^T(\kappa ,\varrho )K\widehat{C_p}{F}^{{\gamma }_1,{\gamma }_2}\Phi (\kappa ,\varrho )\hfill \\ & =& \Gamma \left({\gamma }_1\right)\Gamma \left({\gamma }_2\right){\tilde{\mathsf{\Omega }}}^T\Phi (\kappa ,\varrho ),\hfill \end{array}$$

where $\tilde{\mathsf{\Omega }}=\mathrm{diag}\left(K\widehat{C_p}{F}^{{\gamma }_1,{\gamma }_2}\right)$ is an m-vector. The OMI for mixed variables is obtained by evaluating the following:

$$\begin{array}{ccc}\hfill & & {\int }_0^{\kappa }{\int }_0^{\varrho }{\displaystyle \frac{{\rm Y}\left(\upsilon \right(s,y\left)\right)}{{(\kappa -s)}^{\alpha }}}dyds={\int }_0^{\kappa }{\int }_0^{\varrho }{\displaystyle \frac{{\left[\upsilon (s,y)\right]}^P}{{(\kappa -s)}^{\alpha }}}dyds\approx {\int }_0^{\kappa }{\int }_0^{\varrho }{\displaystyle \frac{C_P^T\Phi (s,y)}{{(\kappa -s)}^{\alpha }}}dyds\hfill \\ & =& C_P^T{\int }_0^{\kappa }{\int }_0^{\varrho }{\displaystyle \frac{\Phi \left(s\right)\otimes \Phi \left(y\right)}{{(\kappa -s)}^{\alpha }}}dyds=C_P^T\left({\int }_0^{\kappa }{\displaystyle \frac{\Phi \left(s\right)}{{(\kappa -s)}^{\alpha }}}ds\right)\otimes \left({\int }_0^{\varrho }\Phi \left(y\right)dy\right)\hfill \\ & =& C_P^T[{\tilde{\rho }}_{\kappa }\Phi \left(\kappa \right)\otimes ({\tau }_{\varrho }\Phi \left(\varrho \right))]=C_P^T({\tilde{\rho }}_{\kappa }\otimes {\tau }_{\varrho })[\Phi \left(\kappa \right)\otimes \Phi \left(\varrho \right)]=C_P^{T}L\Phi (\kappa ,\varrho ),\hfill \end{array}$$

(41)

where $L={\tilde{\rho }}_{\kappa }\otimes {\tau }_{\varrho }$ is the nearly OMI with respect to mixed variables. Inserting Equations (32)–(41) into Equation (1) yields

$$C^T\Phi (\kappa ,\varrho )+Q^T\Phi (\kappa ,\varrho )\approx R^T\Phi (\kappa ,\varrho )+G^T\Phi (\kappa ,\varrho )+\Gamma \left({\gamma }_1\right)\Gamma \left({\gamma }_2\right){\tilde{\mathsf{\Omega }}}^T\Phi (\kappa ,\varrho )+C_P^{T}L\Phi (\kappa ,\varrho ),$$

(42)

which directly implies that

$$C^T+Q^T\approx R^T+G^T+\Gamma \left({\gamma }_1\right)\Gamma \left({\gamma }_2\right){\tilde{\mathsf{\Omega }}}^T+C_P^{T}L.$$

(43)

As an approximate nonlinear system of algebraic equations, Equation (43) can be solved by using Newton’s or other iterative techniques. Consequently, an approximate solution can be computed for Equation (1) as follows:

$$\upsilon (\kappa ,\varrho )\approx C^T\Phi (\kappa ,\varrho ).$$

(44)

6. Numerical Examples

The accuracy and applicability of the proposed EWs OM technique are evaluated in this section for a FWSPVIE through several numerical examples. To solve the resulting nonlinear system of algebraic equations, we employed the Newton–Raphson method, implemented in Maple 2018 software, which provides robust and efficient convergence for the numerical computations.

Example 1.

