Hybrid Functions Approach to Solving Nonlinear Integral Equations in Two Dimensions

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Hybrid Functions Approach to Solving Nonlinear Integral Equations in Two Dimensions

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08 May 2023

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09 May 2023

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Abstract

This study presents the solution of the second type of a two-dimensional nonlinear integral equation in Banach space. Also, the existence and uniqueness of this equation’s solution are discussed. We utilize a numerical approach involving hybrid and block-pulse functions to obtain the approximate solution of a two-dimensional nonlinear integral equation. Nonlinear integral equation in two dimensions is reduced numerically to a system of nonlinear algebraic equations that can be solved using numerical methods. This study focuses on showing the convergence analysis for the numerical approach and obtaining an error estimate. Some numerical examples have been provided to demonstrate the approach’s viability and efficacy

Keywords:

Subject: Computer Science and Mathematics - Mathematics

MSC: 41A30; 45G10; 46B45; 65R20

1. Introduction

Integral equations are used in many disciplines of applied mathematics to explore and solve problems. See [1,5,6,10,24,26,27] for more information on the topics of two-dimensional nonlinear integral equations, which have long been of growing interest in many fields, including medicine, biology, physics, geography, and fuzzy control. According to the references [2,3,7,9,10,11,12], many problems in engineering, applied mathematics, and mathematical physics can be reduced into two-dimensional nonlinear integral equations. The analytical solutions to these equations are typically difficult. Therefore, it is necessary to find approximations. For example, Bernstein polynomials hybrid with functions of block-pulse form [4,22] and Legendre hybrid with functions of block-pulse form [13,21] have both recently been examined as computational approaches for solving two-dimensional nonlinear integral equations. Electrical engineering was originally introduced to block-pulse functions by Harmuth, after which additional academics discussed the topic [8].

Recently, hybrid functions have been considered for solving numerous mathematical models, including [20,23,25]. Combining Legendre polynomials and block-pulse functions yields one of these functions. Using block-pulse functions and Legendre polynomials, [18] described a method for solving mixed-type Hammerstein integral equations, whereas [17] proposed a method for solving optimal control of Volterra integral systems. These hybrid functions have also been applied to solving nonlinear Fredholm-Hammerstein systems. [14,30] obtained a numerical solution of partial differential equations with nonlocal integral conditions; [19] solved Fredholm integral equation of the first kind; [28] discovered the optimal solution of linear time-delay systems; [29] discovered the numerical solutions of stochastic Volterra-Fredholm integral equations; and [13] includes the necessary definitions as well as some properties of Legendre polynomials and hybrid block-pulse functions.

In this study, the second type of two-dimensional nonlinear integral is considered. Under special conditions, the Banach fixed point theorem is used to discuss and prove the existence of a unique solution to two-dimensional nonlinear integral equations. We discuss the properties of hybrid functions, which combine block-pulse functions and Legendre polynomials. These integral equations are solved based on some useful properties of hybrid functions. This technique’s major characteristic is its ability to transform an integral problem into a set of algebraic equations; as a result, the solution processes are correspondingly either reduced or simplified.

The article’s structure is as follows: : In Section 2, the existence and unique solution of Eq. (1) are discussed. Section 3 describes a method for estimating a two-dimensional nonlinear integral equation’s solution. The convergence analysis of the provided method is derived in Section 4, Numerical results are shown in Section 5, and conclusion and Remarks are presented in the last Section 6.

This study aims to present a numerical approach for solving the following two-dimensional nonlinear integral equation approximatively:

\begin{matrix} γ ψ (x, y) & = f (x, y) + λ_{1} \int_{0}^{1} \int_{0}^{1} Φ (x, τ; y, v) μ (τ, v, ψ (τ, v)) d v d τ \\ + λ_{2} \int_{0}^{x} \int_{0}^{1} G (x, τ; y, v) ν (τ, v, ψ (τ, v)) d v d τ, \end{matrix}

(1)

where

λ_{r}, r = 1, 2

are constant scalers having several physical meanings, the function

ψ (x, y)

is unknown in the Banach space

L_{2} [0, 1] \times L_{2} [0, 1] .

The kernels

Φ (x, τ; y, v), G (x, τ; y, v)

are continuous in the same space and the known function

f (x, y)

is continuous in the space

L_{2} [0, 1] \times L_{2} [0, 1]

. In addition the constant

γ

defines the kind of two- dimensional nonlinear integral equations.

2. Existence of a unique solution for the integral equation

The existence of a unique solution of problem (1) will be discussed and proved in this section using the Banach fixed point theorem. For this, we write Eq. (1) in the form of an integral operator:

\bar{V} ψ (x, y) = \frac{1}{γ} f (x, y) + V ψ (x, y); γ \neq 0,

(2)

where

\begin{matrix} V ψ (x, y) & = \frac{λ_{1}}{γ} \int_{0}^{1} \int_{0}^{1} Φ (x, τ; y, v) μ (τ, v, ψ (τ, v)) d v d τ \\ + \frac{λ_{2}}{γ} \int_{0}^{x} \int_{0}^{1} G (x, τ; y, v) ν (τ, v, ψ (τ, v)) d v d τ . \end{matrix}

(3)

Also, we assume the following conditions:

(i): The kernels $Φ (x, τ; y, v)$ and $G (x, τ; y, v)$ satisfy the conditions: $∥ Φ (x, τ; y, v) ∥ \leq A_{1}, ∥ G (x, τ; y, v)) ∥ \leq A_{2},$ where $A_{1}$ and $A_{2}$ are two constants, assume $A = max {A_{1}, A_{2}}$ .
(ii): $∥ f (x, y) ∥ = {[\int_{0}^{1} \int_{0}^{1} {| f (x, y) |}^{2} d x d y]}^{\frac{1}{2}} = D, D$ is a constant.
(iii): The function $μ (x, y, ψ (x, y))$ satisfies the following conditions:

