An Iterative Approximate Method for Solving 2D Weakly Singular Fredholm Integral Equations of the Second Kind

Abstract

This work aims to propose an iterative method for approximating solutions of two-dimensional weakly singular Fredholm integral Equation (2D-WSFIE) by incorporating the product integration technique, an appropriate cubature formula, and the Picard algorithm. This iterative approach is utilized to approximate the solution of the 2D-WSFIE that arises in some contact problems in linear elasticity. Under some sufficient conditions, the existence and uniqueness of the solution are established, an error bound for the approximate solution is estimated, and the order of convergence of the proposed algorithm is discussed. The effectiveness of the proposed method is illustrated through its application to some contact problems involving weakly singular kernels.

1. Introduction

A 2D integral Equation (2D-IE) is an equation that involves an unknown function depending on two independent variables arising within at least one integral sign. This type of equation is rapidly evolving and has many applications, particularly in elasticity. As a result, it is discussed and investigated by many authors. Some authors focus on the sufficient conditions that imply the existence of their solution in certain spaces of functions, while others focus on developing methods to approximate their unknown solution. See [1,2,3] and the references therein.

The integral equation of the Fredholm type (FIE) is commonly encountered in various industrial sectors, including signal processing, inverse problems, linear forward modeling in fluid mechanics, hydrodynamic interactions in elastic interfaces with finite size, radiative transfer processes, and neutron transport theory. For example, the Chandrasekhar H-function, which is defined in terms of FIE, finds applications in modeling radiation transport and photo-electron transport, see [4] for more details. Additionally, Fredholm equations are utilized in computer graphics to simulate light transport from virtual light sources to the image plane for generating photo-realistic images. Moreover, solving Fredholm equations enables the relaxation of experimental spectra to underlying distributions of polymer mass in a polymeric melt and the relaxation of the time distribution in the system, see [5] for more details.

Weakly singular Fredholm integral equations (WSFIEs) arise as boundary integral representations of many physical problems in fluid dynamics, elasticity, and potential theory, where partial differential equations (PDEs) are converted into boundary Fredholm integral equations (BFIEs) to simplify and reduce the dimension of the problems. In this strategy, Green’s function of the governing PDE along with Green’s identities are used to describe the solution by integrals over the boundary of the problem domain. In most cases, these integrals contain kernels that exhibit algebraic or logarithmic singularities at points where coincidence of source and field occurs. If the singularity in the kernel function is integrable, then the kernel is said to have a weak singularity, and the FIE is referred to as a WSFIE of the first or second type based on the problem under consideration. See [6,7] for more details.

However, the majority of these FIEs, particularly WSFIEs, do not have closed-form solutions. As a result, various techniques have been introduced to determine approximate solutions for these problems in different fields. For example, Öztürk et al. [4] used the shifted-Legendre along with Bessel functions to form numerical solutions for the H-function and obtaining its moments as well. Strelnikova et al. [8] approximated the solution of a 1D hypersingular IE arising in potential theory based on the finite element algorithm. Foltran et al. [9] presented an application of the repeated Richardson extrapolation to solve problems in the media of radiative heat transfer. Yang Xiao-lin et al. [10] proposed a new radiative transport Monte-Carlo scheme, based on applying Neumann series of the Fredholm equation to solve the radiative transfer equations. Abdou et al. [11] compared the accuracy of solutions of a nonlinear FIE using the Adomian’s and Homotopy techniques. Katani et al. [12] proposed an algorithm based on Navot’s quadrature rule to approximate the solution of 2D-WSVIE. Iyika [13] approximated the solution of 2D-FIE using the Bernstein Operators technique. Laguardia et al. [14] applied the Nyström technique along with a cubature formula on a curvilinear general domain to approximate the solution of a 2D-FIE. Mezzanotte et al. [15] combined the Nyström method with Jacobi polynomials to approximate the solution of the FIE.

Micula [16] developed a fast and simple iterative algorithm to solve a fractional integral equation of the form

$$\begin{array}{c}\hfill {\displaystyle \phi \left(t\right)-\frac{L_1\left(t\right)}{\mathsf{\Gamma }\left(\beta \right)}{\int }_0^t{L}_2\left(\tau \right){(t-\tau )}^{\beta -1}\phi \left(\tau \right)d\tau =g\left(t\right),}\end{array}$$

(1)

where $t\in [0,a]$, $a<\infty$, $0<\beta <1$, and the functions $L_1$, $L_2$, and g are elements in the space $C\left(\right[0,a],\mathbb{R})$. Micula’s method is based on approximating the integration with a quadrature formula and then applying Picard’s successive approximations to solve Equation (1). Under some smoothness assumptions on the functions $L_1$, $L_2$, and g, Micula derived an estimation for the error. Micula suggested trying out this technique on more intricate problems to evaluate the efficiency of this algorithm. Using the iterative method developed by Micula, we study the numerical solvability of the two-dimensional weakly singular Fredholm model(2D-WSFIE) given by

$$\begin{array}{c}\hfill {\displaystyle \left({\gamma }_1+{\gamma }_2\right)\phi (y,z)-\left({\alpha }_1+{\alpha }_2\right){\int }_0^a{\int }_0^b\mathcal{H}(y-\tau ,z-s)\phi (\tau ,s)d\tau ds=\delta -{\mathcal{G}}_1(y,z)-{\mathcal{G}}_2(y,z),}\end{array}$$

(2)

under the static condition ${\displaystyle {\int }_0^a{\int }_0^b\phi (s,\tau )d\tau ds=P<\infty }$, where $(y,z)\in [0,b]\times [0,a]$ and $a,b<\infty$. The mechanical properties of the contacting bodies are represented by the bed coefficients parameters ${\alpha }_i$ for elastic materials that can be compressed. These parameters, ${\alpha }_i$, are determined experimentally for a variety of materials using the equation ${\alpha }_i={\displaystyle \frac{1-{\kappa }_i^2}{\pi E_i}}$, where ${\kappa }_i$ and $E_i$, $i=1,2,$ represent the Poisson’s coefficients and the elasticity Young modulus, respectively. The resistance to deformation (inherent rigidity) of contacting bodies in the normal direction are represented by the parameters ${\gamma }_i$, $i=1,2$. The surfaces arising in the problem are described by the functions ${\mathcal{G}}_i,i=1,2,$. The contact region between these two surfaces, ${\mathcal{G}}_i,i=1,2$, is denoted by $[0,b]\times [0,a]$. The weakly singular kernel of Equation (2) is the given function $\mathcal{H}$ (see the next section). The unknown function $\phi$ represents the normal stress between the surfaces ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$. The constant $\delta$ is the displacement that occurs between the two bodies when a force P is applied in the vertical downward direction on the upper body. Setting $\alpha ={\displaystyle \frac{{\alpha }_1+{\alpha }_2}{{\gamma }_1+{\gamma }_2}}$ and $\mathcal{G}(y,z)={\displaystyle \frac{\delta -{\mathcal{G}}_1(y,z)-{\mathcal{G}}_2(y,z)}{{\gamma }_1+{\gamma }_2}}$, it is easy to see that Equation (2) takes the form

$$\begin{array}{c}\hfill {\displaystyle \phi (y,z)-\alpha {\int }_0^a{\int }_0^b\mathcal{H}(y-\tau ,z-s)\phi (\tau ,s)d\tau ds=\mathcal{G}(y,z).}\end{array}$$

(3)

Equation (3) represents a 2D-WSFIE of the second kind with a given weakly singular kernel $\mathcal{H}$. This Formula (3) is measured in the space $C\left(\right[0,b]\times [0,a],\mathbb{R})$. In general, we can write the 2D-WSFIE (3) in the form

$$\begin{array}{c}\hfill {\displaystyle \gamma \phi (y,z)-\alpha {\int }_0^a{\int }_0^b\mathcal{H}(y-\tau ,z-s)\phi (\tau ,s)d\tau ds=\mathcal{G}(y,z).}\end{array}$$

(4)

The parameter $\gamma$ defines the kind of the 2D-WSFIE.

While the numerical Nyström algorithm and its current modified versions, to the best of our knowledge, rely essentially on direct linear system solvers to determine the values of the solution at the discretized points and then use the Nyström interpolation formula to generate approximate values of the solution at certain points, the current alternative approach computes approximate values of the solution at any point in the domain without the need for any linear systems. Our proposed method relies on using fixed-point iteration after handling the singularities with the product integration technique along with a suitable cubature formula. Under satisfying some sufficient conditions, the current approach aims to reduce computational cost while maintaining good accuracy, making the proposed algorithm advantageous over Nyström when solving large-scale systems or when repeated solves are needed. In particular, the developed algorithm is worthwhile when we need the solution values at specific points, which is a common requirement in various industrial applications. This practical advantage enhances the algorithm’s utility compared to other methods that rely on direct linear system solvers.

The current work is organized as follows. In Section 2, the basic results about the existence and uniqueness of the 2D-WSFIE (4) when $\gamma \in \mathbb{R}\setminus \left\{0\right\}$ are derived. Section 3 discusses the approximate solution of the proposed model based on combining the product integration technique, appropriate cubature formula, and the Picard algorithm to address the singularity that appears on the diagonal elements of the weakly singular kernel $\mathcal{H}$. The accuracy of the approximate solution and effect of singularity on the convergence rate estimated in Section 4. Section 5 is devoted to the applicability of the results obtained in this work. The proposed algorithm is applied to look for the normal stress $\phi$ between two rigid surfaces with two different materials contacting in the region ${[0,1]}^2$. The obtained results are compared with those obtained using the Nyström method. A discussion of the results and a conclusion appear in Section 6.

2. Basic Results About the Existence and Uniqueness of the 2D-WSFIE

Throughout this work, we consider that conditions (i)–(iii) are verified.

(i)

$\forall (x_1,x_2)\in [0,b]\times [0,a]\subset {\mathbb{R}}_{+}^2,(x_1,x_2)⟼\mathcal{G}(x_1,x_2)\in C([0,b]\times [0,a],\mathbb{R})$, such that $\parallel \mathcal{G}\parallel ={sup}_{(x_1,x_2)\in [0,b]\times [0,a]}\left|\mathcal{G}(x_1,x_2)\right|<\infty$.

(ii)

$\forall ({\zeta }_1,{\zeta }_2)\in [0,b]\times [0,a]$ the weakly singular kernel $({\zeta }_1,{\zeta }_2)⟼\mathcal{H}({\zeta }_1,{\zeta }_2)$ is a real-valued function satisfying

(iia)

${\displaystyle {\displaystyle {sup}_{(x_1,x_2)\in [0,b]\times [0,a]}}{\int }_0^a{\int }_0^b\left|\mathcal{H}(x_1-{\tau }_1,x_2-{\tau }_2)\right|d{\tau }_{1}d{\tau }_2\le d_1,d_1<\infty .}$

(iib)

${\displaystyle {\displaystyle {lim}_{({\overline{x}}_1,{\overline{x}}_2)\to (x_1,x_2)}}{\int }_0^a{\int }_0^b|\mathcal{H}({\overline{x}}_1-{\tau }_1,{\overline{x}}_2-{\tau }_2)-\mathcal{H}(x_1-{\tau }_1,x_2-{\tau }_2)|d{\tau }_{1}d{\tau }_2=0}$.