Consider the following 2DFWSPVIE:

$$\begin{array}{ccc}\hfill {\upsilon }_{\kappa ,\varrho }(\kappa ,\varrho )+{\upsilon }_{\varrho }(\kappa ,\varrho )& =& \upsilon (\kappa ,\varrho )+g(\kappa ,\varrho )+{\displaystyle \frac{1}{\Gamma \left(\frac{3}{2}\right)\Gamma \left(\frac{5}{2}\right)}}{\int }_0^{\kappa }{\int }_0^{\varrho }{(\kappa -s)}^{{\displaystyle \frac{1}{2}}}{(\varrho -y)}^{{\displaystyle \frac{3}{2}}}{\left[\upsilon (s,y)\right]}^{2}dyds\hfill \\ & & +{\int }_0^{\kappa }{\int }_0^{\varrho }{\displaystyle \frac{{\left(\upsilon (s,y)\right)}^2}{{(\kappa -s)}^{\frac{1}{2}}}}dyds,\hfill \end{array}$$

(45)

with

$$\begin{array}{c}\hfill g(\kappa ,\varrho )=6{\kappa }^2\varrho +2{\kappa }^3\varrho -{\kappa }^3{\varrho }^2-{\displaystyle \frac{4194304}{2029052025}}{\displaystyle \frac{{\kappa }^{\frac{15}{2}}{\varrho }^{\frac{13}{2}}}{\pi }}-{\displaystyle \frac{131072}{1616615}}{\kappa }^{{\displaystyle \frac{19}{2}}}{\varrho }^7.\end{array}$$

(46)

The exact solution for this problem is defined as $\upsilon (\kappa ,\varrho )={\kappa }^3{\varrho }^2$, with the initial conditions $\upsilon (\kappa ,0)=\upsilon (0,\varrho )=0$. The results obtained by the EWOMM are presented in

Table 1. Absolute errors at different points for Example 1.

$(\mathit{\kappa },\mathit{\varrho })$	$\mathit{n}=3$	$\mathit{n}=5$	$\mathit{n}=7$
$(0.1,0.1)$	$2.851414\times {10}^{-4}$	$2.654203\times {10}^{-4}$	$5.521303\times {10}^{-7}$
$(0.2,0.2)$	$3.382253\times {10}^{-4}$	$1.753124\times {10}^{-5}$	$1.230241\times {10}^{-7}$
$(0.3,0.3)$	$1.752410\times {10}^{-4}$	$9.030514\times {10}^{-6}$	$3.092219\times {10}^{-6}$
$(0.4,0.4)$	$1.202411\times {10}^{-3}$	$1.175424\times {10}^{-5}$	$3.124385\times {10}^{-6}$
$(0.5,0.5)$	$2.368754\times {10}^{-4}$	$8.002522\times {10}^{-6}$	$2.124924\times {10}^{-7}$
$(0.6,0.6)$	$7.612482\times {10}^{-4}$	$1.655208\times {10}^{-7}$	$2.020188\times {10}^{-7}$
$(0.7,0.7)$	$2.767177\times {10}^{-3}$	$1.575447\times {10}^{-4}$	$1.555310\times {10}^{-6}$
$(0.8,0.8)$	$1.754526\times {10}^{-5}$	$3.975402\times {10}^{-6}$	$3.444507\times {10}^{-7}$
$(0.9,0.9)$	$1.414210\times {10}^{-4}$	$1.432500\times {10}^{-6}$	$9.619903\times {10}^{-7}$

and Figure 1.

Example 2.

Consider the following 2DFWSPVIE:

$$\begin{array}{ccc}\hfill {\upsilon }_{\kappa ,\varrho }(\kappa ,\varrho )+{\upsilon }_{\varrho }(\kappa ,\varrho )& =& \upsilon (\kappa ,\varrho )+g(\kappa ,\varrho )+{\displaystyle \frac{1}{\Gamma \left(\frac{3}{2}\right)\Gamma \left(\frac{3}{2}\right)}}{\int }_0^{\kappa }{\int }_0^{\varrho }{(\kappa -s)}^{{\displaystyle \frac{1}{2}}}{(\varrho -y)}^{{\displaystyle \frac{1}{2}}}\sqrt{\varrho }\left[\upsilon (s,y)\right]dyds\hfill \\ & & +{\int }_0^{\kappa }{\int }_0^{\varrho }{\displaystyle \frac{{\left(\upsilon (s,y)\right)}^2}{{(\kappa -s)}^{\frac{1}{2}}}}dyds,\hfill \end{array}$$