$\begin{matrix} ∥ μ (x, y, ψ (x, y)) ∥ = {[\int_{0}^{1} \int_{0}^{1} {| μ (x, y, ψ (x, y)) |}^{2} d x d y]}^{\frac{1}{2}} \leq M_{1} ∥ ψ (x, y) ∥, (a) \\ ∥ μ (x, y, ψ_{1} (x, y)) - μ (x, y, ψ_{2} (x, y)) ∥ \leq M_{2} ∥ ψ_{1} (x, y) - ψ_{2} (x, y) ∥ . (b) \end{matrix}$
(iv): The function $ν (x, y, ψ (x, y))$ is bounded and satisfy:

$\begin{matrix} ∥ ν (x, y, ψ (x, y)) ∥ = {[\int_{0}^{1} \int_{0}^{1} {| ν (x, y, ψ (x, y)) |}^{2} d x d y]}^{\frac{1}{2}} \leq N_{1} ∥ ψ (x, y) ∥, (a) \\ ∥ ν (x, y, ψ_{1} (x, y)) - ν (x, y, ψ_{2} (x, y)) ∥ \leq N_{2} ∥ ψ_{1} (x, y) - ψ_{2} (x, y) ∥ . (b) \end{matrix}$

Theorem 1.

Assume that the conditions

(i) - i v)

are satisfied. Eq. (1) has an unique solution

ψ (x, y)

in the space,

L_{2} [0, 1] \times L_{2} [0, 1]

. If the condition

η = A |\frac{λ}{γ}| [M + N] < 1; (λ = max {λ_{1}, λ_{2}}, M = max {M_{1}, M_{2}}, N = max {N_{1}, N_{2}})

(4)

is true.

The following two lemmas are necessary for the theorem’s proof:

Lemma 1.

Under the conditions

(i), (i i), (i i i - a)

, and

(i v - a)

, the operator

\bar{V} ψ (x, y)

defined by Eq. (2) maps the space

L_{2} [0, 1] \times L_{2} [0, 1]

into itself.

Proof. In light of formulas (2) and (3), we obtain

\begin{matrix} ∥ \bar{V} ψ (x, y) ∥ & \leq \frac{1}{| γ |} ∥ f (x, y) ∥ + |\frac{λ_{1}}{γ}| ∥\int_{0}^{1} \int_{0}^{1} | Φ (x, τ; y, v) | μ (τ, v, ψ (τ, v)) d v d τ∥ \\ + |\frac{λ_{2}}{γ}| ∥\int_{0}^{x} \int_{0}^{1} | G (x, τ; y, v) | ν (τ, v, ψ (τ, v)) d v d τ∥ . \end{matrix}

Using conditions (i) and (ii), we get

\begin{matrix} ∥ \bar{V} ψ (x, y) ∥ & \leq \frac{D}{| γ |} + A |\frac{λ_{1}}{γ}| ∥\int_{0}^{1} \int_{0}^{1} | μ (τ, v, ψ (τ, v)) d v d τ∥ \\ + A |\frac{λ_{2}}{γ}| ∥\int_{0}^{x} \int_{0}^{1} ν (τ, v, ψ (τ, v)) d v d τ∥ . \end{matrix}

Given conditions

(i i i - a)

and

(i v - a)

, the above inequality takes on the following form:

\begin{matrix} ∥ \bar{V} ψ (x, y) ∥ & \leq \frac{D}{| γ |} + A M ∥ ψ (x, y) ∥ |\frac{λ_{1}}{γ}| ∥\int_{0}^{1} \int_{0}^{1} d v d τ∥ \\ + A N ∥ ψ (x, y) ∥ |\frac{λ_{2}}{γ}| ∥\int_{0}^{x} \int_{0}^{1} d v d τ∥, \end{matrix}

where

max_{0 \leq x \leq 1} | x | = 1

, so that last inequality becomes

\begin{matrix} ∥ \bar{V} ψ (x, y) ∥ \leq \frac{D}{| γ |} + A |\frac{λ}{γ}| [M + N] ∥ ψ (x, y) ∥, \end{matrix}

since

∥ \bar{V} ψ (x, y) ∥ \leq \frac{D}{| γ |} + η ∥ ψ (x, y) ∥; η = A |\frac{λ}{γ}| [M + N] < 1 .

(5)

According to this inequality, the operator

\bar{V}

maps the ball

B_{r} \subset L_{2} [0, 1] \times L_{2} [0, 1]

into itself, where

r = \frac{D}{| γ | (1 - η)},

since,

r > 0, D > 0

, therefore we have

η < 1

. Furthermore, lower bounds for the operators V and

\bar{V}

are involved in the inequality (5).

Lemma 2.

If the conditions

(i), (i i i - b)

, and

(i v - b)

are verified, then the operator

\bar{V} ψ (x, y)

defined by Eq. (2) is continuous in the space

L_{2} [0, 1] \times L_{2} [0, 1]

Proof. For the continuity, Given two functions

Ψ_{1} (x, y)

and

Ψ_{2} (x, y)

in the space

L_{2} [0, 1] \times L_{2} [0, 1]

and satisfy Eq. (2), then

\begin{matrix} \bar{V} ψ_{1} (x, y) - \bar{V} ψ_{2} (x, y) & = \frac{λ_{1}}{γ} \int_{0}^{1} \int_{0}^{1} Φ (x, τ; y, v) [μ (τ, v, ψ_{1} (τ, v)) - μ (τ, v, ψ_{2} (τ, v))] d v d τ \\ + \frac{λ_{2}}{γ} \int_{0}^{x} \int_{0}^{1} G (x, τ; y, v) [ν (τ, v, ψ_{1} (τ, v)) - ν (τ, v, ψ_{2} (τ, v))] d v d τ, \end{matrix}

applying the properties of the norm, we obtain

\begin{matrix} ∥ \bar{V} ψ_{1} (x, y) - \bar{V} ψ_{2} (x, y) ∥ & \leq |\frac{λ_{1}}{γ}| ∥\int_{0}^{1} \int_{0}^{1} | Φ (x, τ; y, v) | | μ (τ, v, ψ_{1} (τ, v)) - μ (τ, v, ψ_{2} (τ, v)) | d v d τ∥ \\ + |\frac{λ_{2}}{γ}| ∥\int_{0}^{x} \int_{0}^{1} | G (x, τ; y, v) | | ν (τ, v, ψ_{1} (τ, v)) - ν (τ, v, ψ_{2} (τ, v)) | d v d τ∥ . \end{matrix}