(iii)

$\left(\left|\gamma \right|-d_1\left|\alpha \right|\right)>0$.

Using Equation (4), when $\gamma \in \mathbb{R}\setminus \left\{0\right\}$, we define the integral operator $\mathsf{{\rm Y}}$ as below.

$$\mathsf{{\rm Y}}\phi (y,z){\displaystyle =\frac{1}{\gamma }\mathcal{G}(y,z)+\frac{\alpha }{\gamma }{\int }_0^a{\int }_0^b\mathcal{H}(y-\tau ,z-s)\phi (\tau ,s)d\tau ds.}$$

(5)

Theorem 1.

Let $\gamma \in \mathbb{R}\setminus \left\{0\right\}$ in Equation (4). Then, under conditions (i)–(ii), the integral operator (5) is completely continuous on the space $C\left(\right[0,b]\times [0,a\left]\right)$.

Proof. 

Let the function $\phi \in C\left(\right[0,b]\times [0,a\left]\right)$, and $\overline{x}=({\overline{x}}_1,{\overline{x}}_2),x=(x_1,x_2)\in [0,b]\times [0,a]$. Therefore, we have

$$\begin{array}{cc}\hfill & |\left(\mathsf{{\rm Y}}\phi \right)({\overline{x}}_1,{\overline{x}}_2)-\left(\mathsf{{\rm Y}}\phi \right)(x_1,x_2)|\le {\displaystyle \frac{1}{\left|\gamma \right|}}\left|\mathcal{G}({\overline{x}}_1,{\overline{x}}_2)-\mathcal{G}(x_1,x_2)\right|\hfill \\ \hfill & {\displaystyle +\frac{\left|\alpha \right|}{\left|\gamma \right|}\parallel \phi \parallel {\int }_0^a{\int }_0^b\left|\mathcal{H}(x_1-\tau ,x_2-s)-\mathcal{H}({\overline{x}}_1-\tau ,{\overline{x}}_2-s)\right|d\tau ds.}\hfill \end{array}$$

(6)

Taking limits of both sides as $({\overline{x}}_1,{\overline{x}}_2)⟶(x_1,x_2)$ implies

$$\begin{array}{c}{\displaystyle \underset{({\overline{x}}_1,{\overline{x}}_2)\to (x_1,x_2)}{lim}}|\left(\mathsf{{\rm Y}}\phi \right)({\overline{x}}_1,{\overline{x}}_2)-\left(\mathsf{{\rm Y}}\phi \right)(x_1,x_2)|\le {\displaystyle \frac{1}{\left|\gamma \right|}}{\displaystyle \underset{({\overline{x}}_1,{\overline{x}}_2)\to (x_1,x_2)}{lim}}\left|\mathcal{G}({\overline{x}}_1,{\overline{x}}_2)-\mathcal{G}(x_1,x_2)\right|\hfill \\ {\displaystyle +\frac{\left|\alpha \right|}{\left|\gamma \right|}\parallel \phi \parallel {\displaystyle \underset{({\overline{x}}_1,{\overline{x}}_2)\to (x_1,x_2)}{lim}}{\int }_0^a{\int }_0^b\left|\mathcal{H}({\overline{x}}_1-\tau ,{\overline{x}}_2-s)-\mathcal{H}(x_1-\tau ,x_2-s)\right|d\tau ds.}\hfill \end{array}$$

(7)

Using conditions (i) and (iib) yields ${lim}_{({\overline{x}}_1,{\overline{x}}_2)\to (x_1,x_2)}|\left(\mathsf{{\rm Y}}\phi \right)\overline{x}-\left(\mathsf{{\rm Y}}\phi \right)x|=0$. Thus, $\mathsf{{\rm Y}}\phi \in C\left(\right[0,b]\times [0,a\left]\right)$ whenever $\phi \in C\left(\right[0,b]\times [0,a\left]\right)$. Moreover, the operator $\mathsf{{\rm Y}}$ is continuous as for any sequence of functions ${\left({\phi }_m(y,z)\right)}_{m=1}^{\infty }$ in $C\left(\right[0,b]\times [0,a\left]\right)$ such that ${lim}_{m\to \infty }\parallel {\phi }_m-\phi \parallel =0$, we have

$${\displaystyle \left|\left(\mathsf{{\rm Y}}{\phi }_m\right)(y,z)-\left(\mathsf{{\rm Y}}\phi \right)(y,z)\right|\le \frac{\left|\alpha \right|}{\left|\gamma \right|}{\int }_0^a{\int }_0^b\left|\mathcal{H}\right(y-\tau ,z-s\left)\right||{\phi }_m(\tau ,s)-\phi (\tau ,s)|d\tau ds.}$$

(8)

By applying condition (iia) and the uniform convergence of ${\left({\phi }_m(y,z)\right)}_{m=1}^{\infty }$, we get

$${\displaystyle \underset{m\to \infty }{lim}}\parallel \mathsf{{\rm Y}}{\phi }_m-\mathsf{{\rm Y}}\phi \parallel \le {\displaystyle \frac{\left|\alpha \right|d_1}{\left|\gamma \right|}}{\displaystyle \underset{m\to \infty }{lim}}\parallel {\phi }_m-\phi \parallel =0.$$

(9)

Furthermore, considering the bounded set ${\mathfrak{S}}_{\mathfrak{B}}=\{\phi \in C([0,b]\times [0,a]): \parallel \phi \parallel \le \mathfrak{B}\}$, $\forall \phi \in {\mathfrak{S}}_{\mathfrak{B}}$ and $\forall \overline{x}=({\overline{x}}_1,{\overline{x}}_2)\in [0,b]\times [0,a]$, applying conditions (i) and (iia) gives

$$\begin{array}{cc}\hfill {\displaystyle \underset{({\overline{x}}_1,{\overline{x}}_2)\in [0,b]\times [0,a]}{sup}}\left|\left(\mathsf{{\rm Y}}\phi \right)({\overline{x}}_1,{\overline{x}}_2)\right|& \le {\displaystyle \frac{1}{\left|\gamma \right|}}{\displaystyle \underset{({\overline{x}}_1,{\overline{x}}_2)\in [0,b]\times [0,a]}{sup}}\left|\mathcal{G}({\overline{x}}_1,{\overline{x}}_2)\right|\hfill \\ \hfill & {\displaystyle +\frac{\left|\alpha \right|}{\left|\gamma \right|}{\displaystyle \underset{({\overline{x}}_1,{\overline{x}}_2)\in [0,b]\times [0,a]}{sup}}{\int }_0^a{\int }_0^b\left|\mathcal{H}\right({\overline{x}}_1-\tau ,{\overline{x}}_2-s\left)\right|\left|\phi (\tau ,s)\right|d\tau ds.}\hfill \\ \hfill & \le {\displaystyle \frac{1}{\left|\gamma \right|}}\left(\parallel \mathcal{G}\parallel +d_1\left|\alpha \right|\mathfrak{B}\right).\hfill \end{array}$$

(10)

Hence, $\parallel \mathsf{{\rm Y}}\phi \parallel$ is finite, and the set $\mathsf{{\rm Y}}\left({\mathfrak{S}}_{\mathfrak{B}}\right)$ is uniformly bounded with the finite upper bound ${\displaystyle \frac{1}{\left|\gamma \right|}}\left(\parallel \mathcal{G}\parallel +d_1\left|\alpha \right|\mathfrak{B}\right)$.

Additionally, from conditions $\left(\mathbf{i}\right)$ and $\left(\mathbf{iib}\right)$, $\forall \phi \in {\mathfrak{S}}_{\mathfrak{B}}$ and $\forall \overline{x}=({\overline{x}}_1,{\overline{x}}_2)$, $x=(x_1,x_2)\in [0,b]\times [0,a]$ and $\forall ϵ>0$, $\exists \delta \left(ϵ\right)>0$ such that $|\overline{x}-x|<\delta$ implies

$$\begin{array}{cc}\hfill {\displaystyle {\int }_0^a{\int }_0^b|\mathcal{H}({\overline{x}}_1-\tau ,{\overline{x}}_2-s)-\mathcal{H}(x_1-\tau ,x_2-s)|d\tau ds}& \le {\displaystyle \frac{\left|\gamma \right|ϵ}{2\left|\alpha \right|\mathfrak{B}}}.\hfill \\ \hfill |\mathcal{G}({\overline{x}}_1,{\overline{x}}_2)-\mathcal{G}(x_1,x_2)|& \le {\displaystyle \frac{\left|\gamma \right|ϵ}{2}}.\hfill \end{array}$$

Thus, we can conclude that

$$\begin{array}{cc}\hfill & |\left(\mathsf{{\rm Y}}\phi \right)({\overline{x}}_1,{\overline{x}}_2)-\left(\mathsf{{\rm Y}}\phi \right)(x_1,x_2)|\le {\displaystyle \frac{1}{\left|\gamma \right|}}|\mathcal{G}({\overline{x}}_1,{\overline{x}}_2)-\mathcal{G}(x_1,x_2)|\hfill \\ \hfill & {\displaystyle +\frac{\left|\alpha \right|\mathfrak{B}}{\left|\gamma \right|}{\int }_0^a{\int }_0^b|\mathcal{H}({\overline{x}}_1-\tau ,{\overline{x}}_2-s)-\mathcal{H}(x_1-\tau ,x_2-s)|d\tau ds}\hfill \\ \hfill & \le {\displaystyle \frac{ϵ}{2}}+{\displaystyle \frac{ϵ}{2}}=ϵ.\hfill \end{array}$$

(11)

The arguments above prove that the set $\mathsf{{\rm Y}}\left({\mathfrak{S}}_{\mathfrak{B}}\right)$ is uniformly equi-continuous. Using the theorem established by Arzela and Ascoli, we conclude that the set $\mathsf{{\rm Y}}\left({\mathfrak{S}}_{\mathfrak{B}}\right)$ is relatively compact. Consequently, the operator $\mathsf{{\rm Y}}$ is compact. Furthermore, since the operator $\mathsf{{\rm Y}}$ is both continuous and compact, it is completely continuous on $C\left(\right[0,b]\times [0,a\left]\right)$. This completes the proof. □

Theorem 2.

Let $\gamma \in \mathbb{R}$ in Equation (4). Then, under conditions (i)–(iii), the integral operator (5) is a contraction on the space $C\left(\right[0,b]\times [0,a\left]\right)$.

Proof. 