(47)

with

$$\begin{array}{ccc}\hfill g(\kappa ,\varrho )& =& 9{\varrho }^2-{\kappa }^2-3{\varrho }^3-{\displaystyle \frac{4}{\pi }}\left({\displaystyle \frac{64}{315}}{\kappa }^{{\displaystyle \frac{3}{2}}}{\varrho }^{{\displaystyle \frac{9}{2}}}+{\displaystyle \frac{32}{315}}{\kappa }^{{\displaystyle \frac{7}{2}}}{\varrho }^{{\displaystyle \frac{3}{2}}}\right)+{\displaystyle \frac{27}{5}}\sqrt{\kappa }{\varrho }^{10}+{\displaystyle \frac{144}{35}}{\kappa }^{{\displaystyle \frac{5}{2}}}{\varrho }^7\hfill \\ & & +{\displaystyle \frac{64}{35}}{\kappa }^{{\displaystyle \frac{9}{2}}}{\varrho }^4+{\displaystyle \frac{2018}{3003}}{\kappa }^{{\displaystyle \frac{13}{2}}}\varrho .\hfill \end{array}$$

(48)

The exact solution is given by $\upsilon (\kappa ,\varrho )={\kappa }^2+3{\varrho }^3$, with the initial conditions $\upsilon (\kappa ,0)={\kappa }^2$ and $\upsilon (0,\varrho )=3{\varrho }^3$. The numerical results obtained by the EWOMM are presented in

Table 2. Absolute errors at different points for Example 2.

$(\mathit{\kappa },\mathit{\varrho })$	$\mathit{n}=3$	$\mathit{n}=5$	$\mathit{n}=7$
$(0.1,0.1)$	$7.412548\times {10}^{-4}$	$2.344124\times {10}^{-6}$	$1.554279\times {10}^{-7}$
$(0.2,0.2)$	$3.274120\times {10}^{-5}$	$9.354202\times {10}^{-5}$	$1.466565\times {10}^{-6}$
$(0.3,0.3)$	$4.357541\times {10}^{-4}$	$6.323414\times {10}^{-5}$	$7.965650\times {10}^{-7}$
$(0.4,0.4)$	$2.275424\times {10}^{-3}$	$1.541452\times {10}^{-4}$	$1.215450\times {10}^{-7}$
$(0.5,0.5)$	$6.220225\times {10}^{-4}$	$7.341201\times {10}^{-5}$	$1.175934\times {10}^{-7}$
$(0.6,0.6)$	$2.254196\times {10}^{-3}$	$7.003102\times {10}^{-5}$	$2.974121\times {10}^{-7}$
$(0.7,0.7)$	$1.234568\times {10}^{-4}$	$5.362101\times {10}^{-6}$	$5.258462\times {10}^{-7}$
$(0.8,0.8)$	$3.555210\times {10}^{-5}$	$6.321210\times {10}^{-5}$	$2.810301\times {10}^{-7}$
$(0.9,0.9)$	$1.254780\times {10}^{-4}$	$8.327415\times {10}^{-5}$	$6.284484\times {10}^{-8}$

and Figure 2.

Example 3.