In view of the conditions (i), (iii-b), and (iv-b), the above inequality becomes

\begin{matrix} ∥ \bar{V} ψ_{1} (x, y) - \bar{V} ψ_{2} (x, y) ∥ \leq A |\frac{λ}{γ}| [M + N] ∥ ψ_{1} (x, y) - ψ_{2} (x, y) ∥, \end{matrix}

since

∥ \bar{V} ψ_{1} (x, y) - \bar{V} ψ_{2} (x, y) ∥ \leq η ∥ ψ_{1} (x, y) - ψ_{2} (x, y) ∥ .

This inequality shows that,

\bar{V}

is a continuous operator in

L_{2} [0, 1] \times L_{2} [0, 1]

. Moreover

\bar{V}

is a contraction operator under the condition

η < 1

The previous two Lemmas 1 and 2 show that the operator

\bar{V}

defined by (2) is a contraction operator in the space

L_{2} [0, 1] \times L_{2} [0, 1]

. Hence, from Banach fixed point theorem,

\bar{V}

has a unique fixed point which is of course, the unique solution of Eq. (1).

3. Method of solution for the main problem

This section applies the collocation method, two-dimensional hybrid functions, and the Gauss quadrature formula to transform the integral equation (1) into nonlinear systems of equations. The following results are obtained by expanding the function

Ψ (x, y)

in Eq. (1) in relation to two-dimensional hybrid functions:

Ψ (x, y) = \sum_{m_{1} = 1}^{\infty} \sum_{n_{1} = 0}^{\infty} \sum_{m_{2} = 1}^{\infty} \sum_{n_{2} = 0}^{\infty} c_{m_{1} n_{1} m_{2} n_{2}} h_{m_{1} n_{1} m_{2} n_{2}} (x, y),

(6)

where the finite series in equation (6) can be written as

Ψ_{S, K} (x, y) = \sum_{m_{1} = 1}^{S} \sum_{n_{1} = 0}^{K - 1} \sum_{m_{2} = 1}^{S} \sum_{n_{2} = 0}^{K - 1} c_{m_{1} n_{1} m_{2} n_{2}} h_{m_{1} n_{1} m_{2} n_{2}} (x, y) .

(7)

where

c_{m_{1} n_{1} m_{2} n_{2}}, m_{1}, m_{2} = 1, 2, \dots, S, n_{1}, n_{2} = 1, 2, \dots, K - 1,

and

S, K

are the unknown hybrid coefficients to be determined.

Substituting Eq. (7) into Eq. (1) yields

\begin{matrix} γ ψ_{S, K} (x, y) & = f (x, y) + λ_{1} \int_{0}^{1} \int_{0}^{1} Φ (x, τ; y, v) μ (τ, v, ψ_{S, K} (τ, v)) d v d τ \\ + λ_{2} \int_{0}^{x} \int_{0}^{1} G (x, τ; y, v) ν (τ, v, ψ_{S, K} (τ, v)) d v d τ, \end{matrix}

(8)

Now, we discretize Eq. (8) at the set of collocation nodes

(x_{m}, y_{n})

for

m, n = 1, 2, \dots, S K,

as follows:

\begin{matrix} γ ψ_{S, K} (x_{m}, y_{n}) & = f (x_{m}, y_{n}) + λ_{1} \int_{0}^{1} \int_{0}^{1} Φ (x_{m}, τ; y_{n}, v) μ (τ, v, ψ_{S, K} (τ, v)) d v d τ \\ + λ_{2} \int_{0}^{x_{m}} \int_{0}^{1} G (x_{m}, τ; y_{n}, v) ν (τ, v, ψ_{S, K} (τ, v)) d v d τ, \end{matrix}

(9)

where

x_{m} = \frac{1}{2} (cos (\frac{(2 m - 1) π}{2 S K}) + 1), m = 1, 2, \dots, S K,

and

y_{n} = \frac{1}{2} (cos (\frac{(2 n - 1) π}{2 S K}) + 1), n = 1, 2, \dots, S K,

The integral operators in Eq. (9) are approximated using the Gauss-Legendre quadrature formula. For this, we use the following transformations to convert the integrals over

[0, 1]

into the integral over

[- 1, 1]

, respectively

\begin{matrix} ξ & = 2 τ - 1; τ \in [0, 1], \\ ϱ & = 2 v - 1; v \in [0, 1] . \end{matrix}

The integral over

[0, x_{m}]

must also be changed into the integral over

[- 1, 1]

, having the following form

\begin{matrix} \bar{ξ} & = \frac{2}{x_{m}} τ - 1; τ \in [0, x_{m}] . \end{matrix}

Then Eq. (9) is converted to

\begin{matrix} γ & ψ_{S, K} (x_{m}, y_{n}) = f (x_{m}, y_{n}) \\ + \frac{λ_{1}}{4} \int_{- 1}^{1} \int_{- 1}^{1} Φ (x_{m}, \frac{1}{2} (ξ + 1); y_{n}, \frac{1}{2} (ϱ + 1)) μ (\frac{1}{2} (ξ + 1), \frac{1}{2} (ϱ + 1), ψ_{S, K} (\frac{1}{2} (ξ + 1), \frac{1}{2} (ϱ + 1))) d ϱ d ξ \\ + \frac{λ_{2} x_{m}}{4} \int_{- 1}^{1} \int_{- 1}^{1} G (x_{m}, \frac{x_{m}}{2} (\bar{ξ} + 1); y_{n}, \frac{1}{2} (ϱ + 1)) \\ \times ν (\frac{x_{m}}{2} (\bar{ξ} + 1), \frac{1}{2} (ϱ + 1), ψ_{S, K} (\frac{x_{m}}{2} (\bar{ξ} + 1), \frac{1}{2} (ϱ + 1))) d ϱ d \bar{ξ} . \end{matrix}