It is clear from Theorem 1 that the operator (5) maps the space $C\left(\right[0,b]\times [0,a\left]\right)$ into itself. Also, condition (iii) guarantees that $\gamma \ne 0$. Let ${\phi }_1,{\phi }_2\in C([0,b]\times [0,a])$, and $\overline{x}=({\overline{x}}_1,{\overline{x}}_2)\in [0,b]\times [0,a]$. So, we have

$$\begin{array}{cc}\hfill & |\left(\mathsf{{\rm Y}}{\phi }_2\right)({\overline{x}}_1,{\overline{x}}_2)-\left(\mathsf{{\rm Y}}{\phi }_1\right)({\overline{x}}_1,{\overline{x}}_2)|\hfill \\ \hfill & {\displaystyle \le \frac{\left|\alpha \right|}{\left|\gamma \right|}{\int }_0^a{\int }_0^b\left|\mathcal{H}\right({\overline{x}}_1-\tau ,{\overline{x}}_2-s\left)\right|\left|{\phi }_2\right(\tau ,s)-{\phi }_1(\tau ,s\left)\right|d\tau ds}\hfill \\ \hfill & {\displaystyle \le \frac{\left|\alpha \right|}{\left|\gamma \right|}\parallel {\phi }_2-{\phi }_1\parallel {\displaystyle \underset{\overline{x}\in [0,b]\times [0,a]}{sup}}{\int }_0^a{\int }_0^b\left|\mathcal{H}({\overline{x}}_1-\tau ,{\overline{x}}_2-s)\right|d\tau ds\le \frac{d_1\left|\alpha \right|}{\left|\gamma \right|}\parallel {\phi }_2-{\phi }_1\parallel .}\hfill \end{array}$$

(12)

Now $\parallel \left(\mathsf{{\rm Y}}{\phi }_2\right)-\left(\mathsf{{\rm Y}}{\phi }_1\right)\parallel \le d_2\parallel {\phi }_2-{\phi }_1\parallel$ with $d_2={\displaystyle \frac{d_1\left|\alpha \right|}{\left|\gamma \right|}}<1$ from condition (iii). So, the operator $\mathsf{{\rm Y}}$ is a $d_2$-contraction on $C\left(\right[0,b]\times [0,a\left]\right)$. This completes the proof. □

Lemma 1.

Let $\gamma \in \mathbb{R}$ in Equation (4). Then, under conditions (i)–(iii), the 2D-WSFIE (4) has only one solution in the space $C\left(\right[0,b]\times [0,a\left]\right)$.

Proof. 

The proof is derived by directly applying the Banach fixed point theorem to the integral operator (5) in the space $C\left(\right[0,b]\times [0,a\left]\right)$. □

3. Numerical Results and Discussion

In this section, we will discuss the numerical solution of the 2D-WSFIE (4) when the conditions of Theorem 2 are verified. In addition, we will assume that the force function, $\mathcal{G}$, and the solution, $\phi$, are elements in the space $C^3([0,b]\times [0,a],\mathbb{R})$. From Equation (4), we have

$$\begin{array}{cc}\hfill \phi (y,z)& {\displaystyle =\frac{1}{\gamma }\mathcal{G}(y,z)+\frac{\alpha }{\gamma }{\int }_0^a{\int }_0^b\mathcal{H}(y-\tau ,z-s)\phi (\tau ,s)d\tau ds.}\hfill \end{array}$$

(13)

Now, the proposed algorithm can be implemented by following these steps:

• Step 1: Weakly Singular Kernel Decomposition

In what follows, we assume the following conditions are met.

(IV)

The weakly singular kernel, $\mathcal{H}(y-\tau ,z-s)$, of Equation (13) can be decomposed into the product of two functions namely $\overline{\mathcal{H}}(y,\tau ;z,s)$ and $\mathsf{P}(y-\tau ,z-s)$. That is, we assume that

$$\mathcal{H}(y-\tau ,z-s)=\mathsf{P}(y-\tau ,z-s)\overline{\mathcal{H}}(y,\tau ;z,s).$$

(14)

(V)

$\overline{\mathcal{H}}(y,\tau ;z,s)\in C^3({[0,b]}^2\times {[0,a]}^2,\mathbb{R})$.

(VI)

All singular characteristics of the weakly singular kernel $\mathcal{H}$ are contained in the part $\mathsf{P}(y-\tau ,z-s)$ such that

$${\displaystyle {\displaystyle \underset{(y,z)\in [0,b]\times [0,a]}{sup}}{\int }_0^a{\int }_0^b\left|\mathsf{P}(y-\tau ,z-s)\right|d\tau ds=N_1<\infty .}$$

(15)

• Step 2: Reformulation of the Integral Equation

Substituting Equation (14) into Equation (13) yields

$$\begin{array}{cc}\hfill \phi (y,z)& {\displaystyle =\frac{1}{\gamma }\mathcal{G}(y,z)+\frac{\alpha }{\gamma }{\int }_0^a{\int }_0^b\mathsf{P}(y-\tau ,z-s)\overline{\mathcal{H}}(y,\tau ;z,s)\phi (\tau ,s)d\tau ds.}\hfill \end{array}$$

(16)

• Step 3: Cubature Approximation

Approximating the double integral in Equation (16) using the cubature formula as shown below in Equation (17).

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_0^a{\int }_0^b\mathsf{P}(y-\tau ,z-s)\overline{\mathcal{H}}(y,\tau ;z,s)\phi (\tau ,s)d\tau ds=\sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j)\phi ({\tau }_l,s_j)+\widehat{R}(y,z).}\hfill \end{array}$$

(17)

Substituting Equation (17) into Equation (16) gives:

$$\begin{array}{cc}\hfill \phi (y,z)& ={\displaystyle \frac{1}{\gamma }}\mathcal{G}(y,z)+{\displaystyle \frac{\alpha }{\gamma }}\sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j)\phi ({\tau }_l,s_j)+{\displaystyle \frac{\alpha }{\gamma }}\widehat{R}(y,z).\hfill \end{array}$$

(18)

The function $\widehat{R}(y,z)$ in Equation (18) is called the error function that depends on the variables y and z and needs to have an upper bound. The functions $w_{l,j}(y,z)$ in Equation (18) are unknown and are called the weights. These functions rely on the variables y and z and need to be determined.

• Step 4: Using Properties of Definite Integration

Let m and n be two even numbers and assume the two partitions $\{0={\tau }_0<{\tau }_1<{\tau }_2<\cdots <{\tau }_m=b\}$ with step size $h={\displaystyle \frac{b}{m}}$ and $\{0=s_0<s_1<s_2<\cdots <s_n=a\}$ with step size $k={\displaystyle \frac{a}{n}}$. By applying the properties of the definite integral, we can derive the following formula below.

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_0^a{\int }_0^b\mathsf{P}(y-\tau ,z-s)\overline{\mathcal{H}}(y,\tau ;z,s)\phi (\tau ,s)d\tau ds}\hfill \\ \hfill & {\displaystyle =\sum _{j=0}^{\frac{n}{2}-1}\sum _{l=0}^{\frac{m}{2}-1}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)\overline{\mathcal{H}}(y,\tau ;z,s)\phi (\tau ,s)d\tau ds.}\hfill \end{array}$$

(19)

Now, from Equations (17) and (19) we have:

$$\begin{array}{cc}\hfill & \sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j)\phi ({\tau }_l,s_j)+\widehat{R}(y,z)\hfill \\ \hfill & {\displaystyle =\sum _{j=0}^{\frac{n}{2}-1}\sum _{l=0}^{\frac{m}{2}-1}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)\overline{\mathcal{H}}(y,\tau ;z,s)\phi (\tau ,s)d\tau ds.}\hfill \end{array}$$

(20)

• Step 5: Polynomial Interpolation of $\overline{\mathcal{H}}(y,\tau ;z,s)\phi (\tau ,s)$

Approximating the function $\overline{\mathcal{H}}(y,\tau ;z,s)\phi (\tau ,s)$ over the region $[{\tau }_{2l},{\tau }_{2l+2}]\times [s_{2j},s_{2j+2}]$ with a Lagrange polynomial of second degree in the independent variables $\tau$ and s as below.

$$\begin{array}{cc}\hfill \overline{\mathcal{H}}(y,\tau ;z,s)\phi (\tau ,s)& =P_2(\tau ,s)+\overline{R}(\tau ,s).\hfill \end{array}$$

(21)

The function $P_2(\tau ,s)$ is a quadratic polynomial in both $\tau$ and s, while the function $\overline{R}(\tau ,s)$ is the remainder term due to the approximation process. They are given as follows:

$$\begin{array}{cc}\hfill P_2(\tau ,s)& =\overline{\mathcal{H}}(y,{\tau }_{2l};z,s_{2j})\phi ({\tau }_{2l},s_{2j})L_{2l,2j}(\tau ,s)\hfill \\ \hfill & +\overline{\mathcal{H}}(y,{\tau }_{2l};z,s_{2j+1})\phi ({\tau }_{2l},s_{2j+1})L_{2l,2j+1}(\tau ,s)\hfill \\ \hfill & +\overline{\mathcal{H}}(y,{\tau }_{2l};z,s_{2j+2})\phi ({\tau }_{2l},s_{2j+2})L_{2l,2j+2}(\tau ,s)\hfill \\ \hfill & +\overline{\mathcal{H}}(y,{\tau }_{2l+1};z,s_{2j})\phi ({\tau }_{2l+1},s_{2j})L_{2l+1,2j}(\tau ,s)\hfill \\ \hfill & +\overline{\mathcal{H}}(y,{\tau }_{2l+1};z,s_{2j+1})\phi ({\tau }_{2l+1},s_{2j+1})L_{2l+1,2j+1}(\tau ,s)\hfill \\ \hfill & +\overline{\mathcal{H}}(y,{\tau }_{2l+1};z,s_{2j+2})\phi ({\tau }_{2l+1},s_{2j+2})L_{2l+1,2j+2}(\tau ,s)\hfill \\ \hfill & +\overline{\mathcal{H}}(y,{\tau }_{2l+2};z,s_{2j})\phi ({\tau }_{2l+2},s_{2j})L_{2l+2,2j}(\tau ,s)\hfill \\ \hfill & +\overline{\mathcal{H}}(y,{\tau }_{2l+2};z,s_{2j+1})\phi ({\tau }_{2l+2},s_{2j+1})L_{2l+2,2j+1}(\tau ,s)\hfill \\ \hfill & +\overline{\mathcal{H}}(y,{\tau }_{2l+2};z,s_{2j+2})\phi ({\tau }_{2l+2},s_{2j+2})L_{2l+2,2j+2}(\tau ,s).\hfill \end{array}$$

(22)

For every $b_1\in \{2l,2l+1,2l+2\}$ and $a_1\in \{2j,2j+1,2j+2\}$, the function $L_{a_1,b_1}(\tau ,s)$ is the Lagrange quadratic polynomial in $(\tau ,s)\in [{\tau }_{2l},{\tau }_{2l+2}]\times [s_{2j},s_{2j+2}]$ and is defined by ${\displaystyle L_{a_1,b_1}(\tau ,s)=\prod _{\begin{array}{c}i=2j\\ i\ne a_1\end{array}}^{2j+2}\prod _{\begin{array}{c}\xi =2l\\ \xi \ne b_1\end{array}}^{2l+2}\frac{(\tau -{\tau }_{\xi })(s-s_i)}{({\tau }_{b_1}-{\tau }_{\xi })(s_{a_1}-s_i)}}$.