Consider the following 2DFWSPVIE:

$$\begin{array}{ccc}\hfill {\upsilon }_{\kappa ,\varrho }(\kappa ,\varrho )+{\upsilon }_{\varrho }(\kappa ,\varrho )& =& \upsilon (\kappa ,\varrho )+g(\kappa ,\varrho )+{\displaystyle \frac{1}{\Gamma \left(\frac{7}{2}\right)\Gamma \left(\frac{11}{2}\right)}}{\int }_0^{\kappa }{\int }_0^{\varrho }{(\kappa -s)}^{{\displaystyle \frac{5}{2}}}{(\varrho -y)}^{{\displaystyle \frac{9}{2}}}{\left[\upsilon (s,y)\right]}^{2}dyds\hfill \\ & & +{\int }_0^{\kappa }{\int }_0^{\varrho }{\displaystyle \frac{{\left(\upsilon (s,y)\right)}^2}{{(\kappa -s)}^{\frac{1}{2}}}}dyds,\hfill \end{array}$$

(49)

with

$$\begin{array}{ccc}\hfill g(\kappa ,\varrho )& =& 6{\kappa }^2\varrho -6\kappa {\varrho }^2+2{\kappa }^3{\varrho }^2-3{\kappa }^2{\varrho }^2-{\kappa }^3{\varrho }^2+{\kappa }^2{\varrho }^3+{\displaystyle \frac{16}{15}}\left(-{\displaystyle \frac{512}{315315}}{\kappa }^{{\displaystyle \frac{7}{2}}}{\varrho }^{{\displaystyle \frac{13}{2}}}+{\displaystyle \frac{512}{218295}}{\kappa }^{{\displaystyle \frac{9}{2}}}{\varrho }^{{\displaystyle \frac{11}{2}}}\right)\hfill \\ & & +{\displaystyle \frac{256}{2205}}{\kappa }^{{\displaystyle \frac{9}{2}}}{\varrho }^7-{\displaystyle \frac{512}{2079}}{\kappa }^{{\displaystyle \frac{11}{2}}}{\varrho }^6+{\displaystyle \frac{2048}{15015}}{\kappa }^{{\displaystyle \frac{13}{2}}}{\varrho }^5.\hfill \end{array}$$

(50)

The exact solution is given by $\upsilon (\kappa ,\varrho )={\kappa }^3{\varrho }^2-{\kappa }^2{\varrho }^3$, which is subject to the initial conditions $\upsilon (\kappa ,0)=\upsilon (0,\varrho )=0$. The results of the EWOMM are presented in

Table 3. Absolute errors at different points for Example 3.

$(\mathit{\kappa },\mathit{\varrho })$	$\mathit{n}=4$	$\mathit{n}=6$	$\mathit{n}=8$
$(0.1,0.1)$	$3.852645\times {10}^{-4}$	$8.523980\times {10}^{-5}$	$9.247079\times {10}^{-7}$
$(0.2,0.2)$	$4.745700\times {10}^{-6}$	$6.521402\times {10}^{-7}$	$4.124022\times {10}^{-8}$
$(0.3,0.3)$	$8.427848\times {10}^{-5}$	$1.056514\times {10}^{-6}$	$1.074554\times {10}^{-7}$
$(0.4,0.4)$	$3.226650\times {10}^{-5}$	$2.675404\times {10}^{-6}$	$3.255147\times {10}^{-7}$
$(0.5,0.5)$	$1.362554\times {10}^{-5}$	$9.750251\times {10}^{-7}$	$6.987955\times {10}^{-8}$
$(0.6,0.6)$	$4.766412\times {10}^{-5}$	$3.868508\times {10}^{-6}$	$3.137858\times {10}^{-7}$
$(0.7,0.7)$	$8.524020\times {10}^{-4}$	$7.425025\times {10}^{-5}$	$6.325110\times {10}^{-8}$
$(0.8,0.8)$	$4.354480\times {10}^{-5}$	$5.635250\times {10}^{-6}$	$6.741401\times {10}^{-7}$
$(0.9,0.9)$	$3.772410\times {10}^{-4}$	$9.115015\times {10}^{-5}$	$7.607223\times {10}^{-7}$

and Figure 3.

Example 4.