The above equation can be expressed as follows using Gauss-Legendre quadrature:

\begin{matrix} γ & ψ_{S, K} (x_{m}, y_{n}) = f (x_{m}, y_{n}) \\ + \frac{λ_{1}}{4} \sum_{j = 1}^{ℓ_{1}} \sum_{i = 1}^{ℓ_{2}} w_{j} w_{i} Φ (x_{m}, \frac{1}{2} (ξ_{i} + 1); y_{n}, \frac{1}{2} (ϱ_{j} + 1)) \\ \times μ (\frac{1}{2} (ξ_{i} + 1), \frac{1}{2} (ϱ_{j} + 1), ψ_{S, K} (\frac{1}{2} (ξ_{i} + 1), \frac{1}{2} (ϱ_{j} + 1))) \\ + \frac{λ_{2} x_{m}}{4} \sum_{j = 1}^{ℓ_{1}} \sum_{i = 1}^{{\bar{ℓ}}_{2}} w_{j} {\bar{w}}_{i} G (x_{m}, \frac{x_{m}}{2} ({\bar{ξ}}_{i} + 1); y_{n}, \frac{1}{2} (ϱ_{j} + 1)) \\ \times ν (\frac{x_{m}}{2} ({\bar{ξ}}_{i} + 1), \frac{1}{2} (ϱ_{j} + 1), ψ_{S, K} (\frac{x_{m}}{2} ({\bar{ξ}}_{i} + 1), \frac{1}{2} (ϱ_{j} + 1))), \\ m = 1, 2, \dots, S K, n = 1, 2, \dots, S K, \end{matrix}

(10)

and

w_{j}, w_{i}

and

{\bar{w}}_{i}

are the corresponding weights.

This technique can be used to transform the two-dimensional nonlinear integral problem (1) into a solvable nonlinear system of algebraic equations.

4. Convergence analysis

The aim of this section is to describe the uniform convergence of the hybrid functions expansion and to determine the maximum absolute truncation error of the function

Ψ

based on hybrid functions.

Theorem 2.

Ψ \in C^{4} [0, 1]

, then the function

Ψ (x, y)

converges uniformly to the infinite sum of the hybrid functions of

Ψ (x, y)

described by (6)

Proof. The hybrid coefficients are defined as

\begin{matrix} c_{m_{1} n_{1} m_{2} n_{2}} = & \frac{\int_{0}^{1} \int_{0}^{1} Ψ (x, y) h_{m_{1} n_{1} m_{2} n_{2}} (x, y) d x d y}{\int_{0}^{1} \int_{0}^{1} h_{m_{1} n_{1} m_{2} n_{2}}^{2} (x, y) d x d y} \\ = & \frac{\int_{\frac{m_{2} - 1}{S}}^{\frac{m_{2}}{S}} \int_{\frac{m_{1} - 1}{S}}^{\frac{m_{1}}{S}} Ψ (x, y) L_{n_{1}} (2 S x - 2 m_{1} + 1) L_{n_{2}} (2 S y - 2 m_{2} + 1) d x d y}{\int_{\frac{m_{1} - 1}{S}}^{\frac{m_{1}}{S}} L_{n_{1}}^{2} (2 S x - 2 m_{1} + 1) d x \int_{\frac{m_{2} - 1}{S}}^{\frac{m_{2}}{S}} L_{n_{2}}^{2} (2 S y - 2 m_{2} + 1) d y} . \end{matrix}

Now, suppose that

2 m_{1} - 1 = \hat{m_{1}}

and

2 S x - \hat{m_{1}} = ℑ

, therefore

\begin{matrix} c_{m_{1} n_{1} m_{2} n_{2}} = & \frac{\int_{\frac{m_{2} - 1}{S}}^{\frac{m_{2}}{S}} (\int_{- 1}^{1} Ψ (\frac{\hat{m_{1}} + ℑ}{2 S}, y) L_{n_{1}} (ℑ) d ℑ) L_{n_{2}} (2 S y - 2 m_{2} + 1) d y}{\int_{- 1}^{1} L_{n_{1}}^{2} (ℑ) d ℑ \int_{\frac{m_{2} - 1}{S}}^{\frac{m_{2}}{S}} L_{n_{2}}^{2} (2 S y - 2 m_{2} + 1) d y} \\ = & \frac{(2 n_{1} + 1)}{2} \frac{\int_{\frac{m_{2} - 1}{S}}^{\frac{m_{2}}{S}} (\int_{- 1}^{1} Ψ (\frac{\hat{m_{1}} + ℑ}{2 S}, y) L_{n_{1}} (ℑ) d ℑ) L_{n_{2}} (2 S y - 2 m_{2} + 1) d y}{\int_{\frac{m_{2} - 1}{S}}^{\frac{m_{2}}{S}} L_{n_{2}}^{2} (2 S y - 2 m_{2} + 1) d y} . \end{matrix}

Using the technique of integration by parts with regard to ℑ and

(2 n + 1) L_{n} (ℑ) = L_{n + 1}^{'} (ℑ) - L_{n - 1}^{'} (ℑ)

, we obtain

c_{m_{1} n_{1} m_{2} n_{2}} = - \frac{1}{2} \frac{\int_{\frac{m_{2} - 1}{S}}^{\frac{m_{2}}{S}} (\int_{- 1}^{1} \frac{\partial}{\partial ℑ} Ψ (\frac{\hat{m_{1}} + ℑ}{2 S}, y) (L_{n_{1} + 1} (ℑ) - L_{n_{1} - 1} (ℑ)) d ℑ) L_{n_{2}} (2 S y - 2 m_{2} + 1) d y}{\int_{\frac{m_{2} - 1}{S}}^{\frac{m_{2}}{S}} L_{n_{2}}^{2} (2 S y - 2 m_{2} + 1) d y} .