The remainder term, $\overline{R}(\tau ,s)$, due to the approximation process is given by:

$$\begin{array}{cc}\hfill & \overline{R}(\tau ,s)={\displaystyle \frac{1}{3!}}{\displaystyle \frac{{\partial }^3\left(\overline{\mathcal{H}}(y,\zeta \left(\tau \right);z,s)\phi (\zeta \left(\tau \right),s)\right)}{\partial {\tau }^3}}(\tau -{\tau }_{2l})(\tau -{\tau }_{2l+1})(\tau -{\tau }_{2l+2})\hfill \\ \hfill & +{\displaystyle \frac{1}{3!}}{\displaystyle \frac{{\partial }^3\left(\overline{\mathcal{H}}(y,{\tau }_{2l};z,\xi \left(s\right))\phi ({\tau }_{2l},\xi \left(s\right))\right)}{\partial s^3}}(s-s_{2j})(s-s_{2j+1})(s-s_{2j+2})L_{2l}\left(\tau \right)\hfill \\ \hfill & +{\displaystyle \frac{1}{3!}}{\displaystyle \frac{{\partial }^3\left(\overline{\mathcal{H}}(y,{\tau }_{2l+1};z,\xi \left(s\right))\phi ({\tau }_{2l+1},\xi \left(s\right))\right)}{\partial s^3}}(s-s_{2j})(s-s_{2j+1})(s-s_{2j+2})L_{2l+1}\left(\tau \right)\hfill \\ \hfill & +{\displaystyle \frac{1}{3!}}{\displaystyle \frac{{\partial }^3\left(\overline{\mathcal{H}}(y,{\tau }_{2l+2};z,\xi \left(s\right))\phi ({\tau }_{2l+2},\xi \left(s\right))\right)}{\partial s^3}}(s-s_{2j})(s-s_{2j+1})(s-s_{2j+2})L_{2l+2}\left(\tau \right).\hfill \end{array}$$

(23)

where $\zeta \left(\tau \right)\in ({\tau }_{2l},{\tau }_{2l+2})$, $\xi \left(s\right)\in (s_{2j},s_{2j+2})$ and ${\displaystyle L_{b_1}\left(\tau \right)=\prod _{\begin{array}{c}\xi =2l\\ \xi \ne b_1\end{array}}^{2l+2}\frac{(\tau -{\tau }_{\xi })}{({\tau }_{b_1}-{\tau }_{\xi })}}$.

Simplifying Equation (23) yields

$$\begin{array}{cc}\hfill & \parallel \overline{R}\parallel ={\displaystyle \underset{(\tau ,s)\in [0,b]\times [0,a]}{sup}}\left|\overline{R}(\tau ,s)\right|\le {\displaystyle \frac{4}{3}}{N}_2{N}_3(h^3+4k^3),\mathrm{where}\hfill \\ \hfill N_2& =max\{\parallel \phi \parallel ,\parallel {\phi }_s\parallel ,\parallel {\phi }_{\tau }\parallel ,\parallel {\phi }_{ss}\parallel ,\parallel {\phi }_{\tau \tau }\parallel ,\parallel {\phi }_{sss}\parallel ,\parallel {\phi }_{\tau \tau \tau }\parallel \}<\infty ,\forall (y,z)\in [0,b]\times [0,a].\hfill \\ \hfill N_3& =max\{\parallel \overline{\mathcal{H}}\parallel ,\parallel {\overline{\mathcal{H}}}_s\parallel ,\parallel {\overline{\mathcal{H}}}_{\tau }\parallel ,\parallel {\overline{\mathcal{H}}}_{ss}\parallel ,\parallel {\overline{\mathcal{H}}}_{\tau \tau }\parallel ,\parallel {\overline{\mathcal{H}}}_{sss}\parallel ,\parallel {\overline{\mathcal{H}}}_{\tau \tau \tau }\parallel \}<\infty ,\forall (y,z)\in [0,b]\times [0,a].\hfill \end{array}$$

(24)

In practice, the constant $N_2$ is estimated as the maximum norm over the approximate numerical solution $\phi$ and its derivatives using finite difference methods (or symbolic derivatives if the analytical solution is known) over the given domain, and similar arguments apply for the constant $N_3$.

• Step 6: Evaluating $w_{l,j}(y,z)$and Estimating an Upper Bound for $\widehat{R}(y,z)$

Substituting Equation (21) into Equation (20) gives

$$\begin{array}{cc}\hfill & \sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j)\phi ({\tau }_l,s_j)+\widehat{R}(y,z)\hfill \\ \hfill & {\displaystyle =\sum _{j=0}^{\frac{n}{2}-1}\sum _{l=0}^{\frac{m}{2}-1}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)\left(P_2(\tau ,s)+\overline{R}(\tau ,s)\right)d\tau ds.}\hfill \end{array}$$

(25)

From Equation (25), we have the following relationships:

$$\begin{array}{c}\hfill {\displaystyle \sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j)\phi ({\tau }_l,s_j)=\sum _{j=0}^{\frac{n}{2}-1}\sum _{l=0}^{\frac{m}{2}-1}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)P_2(\tau ,s)d\tau ds.}\end{array}$$

(26)

$$\begin{array}{c}\hfill {\displaystyle \widehat{R}(y,z)=\sum _{j=0}^{\frac{n}{2}-1}\sum _{l=0}^{\frac{m}{2}-1}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)\overline{R}(\tau ,s)d\tau ds.}\end{array}$$

(27)

As a result of weak singularities appearing at the term $\mathsf{P}(y-\tau ,z-s)$, there exists a constant $N_4>0$ such that

$$\begin{array}{c}\hfill \left|\mathsf{P}(y-\tau ,z-s)\right|\le {\displaystyle \frac{N_4}{{\left|y-\tau -z+s\right|}^{\nu }}},0<\nu <1.\end{array}$$

(28)

Simple computations lead to:

$$\begin{array}{cc}\hfill \parallel \widehat{R}\parallel & ={\displaystyle \underset{(y,z)\in [0,b]\times [0,a]}{sup}}\left|\widehat{R}(y,z)\right|\hfill \\ \hfill & {\displaystyle \le \frac{4}{3}{N}_2{N}_3(h^3+4k^3){\displaystyle \underset{(y,z)\in [0,b]\times [0,a]}{sup}}\sum _{j=0}^{\frac{n}{2}-1}\sum _{l=0}^{\frac{m}{2}-1}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\left|\mathsf{P}(y-\tau ,z-s)\right|d\tau ds}\hfill \\ \hfill & {\displaystyle \le \frac{4}{3}{N}_2{N}_3{N}_4(h^3+4k^3){\displaystyle \underset{(y,z)\in [0,b]\times [0,a]}{sup}}\sum _{j=0}^{\frac{n}{2}-1}\sum _{l=0}^{\frac{m}{2}-1}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}{\left|y-\tau -z+s\right|}^{-\nu }d\tau ds}\hfill \\ \hfill & \le {\displaystyle \frac{2^{3-\nu }ab{N}_2{N}_3{N}_4}{3(1-\nu )(2-\nu )}}(h^2{k}^{-1}+4k^2{h}^{-1})\left({(h+k)}^{2-\nu }-{(h-k)}^{2-\nu }\right).\hfill \end{array}$$

(29)

Remark 1.

From Ineq. (29), we observe that the error, $\parallel \widehat{R}\parallel$, grows as $\nu \to 1$. Additionally, the following observations can be made:

• When $h=k\to 0$, we have

  $$\parallel \widehat{R}\parallel \le {\displaystyle \frac{5ab{N}_2{N}_3{N}_4·2^{5-2\nu }}{3(1-\nu )(2-\nu )}}{h}^{3-\nu }.$$

  (30)
  Thus, the error satisfies $\parallel \widehat{R}\parallel =\mathcal{O}\left(h^{3-\nu }\right)$, indicating a decrease in convergence order caused by the weak singularity $0<\nu <1$, which affects both spatial dimensions.
• When $h\approx k$, then the term ${(h-k)}^{2-\nu }$ can be considered insignificant compared to the leading term ${(h+k)}^{2-\nu }$. Therefore, the error is again estimated as $\parallel \widehat{R}\parallel \approx \mathcal{O}\left(h^{3-\nu }\right)$. Thus, it is advisable to select $h\sim k$ to balance the error contributions from both spatial dimensions.
• The error upper bound indicates that there is interdependence between step sizes h and k because it involves terms $h^2{k}^{-1}$ and $k^2{h}^{-1}$. Therefore, balancing between h and k is required to avoid an increase in errors.
• When $k\ll h$, we can assume that $k=\epsilon h$ where $0<\epsilon \ll 1$. Therefore,

  $$\begin{array}{c}\hfill (h^2{k}^{-1}+4k^2{h}^{-1})={\displaystyle \frac{h}{\epsilon }}(1+4{\epsilon }^3)=\mathcal{O}\left({\displaystyle \frac{h}{\epsilon }}\right).\end{array}$$

  (31)
  Also, from the MacLaurin expansion near $\epsilon =0$, we have:

  $$\begin{array}{c}\hfill {(h+k)}^{2-\nu }-{(h-k)}^{2-\nu }=h^{2-\nu }\left[{(1+\epsilon )}^{2-\nu }-{(1-\epsilon )}^{2-\nu }\right]\approx 2(2-\nu )\epsilon h^{2-\nu }=\mathcal{O}\left(h^{2-\nu }\right).\end{array}$$

  (32)
  Thus

  $$\begin{array}{c}\hfill \left(h^2{k}^{-1}+4k^2{h}^{-1}\right)\left({(h+k)}^{2-\nu }-{(h-k)}^{2-\nu }\right)\approx \mathcal{O}\left(h^{3-\nu }\right).\end{array}$$

  (33)
  Therefore, refining the mesh in the k-direction alone does not lead to a significant improvement in the error unless the mesh size h is also decreased. This highlights the importance of maintaining a balanced mesh size in both dimensions. A similar arguments hold for $h\ll k$.