As our last example, consider the following 2DFWSPVIE:

$$\begin{array}{ccc}\hfill {\upsilon }_{\kappa ,\varrho }(\kappa ,\varrho )+{\upsilon }_{\varrho }(\kappa ,\varrho )& =& \upsilon (\kappa ,\varrho )+g(\kappa ,\varrho )+{\int }_0^{\kappa }{\int }_0^{\varrho }{\displaystyle \frac{{\left[\upsilon (s,y)\right]}^2}{{(\kappa -s)}^{\frac{1}{2}}}}dyds,\hfill \end{array}$$

(51)

with

$$g(\kappa ,\varrho )=2\varrho -{\kappa }^2-{\varrho }^2-{\displaystyle \frac{2}{315}}{\kappa }^{{\displaystyle \frac{1}{2}}}\varrho (128{\kappa }^4+112{\kappa }^2{\varrho }^2+63{\varrho }^4).$$

(52)

The exact solution is given by $\upsilon (\kappa ,\varrho )={\kappa }^2+{\varrho }^2,$ with the initial conditions $\upsilon (\kappa ,0)={\kappa }^2$ and $\upsilon (0,\varrho )={\varrho }^2.$ The comparison of the absolute error function obtained from Bernstein polynomials, Triangular functions, Block pulse, and EWOMM is presented in

Table 4. Comparison of the absolute error for Example 4.

$(\mathit{\kappa },\mathit{\varrho })$	$\mathit{n}=3$
	BPs [48]	BPs [49]	TFs [50]	EWs
$(0.1,0.1)$	$2.1711\times {10}^{-2}$	$6.6482\times {10}^{-6}$	$2.2548\times {10}^{-5}$	$6.3317\times {10}^{-5}$
$(0.2,0.2)$	$3.8287\times {10}^{-2}$	$2.5361\times {10}^{-5}$	$6.3980\times {10}^{-6}$	$9.2574\times {10}^{-5}$
$(0.3,0.3)$	$4.5661\times {10}^{-2}$	$2.6856\times {10}^{-4}$	$7.2902\times {10}^{-6}$	$2.5855\times {10}^{-7}$
$(0.4,0.4)$	$1.0225\times {10}^{-1}$	$1.3289\times {10}^{-3}$	$2.0110\times {10}^{-5}$	$7.7440\times {10}^{-7}$
$(0.5,0.5)$	$1.0038\times {10}^{-1}$	$4.0791\times {10}^{-3}$	$2.0214\times {10}^{-4}$	$8.1412\times {10}^{-5}$
$(0.6,0.6)$	$3.1808\times {10}^{-1}$	$9.3099\times {10}^{-3}$	$2.0240\times {10}^{-4}$	$1.3996\times {10}^{-6}$
$(0.7,0.7)$	$5.7761\times {10}^{-1}$	$1.7106\times {10}^{-2}$	$6.0217\times {10}^{-2}$	$1.1214\times {10}^{-4}$
$(0.8,0.8)$	$4.8987\times {10}^{-1}$	$2.5961\times {10}^{-2}$	$3.3107\times {10}^{-3}$	$5.8510\times {10}^{-3}$
$(0.9,0.9)$	$8.2986\times {10}^{-1}$	$3.1626\times {10}^{-2}$	$0.4748\times {10}^{-4}$	$3.5214\times {10}^{-4}$

.

7. Conclusions and Future Directions

The current study examines the application of EWs in solving a nonlinear fractional two-dimensional partial Volterra integral equation. The technique offers advantages such as reduced costs for setting up the system of equations, without the need for projection techniques like collocation or Galerkin methods, resulting in very low computational expenses. This cost-effectiveness and simplicity are key benefits of the technique from a computational standpoint. The applicability and accuracy of the method were confirmed through several examples. The numerical results demonstrate the high accuracy of the obtained solutions. Moreover, by increasing the parameter n, the technique can be fine-tuned until achieving the desired level of accuracy.

This paper focuses on 2D problems, but many real-world applications, such as those in fluid dynamics or quantum mechanics, involve higher dimensions. Future work could explore the scalability of the method and the computational complexity of constructing OMs for higher-dimensional systems. Another future research is to extend the EWOMM to handle other types of kernels, such as strongly singular kernels, to broaden its applicability. Furthermore, extending the EWOMM to other fractional differential or integral equations could improve its utility in modeling complex systems with memory effects, such as those encountered in viscoelasticity or anomalous diffusion.