Once again, an integration by parts of above relation, results that

\begin{matrix} c_{m_{1} n_{1} m_{2} n_{2}} = \\ \frac{1}{2} \frac{\int_{\frac{m_{2} - 1}{S}}^{\frac{m_{2}}{S}} (\int_{- 1}^{1} \frac{\partial^{2}}{\partial ℑ^{2}} Ψ (\frac{\hat{m_{1}} + ℑ}{2 S}, y) [\frac{- L_{n_{1}} (ℑ)}{2 n_{1} + 3} - \frac{L_{n_{1}} (ℑ) - L_{n_{1} - 2} (ℑ)}{2 n_{1} - 1}] d ℑ) L_{n_{2}} (2 S y - 2 m_{2} + 1) d y}{\int_{\frac{m_{2} - 1}{S}}^{\frac{m_{2}}{S}} L_{n_{2}}^{2} (2 S y - 2 m_{2} + 1) d y} . \end{matrix}

Now, we have

c_{m_{1} n_{1} m_{2} n_{2}} = \frac{1}{2 (2 n_{1} + 3) (2 n_{1} - 1)} \frac{\int_{\frac{m_{2} - 1}{S}}^{\frac{m_{2}}{S}} (\int_{- 1}^{1} \frac{\partial^{2}}{\partial ℑ^{2}} Ψ (\frac{\hat{m_{1}} + ℑ}{2 S}, y) ℵ_{n_{1}} (ℑ) d ℑ) L_{n_{2}} (2 S y - 2 m_{2} + 1) d y}{\int_{\frac{m_{2} - 1}{S}}^{\frac{m_{2}}{S}} L_{n_{2}}^{2} (2 S y - 2 m_{2} + 1) d y},

where

ℵ_{n_{1}} (ℑ) = (2 n_{1} - 1) L_{n_{1} + 2} (ℑ) - (4 n_{1} + 2) L_{n_{1}} (ℑ) + (2 n_{1} + 3) L_{n_{1} - 2} (ℑ) .

Similarly, changing the variable for y as

2 m_{2} - 1 = \hat{m_{2}}

where

2 S y - \hat{m_{2}} = ℘

and integrating by parts with respect to ℘, we get

\begin{matrix} c_{m_{1} n_{1} m_{2} n_{2}} & = \frac{1}{4 (2 n_{1} + 3) (2 n_{1} - 1) (2 n_{2} + 3) (2 n_{2} - 1)} \\ \int_{- 1}^{1} \int_{- 1}^{1} \frac{\partial^{4}}{\partial ℘^{2} \partial ℑ^{2}} Ψ (\frac{\hat{m_{1}} + ℑ}{2 S}, \frac{\hat{m_{2}} + ℘}{2 S}) ℵ_{n_{1}} (ℑ) ℵ_{n_{2}} (℘) d ℑ d ℘, \end{matrix}

where

ℵ_{n_{2}} (℘) = (2 n_{2} - 1) L_{n_{2} + 2} (℘) - (4 n_{2} + 2) L_{n_{2}} (℘) + (2 n_{2} + 3) L_{n_{2} - 2} (℘) .

Using the chain derivatives and

σ = {max}_{(x, y) \in [0, 1]} |\frac{\partial^{4} Ψ (x, y)}{\partial x^{2} \partial y^{2}}|

, it follows that

\begin{matrix} c_{m_{1} n_{1} m_{2} n_{2}} & \leq \frac{σ}{64 S^{4} (2 n_{1} + 3) (2 n_{1} - 1) (2 n_{2} + 3) (2 n_{2} - 1)} \int_{- 1}^{1} \int_{- 1}^{1} | ℵ_{n_{1}} (ℑ) | | ℵ_{n_{2}} (℘) | d ℑ d ℘, \\ \leq \frac{σ}{64 m_{1}^{2} m_{2}^{2} (2 n_{1} + 3) (2 n_{1} - 1) (2 n_{2} + 3) (2 n_{2} - 1)} \int_{- 1}^{1} \int_{- 1}^{1} | ℵ_{n_{1}} (ℑ) | | ℵ_{n_{2}} (℘) | d ℑ d ℘ . \end{matrix}

(11)

However

\begin{matrix} {(\int_{- 1}^{1} | ℵ_{n_{1}} (ℑ) | d ℑ)}^{2} = & (\int_{- 1}^{1} | (2 n_{1} - 1) L_{n_{1} + 2} (ℑ) - (4 n_{1} + 2) L_{n_{1}} (ℑ) + (2 n_{1} + 3) L_{n_{1} - 2} (ℑ) | d ℑ), \\ \leq & 2 \int_{- 1}^{1} | {(2 n_{1} - 1)}^{2} L_{n_{1} + 2}^{2} (ℑ) + {(4 n_{1} + 2)}^{2} L_{n_{1}}^{2} (ℑ) + {(2 n_{1} + 3)}^{2} L_{n_{1} - 2}^{2} (ℑ) | d ℑ, \end{matrix}

Using the Legendre polynomials’ orthogonality property, we determine that

{(\int_{- 1}^{1} | ℵ_{n_{1}} (ℑ) | d ℑ)}^{2} \leq \frac{24 {(2 n_{1} + 3)}^{2}}{2 n_{1} - 3},

(12)

thus

\int_{- 1}^{1} | ℵ_{n_{1}} (ℑ) | d ℑ \leq \frac{2 \sqrt{6} (2 n_{1} + 3)}{\sqrt{2 n_{1} - 3}},

(13)

and

\int_{- 1}^{1} | ℵ_{n_{2}} (℘) | d ℘ \leq \frac{2 \sqrt{6} (2 n_{2} + 3)}{\sqrt{2 n_{2} - 3}} .