Further, substituting Equation (22) into Equation (26), then equating the coefficients of $\overline{\mathcal{H}}(y,{\tau }_l;z,s_j)\phi ({\tau }_l,s_j)$ for $l=0,1,\cdots ,m$ and $j=0,1,\cdots ,n$, the weight functions can be deduced as follows.

$$\begin{array}{cc}\hfill w_{0,0}(y,z)& {\displaystyle =\frac{1}{4h^2{k}^2}{\int }_{s_0}^{s_2}{\int }_{{\tau }_0}^{{\tau }_2}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_1)(\tau -{\tau }_2)(s-s_1)(s-s_2)d\tau ds.}\hfill \\ \hfill w_{0,n}(y,z)& {\displaystyle =\frac{1}{4h^2{k}^2}{\int }_{s_{n-2}}^{s_n}{\int }_{{\tau }_0}^{{\tau }_2}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_1)(\tau -{\tau }_2)(s-s_{n-1})(s-s_{n-2})d\tau ds.}\hfill \\ \hfill w_{m,0}(y,z)& {\displaystyle =\frac{1}{4h^2{k}^2}{\int }_{s_0}^{s_2}{\int }_{{\tau }_{m-2}}^{{\tau }_m}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{m-1})(\tau -{\tau }_{m-2})(s-s_1)(s-s_2)d\tau ds.}\hfill \\ \hfill w_{m,n}(y,z)& {\displaystyle =\frac{1}{4h^2{k}^2}{\int }_{s_{n-2}}^{s_n}{\int }_{{\tau }_{m-2}}^{{\tau }_m}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{m-1})(\tau -{\tau }_{m-2})(s-s_{n-1})(s-s_{n-2})d\tau ds.}\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & {\displaystyle w_{0,2j}(y,z)=\frac{1}{4h^2{k}^2}{\int }_{s_{2j-2}}^{s_{2j}}{\int }_{{\tau }_0}^{{\tau }_2}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_1)(\tau -{\tau }_2)(s-s_{2j-1})(s-s_{2j-2})d\tau ds}\hfill \\ \hfill & {\displaystyle +\frac{1}{4h^2{k}^2}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_0}^{{\tau }_2}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_1)(\tau -{\tau }_2)(s-s_{2j+1})(s-s_{2j+2})d\tau ds,}\hfill \\ \hfill & {\displaystyle w_{m,2j}(y,z)=\frac{1}{4h^2{k}^2}{\int }_{s_{2j-2}}^{s_{2j}}{\int }_{{\tau }_{m-2}}^{{\tau }_m}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{m-1})(\tau -{\tau }_{m-2})(s-s_{2j-1})(s-s_{2j-2})d\tau ds}\hfill \\ \hfill & {\displaystyle +\frac{1}{4h^2{k}^2}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{m-2}}^{{\tau }_m}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{m-1})(\tau -{\tau }_{m-2})(s-s_{2j+1})(s-s_{2j+2})d\tau ds,}\hfill \\ \hfill & \mathrm{where}j=1,2,\cdots ,\frac{n}{2}-1.\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & {\displaystyle w_{2l,0}(y,z)=\frac{1}{4h^2{k}^2}{\int }_{s_0}^{s_2}{\int }_{{\tau }_{2l-2}}^{{\tau }_{2l}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l-1})(\tau -{\tau }_{2l-2})(s-s_1)(s-s_2)d\tau ds}\hfill \\ \hfill & {\displaystyle +\frac{1}{4h^2{k}^2}{\int }_{s_0}^{s_2}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l+1})(\tau -{\tau }_{2l+2})(s-s_1)(s-s_2)d\tau ds.}\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & {\displaystyle w_{2l,n}(y,z)=\frac{1}{4h^2{k}^2}{\int }_{s_{n-2}}^{s_n}{\int }_{{\tau }_{2l-2}}^{{\tau }_{2l}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l-1})(\tau -{\tau }_{2l-2})(s-s_{n-1})(s-s_{n-2})d\tau ds}\hfill \\ \hfill & {\displaystyle +\frac{1}{4h^2{k}^2}{\int }_{s_{n-2}}^{s_n}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l+1})(\tau -{\tau }_{2l+2})(s-s_{n-1})(s-s_{n-2})d\tau ds,}\hfill \\ \hfill & \mathrm{where}l=1,2,\cdots ,\frac{m}{2}-1.\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & {\displaystyle w_{0,2j+1}(y,z)=\frac{-1}{2h^2{k}^2}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_0}^{{\tau }_2}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_1)(\tau -{\tau }_2)(s-s_{2j})(s-s_{2j+2})d\tau ds,}\hfill \\ \hfill & {\displaystyle w_{m,2j+1}(y,z)=\frac{-1}{2h^2{k}^2}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{m-2}}^{{\tau }_m}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{m-1})(\tau -{\tau }_{m-2})(s-s_{2j})(s-s_{2j+2})d\tau ds,}\hfill \\ \hfill & \mathrm{where}j=0,1\cdots ,\frac{n}{2}-1.\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & {\displaystyle w_{2l+1,0}(y,z)=\frac{-1}{2h^2{k}^2}{\int }_{s_0}^{s_2}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l})(\tau -{\tau }_{2l+2})(s-s_1)(s-s_2)d\tau ds,}\hfill \\ \hfill & {\displaystyle w_{2l+1,n}(y,z)=\frac{-1}{2h^2{k}^2}{\int }_{s_{n-2}}^{s_n}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l})(\tau -{\tau }_{2l+2})(s-s_{n-1})(s-s_{n-2})d\tau ds,}\hfill \\ \hfill & \mathrm{where}l=0,1,\cdots ,\frac{m}{2}-1.\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & {\displaystyle w_{2l+1,2j+1}(y,z)=\frac{1}{h^2{k}^2}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l})(\tau -{\tau }_{2l+2})(s-s_{2j})(s-s_{2j+2})d\tau ds,}\hfill \\ \hfill & \mathrm{where}l=0,1,2,\cdots ,\frac{m}{2}-1,\mathrm{and}j=0,1,2,\cdots ,\frac{n}{2}-1.\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & {\displaystyle w_{2l,2j+1}(y,z)=\frac{-1}{2h^2{k}^2}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{2l-2}}^{{\tau }_{2l}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l-1})(\tau -{\tau }_{2l-2})(s-s_{2j})(s-s_{2j+2})d\tau ds}\hfill \\ \hfill & {\displaystyle +\frac{-1}{2h^2{k}^2}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l+1})(\tau -{\tau }_{2l+2})(s-s_{2j})(s-s_{2j+2})d\tau ds,}\hfill \\ \hfill & \mathrm{where}l=1,2,\cdots ,\frac{m}{2}-1,\mathrm{and}j=0,1,2,\cdots ,\frac{n}{2}-1.\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & {\displaystyle w_{2l+1,2j}(y,z)=\frac{-1}{2h^2{k}^2}{\int }_{s_{2j-2}}^{s_{2j}}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l})(\tau -{\tau }_{2l+2})(s-s_{2j-1})(s-s_{2j-2})d\tau ds}\hfill \\ \hfill & {\displaystyle +\frac{-1}{2h^2{k}^2}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l})(\tau -{\tau }_{2l+2})(s-s_{2j+1})(s-s_{2j+2})d\tau ds,}\hfill \\ \hfill & \mathrm{where}l=0,1,2,\cdots ,\frac{m}{2}-1,\mathrm{and}j=1,2,\cdots ,\frac{n}{2}-1.\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & {\displaystyle w_{2l,2j}(y,z)=\frac{1}{4h^2{k}^2}{\int }_{s_{2j-2}}^{s_{2j}}{\int }_{{\tau }_{2l-2}}^{{\tau }_{2l}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l-1})(\tau -{\tau }_{2l-2})(s-s_{2j-1})(s-s_{2j-2})d\tau ds}\hfill \\ \hfill & {\displaystyle +\frac{1}{4h^2{k}^2}{\int }_{s_{2j-2}}^{s_{2j}}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l+1})(\tau -{\tau }_{2l+2})(s-s_{2j-1})(s-s_{2j-2})d\tau ds}\hfill \\ \hfill & {\displaystyle +\frac{1}{4h^2{k}^2}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{2l-2}}^{{\tau }_{2l}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l-1})(\tau -{\tau }_{2l-2})(s-s_{2j+1})(s-s_{2j+2})d\tau ds}\hfill \\ \hfill & {\displaystyle +\frac{1}{4h^2{k}^2}{\int }_{s_{2j}}^{s_{2j+2}}{\int }_{{\tau }_{2l}}^{{\tau }_{2l+2}}\mathsf{P}(y-\tau ,z-s)(\tau -{\tau }_{2l+1})(\tau -{\tau }_{2l+2})(s-s_{2j+1})(s-s_{2j+2})d\tau ds.}\hfill \\ \hfill & \mathrm{where}l=1,2,\cdots ,\frac{m}{2}-1,\mathrm{and}j=1,2,\cdots ,\frac{n}{2}-1.\hfill \end{array}$$

(43)

To simplify the calculation of the weight functions, we introduce the following functions:

$$\begin{array}{cc}\hfill & {\displaystyle {\mathsf{\Psi }}_1(y,z,l,j)=\frac{hk}{4}{\int }_0^2{\int }_0^2\mathsf{P}(y-(2l-2+\eta )h,z-(2j-2+\xi )k)(\eta -1)(\eta -2)(\xi -1)(\xi -2)d\eta d\xi .}\hfill \\ \hfill & {\displaystyle {\mathsf{\Psi }}_2(y,z,l,j)=\frac{hk}{4}{\int }_0^2{\int }_0^2\mathsf{P}(y-(2l-2+\eta )h,z-(2j-2+\xi )k)\eta (\eta -2)(\xi -1)(\xi -2)d\eta d\xi .}\hfill \\ \hfill & {\displaystyle {\mathsf{\Psi }}_3(y,z,l,j)=\frac{hk}{4}{\int }_0^2{\int }_0^2\mathsf{P}(y-(2l-2+\eta )h,z-(2j-2+\xi )k)\eta (\eta -1)(\xi -1)(\xi -2)d\eta d\xi .}\hfill \\ \hfill & {\displaystyle {\mathsf{\Psi }}_4(y,z,l,j)=\frac{hk}{4}{\int }_0^2{\int }_0^2\mathsf{P}(y-(2l-2+\eta )h,z-(2j-2+\xi )k)(\eta -1)(\eta -2)\xi (\xi -2)d\eta d\xi .}\hfill \\ \hfill & {\displaystyle {\mathsf{\Psi }}_5(y,z,l,j)=\frac{hk}{4}{\int }_0^2{\int }_0^2\mathsf{P}(y-(2l-2+\eta )h,z-(2j-2+\xi )k)\eta (\eta -2)\xi (\xi -2)d\eta d\xi .}\hfill \\ \hfill & {\displaystyle {\mathsf{\Psi }}_6(y,z,l,j)=\frac{hk}{4}{\int }_0^2{\int }_0^2\mathsf{P}(y-(2l-2+\eta )h,z-(2j-2+\xi )k)\eta (\eta -1)\xi (\xi -2)d\eta d\xi .}\hfill \\ \hfill & {\displaystyle {\mathsf{\Psi }}_7(y,z,l,j)=\frac{hk}{4}{\int }_0^2{\int }_0^2\mathsf{P}(y-(2l-2+\eta )h,z-(2j-2+\xi )k)(\eta -1)(\eta -2)\xi (\xi -1)d\eta d\xi .}\hfill \\ \hfill & {\displaystyle {\mathsf{\Psi }}_8(y,z,l,j)=\frac{hk}{4}{\int }_0^2{\int }_0^2\mathsf{P}(y-(2l-2+\eta )h,z-(2j-2+\xi )k)\eta (\eta -2)\xi (\xi -1)d\eta d\xi .}\hfill \\ \hfill & {\displaystyle {\mathsf{\Psi }}_9(y,z,l,j)=\frac{hk}{4}{\int }_0^2{\int }_0^2\mathsf{P}(y-(2l-2+\eta )h,z-(2j-2+\xi )k)\eta (\eta -1)\xi (\xi -1)d\eta d\xi .}\hfill \end{array}$$

(44)

Using the following substitutions:

$$\begin{array}{c}\hfill \eta ={\displaystyle \frac{1}{h}}\left(\tau -{\tau }_{2l-2}\right),\mathrm{and}\xi ={\displaystyle \frac{1}{k}}\left(s-s_{2j-2}\right),\mathrm{where}\tau \in [{\tau }_{2l-2},{\tau }_{2l}],\mathrm{and}s\in [s_{2j-2},s_{2j}].\end{array}$$

(45)

Therefore, the weight functions take the following compact form as shown below.

$$\begin{array}{cc}\hfill & w_{2l,2j}(y,z)={\mathsf{\Psi }}_9(y,z,l,j)+{\mathsf{\Psi }}_7(y,z,l+1,j)+{\mathsf{\Psi }}_3(y,z,l,j+1)+{\mathsf{\Psi }}_1(y,z,l+1,j+1),\hfill \\ \hfill & \mathrm{where}l=1,2,\cdots ,\frac{m}{2}-1,\mathrm{and}j=1,2,\cdots ,\frac{n}{2}-1.\hfill \\ \hfill & w_{2l+1,2j}(y,z)=-2\left[{\mathsf{\Psi }}_8(y,z,l+1,j)+{\mathsf{\Psi }}_2(y,z,l+1,j+1)\right],\hfill \\ \hfill & \mathrm{where}l=0,1,\cdots ,\frac{m}{2}-1,j=1,2,\cdots ,\frac{n}{2}-1.\hfill \\ \hfill & w_{2l,2j+1}(y,z)=-2\left[{\mathsf{\Psi }}_6(y,z,l,j+1)+{\mathsf{\Psi }}_4(y,z,l+1,j+1)\right],\hfill \\ \hfill & \mathrm{where}l=1,2,\cdots ,\frac{m}{2}-1,j=0,1,\cdots ,\frac{n}{2}-1.\hfill \\ \hfill & w_{2l+1,2j+1}(y,z)=4{\mathsf{\Psi }}_5(y,z,l+1,j+1),l=0,1,\cdots ,\frac{m}{2}-1,j=0,1,\cdots ,\frac{n}{2}-1.\hfill \\ \hfill & w_{2l,n}(y,z)={\mathsf{\Psi }}_9(y,z,l,\frac{n}{2})+{\mathsf{\Psi }}_1(y,z,l+1,\frac{n}{2}),l=1,2,\cdots ,\frac{m}{2}-1.\hfill \\ \hfill & w_{2l,0}(y,z)={\mathsf{\Psi }}_3(y,z,l,1)+{\mathsf{\Psi }}_1(y,z,l+1,1),l=1,2,\cdots ,\frac{m}{2}-1.\hfill \\ \hfill & w_{m,2j}(y,z)={\mathsf{\Psi }}_9(y,z,\frac{m}{2},j)+{\mathsf{\Psi }}_3(y,z,\frac{m}{2},j+1),j=1,2,\cdots ,\frac{n}{2}-1.\hfill \\ \hfill & w_{0,2j}(y,z)={\mathsf{\Psi }}_7(y,z,1,j)+{\mathsf{\Psi }}_1(y,z,1,j+1),j=1,2,\cdots ,\frac{n}{2}-1.\hfill \\ \hfill & w_{2l+1,n}(y,z)=-2{\mathsf{\Psi }}_8(y,z,l+1,\frac{n}{2}),l=0,1,2,\cdots ,\frac{m}{2}-1.\hfill \\ \hfill & w_{2l+1,0}(y,z)=-2{\mathsf{\Psi }}_2(y,z,l+1,1),l=0,1,2,\cdots ,\frac{m}{2}-1.\hfill \\ \hfill & w_{m,2j+1}(y,z)=-2{\mathsf{\Psi }}_6(y,z,\frac{m}{2},j+1),j=0,1,2,\cdots ,\frac{n}{2}-1.\hfill \\ \hfill & w_{0,2j+1}(y,z)=-2{\mathsf{\Psi }}_4(y,z,1,j+1),j=0,1,2,\cdots ,\frac{n}{2}-1.\hfill \\ \hfill & w_{0,0}(y,z)={\mathsf{\Psi }}_1(y,z,1,1).\hfill \\ \hfill & w_{0,n}(y,z)={\mathsf{\Psi }}_7(y,z,1,\frac{n}{2}).\hfill \\ \hfill & w_{m,0}(y,z)={\mathsf{\Psi }}_3(y,z,\frac{m}{2},1).\hfill \\ \hfill & w_{m,n}(y,z)={\mathsf{\Psi }}_9(y,z,\frac{m}{2},\frac{n}{2}).\hfill \end{array}$$

(46)

• Step 7: Iterative Solution via the Fixed Point Algorithm

After evaluating the weight function, we determine the approximate solution for the proposed problem (4) using the recursive relation below.

$$\begin{array}{cc}\hfill {\phi }_0(y,z)& =\widehat{\mathcal{G}}(y,z),\mathrm{where}\widehat{\mathcal{G}}(y,z)={\displaystyle \frac{1}{\gamma }}\mathcal{G}(y,z).\hfill \\ \hfill {\widehat{\phi }}_M(y,z)& ={\displaystyle \frac{\alpha }{\gamma }}\sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j){\widehat{\phi }}_{M-1}({\tau }_l,s_j)+\widehat{\mathcal{G}}(y,z),M=1,2,\cdots .\hfill \end{array}$$

(47)

4. Accuracy and Convergence Rate of the Approximate Solution

Estimating an upper bound of the error resulting from using the approximated solution generated by the proposed algorithm is important for evaluating the precision and reliability of the proposed method. Therefore, this section focuses on determining the maximum difference between the approximate and exact solution on the domain. Additionally, the effect of the weak singularity in the proposed weakly singular kernel on the convergence of the approximate solution is demonstrated. In the following, let us denote

$$\begin{array}{cc}\hfill {\mathcal{E}}_{\xi }& =sup\left\{\right|{\mathsf{\Psi }}_{\xi }(y,z,l,j)|:(y,z,l,j)\in [0,b]\times [0,a]\times \{0,\cdots ,\frac{m}{2}\}\times \{0,\cdots ,\frac{n}{2}\}\},\xi =1,\cdots ,9.\hfill \\ \hfill \mathcal{E}& =max\{{\mathcal{E}}_{\xi }:\xi =1,\cdots ,9\}.\hfill \\ \hfill \sigma & ={\displaystyle \frac{9\left|\alpha \right|N_3\mathcal{E}ab}{\left|\gamma \right|}}.\hfill \end{array}$$

(48)

Theorem 3.

Let the conditions of Theorem 2 be met and $\sigma <1$. Then, the estimation of the error between the exact solution φ of the 2D-WSFIE (4) and approximate solution ${\widehat{\phi }}_M$ evaluated by Equation (47) satisfies the inequality below.

$$\parallel \phi -{\widehat{\phi }}_M\parallel \le {\displaystyle \frac{d_2^M}{1-d_2}}\parallel {\phi }_1-{\phi }_0\parallel +{\displaystyle \frac{2^{3-\nu }ab{N}_2{N}_3{N}_4\left|\alpha \right|(h^2{k}^{-1}+4k^2{h}^{-1})\left({(h+k)}^{2-\nu }-{(h-k)}^{2-\nu }\right)}{3\left|\gamma \right|(1-\nu )(2-\nu )(1-\sigma )}}.$$

(49)

Proof. 

Let us consider the following recursive relation.

$$\begin{array}{c}\hfill {\displaystyle {\phi }_M(y,z)=\frac{\alpha }{\gamma }{\int }_0^a{\int }_0^b\mathsf{P}(y-\tau ,z-s)\overline{\mathcal{H}}(y,\tau ;z,s){\phi }_{M-1}(\tau ,s)d\tau ds+\widehat{\mathcal{G}}(y,z),M=1,2,\cdots .}\end{array}$$

(50)

Setting $M=1$ in Equation (50) and utilizing the approximation (17) implies

$$\begin{array}{cc}\hfill {\phi }_1(y,z)& {\displaystyle =\frac{\alpha }{\gamma }{\int }_0^a{\int }_0^b\mathsf{P}(y-\tau ,z-s)\overline{\mathcal{H}}(y,\tau ;z,s)\widehat{\mathcal{G}}(\tau ,s)d\tau ds+\widehat{\mathcal{G}}(y,z)}\hfill \\ \hfill & ={\displaystyle \frac{\alpha }{\gamma }}\left(\sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j)\widehat{\mathcal{G}}({\tau }_l,s_j)+{\widehat{R}}_0(y,z)\right)+\widehat{\mathcal{G}}(y,z).\hfill \\ \hfill & ={\displaystyle \frac{\alpha }{\gamma }}\sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j)\widehat{\mathcal{G}}({\tau }_l,s_j)+\widehat{\mathcal{G}}(y,z)+{\displaystyle \frac{\alpha }{\gamma }}{\widehat{R}}_0(y,z).\hfill \end{array}$$

(51)

Substituting from Equation (47), when $M=1$, into Equation (51) gives

$$\begin{array}{c}\hfill {\phi }_1(y,z)={\widehat{\phi }}_1(y,z)+{\displaystyle \frac{\alpha }{\gamma }}{\widehat{R}}_0(y,z).\end{array}$$

(52)

As a result, we have:

$$\begin{array}{c}\hfill {\displaystyle \underset{(y,z)\in [0,b]\times [0,a]}{sup}}|{\phi }_1(y,z)-{\widehat{\phi }}_1(y,z)|\le {\displaystyle \frac{\left|\alpha \right|}{\left|\gamma \right|}}{\displaystyle \underset{(y,z)\in [0,b]\times [0,a]}{sup}}\left|{\widehat{R}}_0(y,z)\right|\end{array}$$

(53)

Now, let $\parallel {\phi }_1-{\widehat{\phi }}_1\parallel ={sup}_{(y,z)\in [0,b]\times [0,a]}|{\phi }_1(y,z)-{\widehat{\phi }}_1(y,z)|$ and noting that $\parallel {\widehat{R}}_0\parallel \le \parallel \widehat{R}\parallel$. Then, with the help of (29), we can deduce that

$$\begin{array}{cc}\hfill \parallel {\phi }_1-{\widehat{\phi }}_1\parallel & \le {\displaystyle \frac{\left|\alpha \right|}{\left|\gamma \right|}}\parallel {\widehat{R}}_0\parallel \hfill \\ \hfill & \le {\displaystyle \frac{2^{3-\nu }ab{N}_2{N}_3{N}_4\left|\alpha \right|(h^2{k}^{-1}+4k^2{h}^{-1})\left({(h+k)}^{2-\nu }-{(h-k)}^{2-\nu }\right)}{3\left|\gamma \right|(1-\nu )(2-\nu )}}.\hfill \end{array}$$

(54)