(14)

By substituting (13) and (14) into (11), we obtain

\begin{matrix} c_{m_{1} n_{1} m_{2} n_{2}} & \leq \frac{24 σ}{64 m_{1}^{2} m_{2}^{2} (2 n_{1} - 1) (2 n_{2} - 1) \sqrt{(2 n_{1} - 3)} \sqrt{(2 n_{2} - 3)}}, \\ \leq \frac{3 σ}{8 m_{1}^{2} m_{2}^{2} {(2 n_{1} - 3)}^{\frac{3}{2}} {(2 n_{2} - 3)}^{\frac{3}{2}}} . \end{matrix}

(15)

Therefore, the series

\sum_{m_{1} = 1}^{\infty} \sum_{n_{1} = 0}^{\infty} \sum_{m_{2} = 1}^{\infty} \sum_{n_{2} = 0}^{\infty}

is absolutely convergent. Also,

\begin{matrix} | Ψ (x, y) | & = |\sum_{m_{1} = 1}^{\infty} \sum_{n_{1} = 0}^{\infty} \sum_{m_{2} = 1}^{\infty} \sum_{n_{2} = 0}^{\infty} c_{m_{1} n_{1} m_{2} n_{2}} h_{m_{1} n_{1} m_{2} n_{2}} (x, y)| \\ \leq \sum_{m_{1} = 1}^{\infty} \sum_{n_{1} = 0}^{\infty} \sum_{m_{2} = 1}^{\infty} \sum_{n_{2} = 0}^{\infty} | c_{m_{1} n_{1} m_{2} n_{2}} | \\ \leq \infty, \end{matrix}

and the series (6) converges to the function

Ψ (x, y)

uniformly.

Theorem 3.

The maximum absolute truncation error of the series solution (6) to two- dimensional nonlinear integral equation

()

is estimated to be,

∥ Ψ (x, y) - Ψ_{S, K} (x, y) ∥ \leq \frac{3 σ}{8 S} {(\sum_{m_{1} = S + 1}^{\infty} \frac{1}{m_{1}^{4}} \sum_{n_{1} = K}^{\infty} \frac{1}{{(2 n_{1} - 3)}^{4}} \sum_{m_{2} = S + 1}^{\infty} \frac{1}{m_{2}^{4}} \sum_{n_{2} = K}^{\infty} \frac{1}{{(2 n_{2} - 3)}^{4}})}^{\frac{1}{2}} .

Proof.

\begin{matrix} ∥ & Ψ (x, y) - Ψ_{S, K} (x, y) ∥ \\ = {(\int_{0}^{1} \int_{0}^{1} {(Ψ (x, y) - \sum_{m_{1} = 1}^{S} \sum_{n_{1} = 0}^{K - 1} \sum_{m_{2} = 1}^{S} \sum_{n_{2} = 0}^{K - 1} c_{m_{1} n_{1} m_{2} n_{2}} h_{m_{1} n_{1} m_{2} n_{2}} (x, y))}^{2} d x d y)}^{\frac{1}{2}}, \\ \leq {(\sum_{m_{1} = S + 1}^{\infty} \sum_{n_{1} = K}^{\infty} \sum_{m_{2} = S + 1}^{\infty} \sum_{n_{2} = K}^{\infty} c_{m_{1} n_{1} m_{2} n_{2}}^{2} \int_{0}^{1} \int_{0}^{1} h_{m_{1} n_{1} m_{2} n_{2}}^{2} (x, y) d x d y)}^{\frac{1}{2}} . \end{matrix}

Using the hybrid functions’ orthogonality property and taking relation (15) into consideration, we are able to

\begin{matrix} ∥ Ψ (x, y) - Ψ_{S, K} (x, y) ∥ \leq & {(\sum_{m_{1} = S + 1}^{\infty} \sum_{n_{1} = K}^{\infty} \sum_{m_{2} = S + 1}^{\infty} \sum_{n_{2} = K}^{\infty} c_{m_{1} n_{1} m_{2} n_{2}}^{2} \frac{1}{S^{2} (2 n_{1} + 1) (2 n_{2} + 1)})}^{\frac{1}{2}} \\ \leq & \frac{3 σ}{8 S} {(\sum_{m_{1} = S + 1}^{\infty} \frac{1}{m_{1}^{4}} \sum_{n_{1} = K}^{\infty} \frac{1}{{(2 n_{1} - 3)}^{4}} \sum_{m_{2} = S + 1}^{\infty} \frac{1}{m_{2}^{4}} \sum_{n_{2} = K}^{\infty} \frac{1}{{(2 n_{2} - 3)}^{4}})}^{\frac{1}{2}} . \end{matrix}

5. Application and numerical results

In order to show the accuracy and efficiency of the proposed method, some numerical examples are given in this section. We introduce the following notation to study the absolute values of this method’s errors:

R_{S, K} = | Ψ (x, y) - Ψ_{S, K} (x, y) |,

where

Ψ (x, y)

and

Ψ_{S, K} (x, y)

are the exact solution and the approximate solution of the integral equations, respectively.

Example 5.1.

Consider the following two- dimensional nonlinear integral equation:

16 ψ (x, y) = f (x, y) + \int_{0}^{1} \int_{0}^{1} (x τ + y v) ψ^{2} (τ, v) d v d τ + \int_{0}^{x} \int_{0}^{1} (x^{2} τ^{2} + y v) ψ^{2} (τ, v) d v d τ,

(16)

where

f (x, y) = \frac{- 7}{24} - \frac{28 y x}{45} + 16 (x^{2} + y^{2}) - \frac{x^{2} y^{2}}{360} (30 y^{4} + 72 y^{5} x + 45 y^{2} x^{2} + 80 y^{3} x^{3} + 30 x^{4} + 72 y x^{5}) .

The exact solution is

ψ (x, y) = x^{2} + y^{2} .