Proceeding as we did above by setting $M=2$ in Equation (50) and utilizing the approximation, (17) implies that

$$\begin{array}{cc}\hfill {\phi }_2(y,z)& {\displaystyle =\frac{\alpha }{\gamma }{\int }_0^a{\int }_0^b\mathsf{P}(y-\tau ,z-s)\overline{\mathcal{H}}(y,\tau ;z,s){\phi }_1(\tau ,s)d\tau ds+\widehat{\mathcal{G}}(y,z)}\hfill \\ \hfill & ={\displaystyle \frac{\alpha }{\gamma }}\left(\sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j){\phi }_1({\tau }_l,s_j)+{\widehat{R}}_1(y,z)\right)+\widehat{\mathcal{G}}(y,z)\hfill \\ \hfill & ={\displaystyle \frac{\alpha }{\gamma }}\sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j){\phi }_1({\tau }_l,s_j)+\widehat{\mathcal{G}}(y,z)+{\displaystyle \frac{\alpha }{\gamma }}{\widehat{R}}_1(y,z).\hfill \end{array}$$

(55)

Substituting from Equation (52) into Equation (55) gives

$$\begin{array}{cc}\hfill {\phi }_2(y,z)& ={\displaystyle \frac{\alpha }{\gamma }}\sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j)\left({\widehat{\phi }}_1({\tau }_l,s_j)+{\displaystyle \frac{\alpha }{\gamma }}{\widehat{R}}_0({\tau }_l,s_j)\right)+\widehat{\mathcal{G}}(y,z)+{\displaystyle \frac{\alpha }{\gamma }}{\widehat{R}}_1(y,z)\hfill \\ \hfill & ={\displaystyle \frac{\alpha }{\gamma }}\sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j){\widehat{\phi }}_1({\tau }_l,s_j)+\widehat{\mathcal{G}}(y,z)\hfill \\ \hfill & +{\displaystyle \frac{{\alpha }^2}{{\gamma }^2}}\sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j){\widehat{R}}_0({\tau }_l,s_j)+{\displaystyle \frac{\alpha }{\gamma }}{\widehat{R}}_1(y,z).\hfill \\ \hfill & ={\widehat{\phi }}_2(y,z)+{\displaystyle \frac{{\alpha }^2}{{\gamma }^2}}\sum _{j=0}^n\sum _{l=0}^m{w}_{l,j}(y,z)\overline{\mathcal{H}}(y,{\tau }_l;z,s_j){\widehat{R}}_0({\tau }_l,s_j)+{\displaystyle \frac{\alpha }{\gamma }}{\widehat{R}}_1(y,z).\hfill \end{array}$$

(56)

Furthermore,

$$\parallel {\phi }_2-{\widehat{\phi }}_2\parallel \le {\displaystyle \frac{\left|\alpha \right|}{\left|\gamma \right|}}\left({\displaystyle \frac{\left|\alpha \right|N_3}{\left|\gamma \right|}}{\displaystyle \underset{(y,z)\in [0,b]\times [0,a]}{sup}}\sum _{j=0}^n\sum _{l=0}^m\left|w_{l,j}(y,z)\right|+1\right)\parallel \widehat{R}\parallel .$$

(57)

From Equations (46) and (48), it is easy to see that

$$\begin{array}{cc}\hfill & {\displaystyle \underset{(y,z)\in [0,b]\times [0,a]}{sup}}\sum _{j=0}^n\sum _{l=0}^m|w_{l,j}(y,z)|\le hk{\displaystyle \underset{(y,z)\in [0,b]\times [0,a]}{sup}}\sum _{j=0}^{\frac{n}{2}}\sum _{l=0}^{\frac{m}{2}}\sum _{\xi =1}^9\left|{\mathsf{\Psi }}_{\xi }(y,z,j,l)\right|=9\mathcal{E}ab.\hfill \end{array}$$

(58)

Substituting Ineqs. (58) and (29) into (57) gives

$$\parallel {\phi }_2-{\widehat{\phi }}_2\parallel \le {\displaystyle \frac{2^{3-\nu }ab{N}_2{N}_3{N}_4\left|\alpha \right|(h^2{k}^{-1}+4k^2{h}^{-1})\left({(h+k)}^{2-\nu }-{(h-k)}^{2-\nu }\right)}{3\left|\gamma \right|(1-\nu )(2-\nu )}}\left(1+{\displaystyle \frac{9\left|\alpha \right|N_3\mathcal{E}ab}{\left|\gamma \right|}}\right).$$

(59)

Using $\sigma ={\displaystyle \frac{9\left|\alpha \right|N_3\mathcal{E}ab}{\left|\gamma \right|}}$ and proceeding by the same way as above gives

$$\parallel {\phi }_3-{\widehat{\phi }}_3\parallel \le {\displaystyle \frac{2^{3-\nu }ab{N}_2{N}_3{N}_4\left|\alpha \right|(h^2{k}^{-1}+4k^2{h}^{-1})\left({(h+k)}^{2-\nu }-{(h-k)}^{2-\nu }\right)}{3\left|\gamma \right|(1-\nu )(2-\nu )}}\left(1+\sigma +{\sigma }^2\right).$$

(60)

Therefore, by induction we have

$$\begin{array}{cc}\hfill & \parallel {\phi }_M-{\widehat{\phi }}_M\parallel \hfill \\ \hfill & \le {\displaystyle \frac{2^{3-\nu }ab{N}_2{N}_3{N}_4\left|\alpha \right|(h^2{k}^{-1}+4k^2{h}^{-1})\left({(h+k)}^{2-\nu }-{(h-k)}^{2-\nu }\right)}{3\left|\gamma \right|(1-\nu )(2-\nu )}}\left(1+\sigma +{\sigma }^2+\cdots +{\sigma }^{M-1}\right).\hfill \end{array}$$

(61)

Now, for a sufficiently large M and using the assumption that $\sigma <1$ along with (61), we have:

$$\begin{array}{c}\hfill \parallel {\phi }_M-{\widehat{\phi }}_M\parallel \le {\displaystyle \frac{2^{3-\nu }ab{N}_2{N}_3{N}_4\left|\alpha \right|(h^2{k}^{-1}+4k^2{h}^{-1})\left({(h+k)}^{2-\nu }-{(h-k)}^{2-\nu }\right)}{3\left|\gamma \right|(1-\nu )(2-\nu )(1-\sigma )}}.\end{array}$$

(62)

Also, from the approximation algorithm due to the Banach fixed point theorem, see [17], we have

$$\begin{array}{c}\hfill \parallel \phi -{\phi }_M\parallel \le {\displaystyle \frac{d_2^M}{1-d_2}}\parallel {\phi }_1-{\phi }_0\parallel ,\forall M=1,2\cdots .\end{array}$$

(63)

As a result, we have:

$$\parallel \phi -{\widehat{\phi }}_M\parallel \le \parallel \phi -{\phi }_M\parallel +\parallel {\phi }_M-{\widehat{\phi }}_M\parallel .$$

(64)

Substituting Ineqs. (62) and (63) into Ineq. (64), then $\forall M=1,2\cdots$ we have

$$\begin{array}{c}\hfill \parallel \phi -{\widehat{\phi }}_M\parallel \le {\displaystyle \frac{d_2^M}{1-d_2}}\parallel {\phi }_1-{\phi }_0\parallel +{\displaystyle \frac{2^{3-\nu }ab{N}_2{N}_3{N}_4\left|\alpha \right|(h^2{k}^{-1}+4k^2{h}^{-1})\left({(h+k)}^{2-\nu }-{(h-k)}^{2-\nu }\right)}{3\left|\gamma \right|(1-\nu )(2-\nu )(1-\sigma )}}.\end{array}$$

(65)

This completes the proof. □

Remark 2.

From Ineq. (49), we observe that the error estimate shows the impact of three important key parameters:

• The effect of iterating number M appears in the term ${\displaystyle \frac{d_2^M}{1-d_2}}\parallel {\phi }_1-{\phi }_0\parallel$. Since $0<d_2<1$, increasing M leads to exponential decay in the error and thus enhances accuracy. The smaller $d_2$ is (i.e., $d_2\ll 1$), the fewer iterations we need to obtain good accuracy.
• As the parameter σ gets close to 1, the denominator $(1-\sigma )$ becomes small, causing the second term in the error upper bound to increase. This increase slows the convergence rate of the algorithm. Hence, it is crucial to maintain $\sigma \ll 1$ to guarantee the stability, robustness, and accuracy of the approximate solution, and to obtain rapid convergence.
• As the parameter ν approaches 1, the weak singularity in the kernel becomes stronger. This causes the denominators $(1-\nu )$ and $(2-\nu )$ to decrease, raising the overall error upper bound and leading to increased numerical errors with the same discretization parameters and iteration count.
• The impact of the scaling factor ${(h+k)}^{2-\nu }-{(h-k)}^{2-\nu }$ on the interaction between the discretization parameters h and k, and how to minimize numerical errors, is addressed in Remark 1.

5. Application

This section is devoted to demonstrating the applicability of the results obtained in this work. We are looking for the normal stress $\phi$ between two rigid surfaces, each made of a different material, in contact in the region ${[0,1]}^2$. This normal stress is modeled by a 2D-WSFIEs with weakly singular kernels capturing the singularities near the contact region, as shown below.

Example 1.

We consider the 2D-WSFIE with a logarithmic weakly singular kernel as demonstrated below.

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}\phi (y,z)-\alpha {\int }_0^1{\int }_0^{1}log|y-\tau |log|z-s|\phi (\tau ,s)d\tau ds=\mathcal{G}(y,z).}\end{array}$$

(66)

Comparing Equation (66) with the main Equation (4), we can see that $a=b=1$, $\gamma =\frac{1}{2}$, and the weakly singular kernel $\mathcal{H}(y-\tau ,z-s)$ is of a logarithmic type as demonstrated in Equation (67) below.

$$\begin{array}{c}\hfill \mathcal{H}(y-\tau ,z-s)=log|y-\tau |log|z-s|.\end{array}$$

(67)

Moreover, using some simple computations, we can see that

$$\begin{array}{c}\hfill {\displaystyle {\displaystyle \underset{(y,z)\in [0,1]\times [0,1]}{sup}}{\int }_0^1{\int }_0^{1}log|y-\tau |log|z-s|d\tau ds={\left(1+log2\right)}^2.}\end{array}$$

(68)

Therefore, we can assume that $d_1={\left(1+log2\right)}^2$. Also, we have

$$\begin{array}{c}\hfill {\displaystyle {\displaystyle \underset{(\overline{y},\overline{z})\to (y,z)}{lim}}{\int }_0^1{\int }_0^1|log|\overline{y}-\tau |log|\overline{z}-s|-log|y-\tau |log|z-s\left|\right|d\tau ds=0.}\end{array}$$

(69)

So, conditions (iia) and (iib) are verified. To find $\alpha$, we consider two rigid surfaces in contact in the region ${[0,1]}^2$ with Poisson’s ratios of ${\kappa }_1={\displaystyle \frac{2}{7}}$ and ${\kappa }_2={\displaystyle \frac{3}{7}}$, respectively. Using the well-known relations mentioned below in Equation (70), we can determine $\alpha$.

$$\begin{array}{cc}\hfill \alpha & ={\displaystyle \frac{{\alpha }_1+{\alpha }_2}{{\gamma }_1+{\gamma }_2}},\mathrm{where}{\alpha }_i={\displaystyle \frac{1-{\kappa }_i^2}{\pi E_i}},\mathrm{and}{E}_i=2\gamma (1+{\kappa }_i),\mathrm{and}{\gamma }_i={\displaystyle \frac{2\gamma {\kappa }_i}{1-2{\kappa }_i}},i=1,2.\hfill \end{array}$$

(70)

We can find that $\alpha =\frac{123}{1102}$. As a result, we can see that condition (iii) is satisfied, because $\left(\left|\gamma \right|-d_1\left|\alpha \right|\right)=\left(\frac{1}{2}-{\left(1+log2\right)}^2\times \frac{123}{1102}\right)\approx 0.18>0.$ On the other hand, consider $\mathsf{\Lambda }\left(\theta \right)=2\left({\theta }^{2}log\left|\theta \right|+(1-{\theta }^2)log|1-\theta |\right)-1$ and define the force function $\mathcal{G}$ as follows:

$$\mathcal{G}(y,z)=\frac{-41}{70528}\left(\mathsf{\Lambda }\left(y\right)\mathsf{\Lambda }\left(z\right)-2\left(y\mathsf{\Lambda }\right(z)+z\mathsf{\Lambda }(y\left)\right)\right)+\frac{2081}{52896}yz$$

(71)

It is easy to see that $\parallel \mathcal{G}\parallel$ finite on ${[0,1]}^2$, and hence all conditions of Lemma 1 are met. As a result, the 2D-WSIE (66) has a unique solution $\mathcal{G}(y,z)=\frac{1}{12}yz$ in the space $C{[0,1]}^2$.

The tables above show the numerical results generated using MATLAB R2019a software.

Table 1. Comparison of absolute errors and CPU times (in seconds) for the proposed algorithm with $M=6$ and the Nyström method for $m=4$, $n=8$ at some selected points in the domain.

Points		Exact Sol.	Approx. Sol. (6th Iter.)			Approx. Sol. (Nyström)
y	z	Value	Prop. Alg.	Abs. Err.	CPU	Value	Abs. Err.	CPU
0.0	0.0	0.0000	$2.3253\times {10}^{-4}$	$2.3253\times {10}^{-4}$	$13.3864$	$2.0840\times {10}^{-4}$	$2.0840\times {10}^{-4}$	$12.4890$
0.2	0.2	$3.3333\times {10}^{-3}$	$3.0850\times {10}^{-3}$	$2.4833\times {10}^{-4}$	$13.4987$	$3.1210\times {10}^{-3}$	$2.1233\times {10}^{-4}$	$13.8993$
0.4	0.4	$1.3333\times {10}^{-2}$	$1.3181\times {10}^{-2}$	$1.5200\times {10}^{-4}$	$13.4997$	$1.2948\times {10}^{-2}$	$3.8519\times {10}^{-4}$	$13.8993$
0.6	0.6	$3.0000\times {10}^{-2}$	$2.9632\times {10}^{-2}$	$3.6824\times {10}^{-4}$	$13.5194$	$2.9207\times {10}^{-2}$	$7.9296\times {10}^{-4}$	$13.8993$
0.8	0.8	$5.3333\times {10}^{-2}$	$5.1549\times {10}^{-2}$	$1.7838\times {10}^{-3}$	$13.5194$	$5.1034\times {10}^{-2}$	$2.2992\times {10}^{-3}$	$13.8993$
1.0	1.0	$8.3333\times {10}^{-2}$	$8.0628\times {10}^{-2}$	$2.7050\times {10}^{-3}$	$13.5194$	$8.1016\times {10}^{-2}$	$2.3175\times {10}^{-3}$	$12.4890$

,

Table 2. Comparison of absolute errors and CPU times (in seconds) for the proposed algorithm with $M=6$ and the Nyström method for $m=8$, $n=8$ at some selected points in the domain.

Points		Exact Sol.	Approx. Sol. (6th Iter.)			Approx. Sol. (Nyström)
y	z	Value	Prop. Alg.	Abs. Err.	CPU	Value	Abs. Err.	CPU
0.0	0.0	0.0000	$1.0814\times {10}^{-4}$	$1.0814\times {10}^{-4}$	$26.0123$	$1.1538\times {10}^{-4}$	$1.1538\times {10}^{-4}$	$25.4398$
0.2	0.2	$3.3333\times {10}^{-3}$	$3.2171\times {10}^{-3}$	$1.1623\times {10}^{-4}$	$26.2361$	$3.2132\times {10}^{-3}$	$1.2006\times {10}^{-4}$	$27.1259$
0.4	0.4	$1.3333\times {10}^{-2}$	$1.3229\times {10}^{-2}$	$1.0434\times {10}^{-4}$	$26.2366$	$1.3118\times {10}^{-2}$	$2.1456\times {10}^{-4}$	$27.1259$
0.6	0.6	$3.0000\times {10}^{-2}$	$2.9803\times {10}^{-2}$	$1.9709\times {10}^{-4}$	$26.3679$	$2.9580\times {10}^{-2}$	$4.1977\times {10}^{-4}$	$27.1259$
0.8	0.8	$5.3333\times {10}^{-2}$	$5.2283\times {10}^{-2}$	$1.0502\times {10}^{-3}$	$26.3891$	$5.2022\times {10}^{-2}$	$1.3112\times {10}^{-3}$	$27.1259$
1.0	1.0	$8.3333\times {10}^{-2}$	$8.1954\times {10}^{-2}$	$1.3792\times {10}^{-3}$	$26.4965$	$8.1945\times {10}^{-2}$	$1.3879\times {10}^{-3}$	$25.5682$

and

Table 3. Comparison of absolute errors and CPU times (in seconds) for the proposed algorithm with $M=6$ and the Nyström method for $m=8$, $n=16$ at some selected points in the domain.

Points		Exact Sol.	Approx. Sol. (6th Iter.)			Approx. Sol. (Nyström)
y	z	Value	Prop. Alg.	Abs. Err.	CPU	Value	Abs. Err.	CPU
0.0	0.0	0.0000	$1.0650\times {10}^{-4}$	$1.0650\times {10}^{-4}$	$71.0812$	$1.3554\times {10}^{-3}$	$1.3554\times {10}^{-3}$	$75.8475$
0.2	0.2	$3.3333\times {10}^{-3}$	$3.2406\times {10}^{-3}$	$9.2744\times {10}^{-5}$	$71.1478$	$3.2284\times {10}^{-3}$	$1.0486\times {10}^{-4}$	$76.9601$
0.4	0.4	$1.3333\times {10}^{-2}$	$1.3263\times {10}^{-2}$	$6.9789\times {10}^{-5}$	$71.5582$	$1.3168\times {10}^{-2}$	$1.6505\times {10}^{-4}$	$76.9984$
0.6	0.6	$3.0000\times {10}^{-2}$	$3.9890\times {10}^{-2}$	$1.0996\times {10}^{-4}$	$71.6889$	$2.9698\times {10}^{-2}$	$3.0207\times {10}^{-4}$	$77.0386$
0.8	0.8	$5.3333\times {10}^{-2}$	$5.2734\times {10}^{-2}$	$5.9902\times {10}^{-4}$	$71.8869$	$5.2305\times {10}^{-2}$	$1.0278\times {10}^{-3}$	$77.2591$
1.0	1.0	$8.3333\times {10}^{-2}$	$8.2372\times {10}^{-2}$	$9.6141\times {10}^{-4}$	$72.1236$	$8.1415\times {10}^{-2}$	$1.9184\times {10}^{-3}$	$76.1483$

provide a comparison between the proposed algorithm and the Nyström method for various discretization meshes. The comparison involves evaluating the methods at specific points $(y,z)$ in the computational domain. The tables present the absolute errors and CPU times (in seconds) for both methods.

The results in Table 1 show that the proposed algorithm performs slightly better than the Nyström method by achieving lower absolute errors at interior points. Both methods have similar performance near the origin. Despite higher errors near the corner $(1,1)$ due to singularity effects, the proposed method maintains a small accuracy advantage. This improved accuracy is achieved without a significant increase in CPU time, indicating an efficient balance between precision and computation. Figure 1 and Figure 2 confirm these results, displaying well-controlled errors for both methods throughout the domain, with only minor increases near the corner $(1,1)$.

Increasing the discretization meshes to $m=8$ and $n=8$ results in a significant decrease in absolute errors for both methods. The proposed algorithm consistently shows slightly better accuracy compared to the other method, as indicated in Table 2 and corresponding Figure 3 and Figure 4. Despite the improved precision, the CPU times for both approaches remain similar, highlighting the efficiency of the proposed method.

With $m=8$ and $n=16$, the proposed algorithm consistently achieves approximately half the absolute error of the Nyström method, as shown in Table 3. Despite both methods requiring a significant increase in CPU time due to finer discretization, the Nyström method’s coefficient matrix condition number increases to $2.01\times {10}^3$, indicating moderate conditioning issues due to continued increase in the differences in step sizes between both spatial dimensions. In contrast, the proposed method remains stable but exhibits a slight decrease in convergence rate, as seen in the smooth error distributions for the proposed algorithm in Figure 5 compared to the clear increase in absolute error at corners $(0,0)$ and $(1,1)$ for the Nyström method in Figure 6.

Based on the numerical results presented in Table 1, Table 2 and Table 3, the proposed algorithm shows better accuracy than the Nyström method in most tested configurations. It consistently achieves lower absolute errors at interior and boundary points, effectively handling the weak singularities in the integral equation. In comparison to the Nyström algorithm, the proposed method reduces computational cost while maintaining good accuracy. Moreover, the proposed algorithm remains stable even under challenging discretization settings where the Nyström method’s system shows moderate conditioning issues, as seen in the condition number growth in Table 3. Overall, the proposed approach provides a robust and reliable alternative, offering improved accuracy and comparable efficiency for solving WSIEs.

6. Conclusions and Discussion

In this work, we have studied the existence and uniqueness of a continuous solution for a class of 2D-WSFIE of the second type based on utilizing the fixed-point principle. The approximate solution for this class has been investigated by incorporating the product integration and Picard technique along with a cubature formula to tackle the issues resulting from the singularity appearing on diagonal elements of the weakly singular kernel. We have estimated an error bound for the absolute error, discussing the convergence rate for the developed technique and the effect of singularity on the convergence rate. The applicability of the results obtained in this work is demonstrated by applying the proposed algorithm to approximately evaluate the normal stress $\phi$ between two rigid surfaces, each made of a different material, in contact in the region ${[0,1]}^2$ and modeled by 2D-WSFIEs. Our derived results are compared with those obtained by the Nyström method. In comparison to the results obtained by the Nyström algorithm, the proposed method reduces computational cost while maintaining good accuracy. Moreover, the proposed algorithm remains stable even under challenging discretization settings where the Nyström method’s system shows moderate conditioning issues. Overall, the proposed approach provides a robust and reliable alternative, offering improved accuracy and comparable efficiency for solving WSIEs. The developed algorithm is useful when we need the solution values at specific points, a common requirement in various industrial applications. This practical advantage enhances the algorithm’s utility compared to other methods that rely on direct linear system solvers.