Using the proposed numerical technique, where

S = 2

and

K = 2, 4, 6, 8

in the interval

[0, 1)

In Table 1, we presented the absolute error

| Ψ (x, y) - Ψ_{S, K} (x, y) |

, using the introduced numerical method with

S = 2

and

K = 2, 4, 6, 8

in the interval

[0, 1)

. Table 2, shows the maximum absolute errors of the given method.

Moreover, in Figure 1, Figure 2, Figure 3 and Figure 4, we showed a comparison between the exact solution and the approximate solution using the presented numerical technique with different values of

K = 2, 4, 6, 8

with

S = 2

in the interval

[0, 1)

Example 5.2.

Consider the nonlinear integral equation:

ψ (x, y) = f (x, y) + 3 \int_{0}^{1} \int_{0}^{1} (x τ^{2} + v cos y) ψ^{3} (τ, v) d v d τ + 3 \int_{0}^{x} \int_{0}^{1} (x^{2} τ + y v) ψ (τ, v) d v d τ,

(17)

where

\begin{matrix} f (x, y) & = \frac{1}{12} (27 + 16 cos 1 - 16 cos 2 - 18 cos 3 + 7 cos 4 - 12 t^{2} x (2 + cos 1) sin {(\frac{1}{4})}^{4} \\ - 12 sin 1 - 36 sin 2 + 6 sin 3 + 6 sin 4) + x sin y + \frac{1}{2} y^{3} (- 3 x^{2} + x (2 + 3 x) cos x - 2 sin x) . \end{matrix}

The exact solution is

ψ (x, y) = x sin y .

Using the presented numerical technique with

S = 2

and

K = 3, 5, 7, 9

in the interval

[0, 1)

In Table 3, we showed the absolute error

| Ψ (x, y) - Ψ_{S, K} (x, y) |

, using the introduced numerical method with

S = 2

and

K = 3, 5, 7, 9

in the interval

[0, 1)

. Table 4, the maximum absolute errors of the given method are obtained.

Furthermore, in Figure 5, Figure 6, Figure 7 and Figure 8, we presented a comparison between the exact solution and the approximate solution using the introduced numerical method with different values of

K = 3, 5, 7, 9

with

S = 2

in the interval

[0, 1)

6. Conclusions and Remarks

The following can be deduced from the above analysis and discussion:

Under some conditions, the equation (1) has a unique solution $Ψ (x, y)$ in the space $L_{2} [0, 1] \times L_{2} [0, 1]$ .
After applying the proposed method, a two-dimensional integral equation of the second kind, in time and position, tends to result in an algebraic system of equations.
A nonlinear system of algebraic equations has a solution.
Maximum error obtained by proposed method is decreasing when number of $(K)$ is increasing.
Illustrative examples are provided to evaluate and validate the effectiveness and dependability of the proposed method.

References

Abdou, M.A.; Soliman, A.A.; Abdel–Aty, M.A. On a discussion of Volterra–Fredholm integral equation with discontinuous kernel. J. Egypt Math. Soc. 2020, 28. [Google Scholar] [CrossRef]
Abdou, M.A.; Nasr, M.E.; Abdel-Aty, M.A. A study of normality and continuity for mixed integral equations. J. of Fixed Point Theory Appl. 2018, 20. [Google Scholar] [CrossRef]
Al-Bugami, A.M. Numerical treating of mixed Integral equation two-dimensional in surface cracks in finite layers of materials. Advanced in math. Physics 2022, 25, 1–12. [Google Scholar] [CrossRef]
Alipour, M.; Baleanu, D.; Babaei, F. Hybrid Bernstein block-pulse functions method for second kind integral equations with convergence analysis. Abstr. Appl. Anal. 2014, 2014, 1–8. [Google Scholar] [CrossRef]
Bakhshayesh, S.J. , Discontinuous Galerkin approximations for Volterra integral equations of the first kind with convolution kernel. Indian Journal of Science and Technology 2015, 8, 1–4. [Google Scholar] [CrossRef]
Brezinski, C.; Redivo-Zalglia, M. Extrapolation methods for the numerical solution of nonlinear Fredholm integral equations. J. Integral Equations Appl. 2019, 31, 29–57. [Google Scholar] [CrossRef]
Chen, Z.; Jiang, W. An Efficient Algorithm for Solving Nonlinear Volterra-Fredholm Integral Equations. Appl. Math. Comput. 2015, 259, 614–619. [Google Scholar] [CrossRef]
Datta, K.B.; Mohan, B.M. Orthogonal Function in Systems and Control, 9, World Scientific, 1995.
Elzaki, T.M.; Alamri, A.S. Note on new homotopy perturbation method for solving nonlinear integral equations. J. Math. Comput. Sci. 2016, 6, 149–155. [Google Scholar]
Golberg, M.A.; Chen, C.S. Discrete Projection Methods for Integral Equation Computational Mechanics Publications, 1997.
Gouyandeh, Z.; Allahviranloo, T.; Armand, A. Numerical Solution of Nonlinear Volterra-Fredholm-Hammerstein Integral Equations via Tau-Collocation Method With Convergence Analysis. J. Comput. Appl. Math. 2016, 308, 435–446. [Google Scholar] [CrossRef]
Hafez, R.M.; Youssri, Y.H. Spectral Legendre-Chebyshev treatment of 2D linear and nonlinear mixed Volterra-Fredholm integral equation. Math. Sci. Lett. 2020, 9, 37–47. [Google Scholar]
Hashemzadeh, E.; Maleknejad, K.; Basirat, B. Hybrid functions approach for the nonlinear Volterra–Fredholm integral equations. Proc. Comput. Sci. 2011, 3, 1189–1194. [Google Scholar] [CrossRef]
Hesameddini, E.; Riahi, M. Hybrid Legendre Block-pulse functions method for solving partial differential equations with non-local integral boundary conditions. J. Inf. Optim. Sci. 2019, 40, 1391–1403. [Google Scholar] [CrossRef]
Katani, R. Numerical solution of the Fredholm integral equations with a quadrature method. SeMA J. 2019, 76, 449–452. [Google Scholar] [CrossRef]
Kreyszig, E. Introductory Functional Analysis with Applications, John Wiley and Sons. Inc., New York, 1989.
Maleknejad, K.; Ebrahimzadeh, A. An efficient hybrid pseudo-spectral method for solving optimal control of Volterra integral systems. Math. Commun. 2014, 19, 417–435. [Google Scholar]
Maleknejad, K.; Hashemizadeh, E. Numerical solution of the dynamic model of a chemical reactor by hybrid functions. Procedia. Comput. Sci. 2011, 3, 908–912. [Google Scholar] [CrossRef]
Maleknejad, K.; Saeedipoor, E. An efficient method based on hybrid functions for Fredholm integral equation of the first kind with convergence analysis. Appl. Math. Comput. 2017, 304, 93–102. [Google Scholar] [CrossRef]
Mashayekhi, S.; Razzaghi, M. Numerical solution of distributed order fractional differential equations by hybrid functions. J. Comput. Phys. 2016, 315, 169–181. [Google Scholar] [CrossRef]
Marzban, H.R.; Tabrizidooz, H.R.; Razzaghi, M. A composite collection method for the nonlinear mixed Volterra–Fredholm–Hammerstein integral equation, Commun. Nonlinear Sci. Numer. Simul. 2011, 16, 1186–1194. [Google Scholar] [CrossRef]
Masouri, Z. Numerical expansion-iterative method for solving second kind Volterra and Fredholm integral equations using block-pulse functions, Adv. Comput. Tech. Electromagn. 2021, 20, 7–17. [Google Scholar]
Mirzaee, F.; Alipour, S.; Samadyar, N. Numerical solution based on hybrid of Block-pulse and parabolic functions for solving a system of nonlinear stochastic Ito-Volterra integral equations of fractional order. J. Comput. Appl. Math. 2019, 349, 157–171. [Google Scholar] [CrossRef]
Mirzaee, F. Numerical solution of system of linear integral equations via improvement of block-pulse functions. J. Math. Model. 2016, 4, 133–159. [Google Scholar]
Mohammadi, F.; Moradi, L.; Baleanu, D.; Jajarmi, A. A hybrid functions numerical scheme for fractional optimal control problems: Application to non analytic dynamic systems. J. Vib. Control. 2017, 24, 5030–5043. [Google Scholar] [CrossRef]
Nasr, M.E.; Abdel-Aty, M.A. Analytical discussion for the mixed integral equations. J. of Fixed Point Theory Appl. 2018, 20. [Google Scholar] [CrossRef]
Nasr, M.E.; Abdel-Aty, M.A. A new techniques applied to Volterra–Fredholm integral equations with discontinuous kernel. J. of Computational Analysis and Appl. 2021, 29, 11–24. [Google Scholar]
Rafiei, Z.; Kafash, B.; Karbasi, S.M. State-control parameterization method based on using hybrid functions of Block-pulse and Legendre polynomials for optimal control of linear time delay systems. Appl. Math. Model. 2017, 45, 1008–1019. [Google Scholar] [CrossRef]
Ray, S.S.; Singh, S. Numerical solution of stochastic Volterra-Fredholm integral equations by hybrid Legendre Block-pulse functions. Int. J. Nonlinear Sci. Numer. Simul. 2018, 19, 1–9. [Google Scholar]
Sahu, P.K.; Ray, S.S. Hybrid Legendre Block-pulse functions for the numerical solutions of system of nonlinear Fredholm-Hammerstein integral equations. Appl. Math. Comput. 2015, 270, 871–878. [Google Scholar] [CrossRef]
Wang, K.; Wang, Q. Taylor polynomial method and error estimation for a kind of mixed Volterra-Fredholm integral equations. Appl. Math. Comput. 2014, 229, 53–59. [Google Scholar] [CrossRef]

Figure 1. Exact and approximate solution of Eq. (16) with

S = 2

and

K = 2 .

Figure 1. Exact and approximate solution of Eq. (16) with

S = 2

and

K = 2 .

Figure 2. Exact and approximate solution of Eq. (16) with

S = 2

and

K = 4 .

Figure 2. Exact and approximate solution of Eq. (16) with

S = 2

and

K = 4 .

Figure 3. Exact and approximate solution of Eq. (16) with

S = 2

and

K = 6 .

Figure 3. Exact and approximate solution of Eq. (16) with

S = 2

and

K = 6 .

Figure 4. Exact and approximate solution of Eq. (16) with

S = 2

and

K = 8 .

Figure 4. Exact and approximate solution of Eq. (16) with

S = 2

and

K = 8 .

Figure 5. Exact and approximate solution of Eq. (17) with

S = 2

and

K = 3 .

Figure 5. Exact and approximate solution of Eq. (17) with

S = 2

and

K = 3 .

Figure 6. Exact and approximate solution of Eq. (17) with

S = 2

and

K = 5 .

Figure 6. Exact and approximate solution of Eq. (17) with

S = 2

and

K = 5 .

Figure 7. Exact and approximate solution of Eq. (17) with

S = 2

and

K = 7 .

Figure 7. Exact and approximate solution of Eq. (17) with

S = 2

and

K = 7 .

Figure 8. Exact and approximate solution of Eq. (17) with

S = 2

and

K = 9 .

Figure 8. Exact and approximate solution of Eq. (17) with

S = 2

and

K = 9 .

Table 1. Absolute error of solution of Eq. (16) by using present method with

S = 2

and

K = 2, 4, 6, 8

Table 1. Absolute error of solution of Eq. (16) by using present method with

S = 2

and

K = 2, 4, 6, 8

$(x_{i}, t_{i})$	$S = 2, K = 2$	$S = 2, K = 4$	$S = 2, K = 6$	$S = 2, K = 8$
(0,0)	5.62845 $\times 10^{- 9}$	3.25447 $\times 10^{- 10}$	2.36512 $\times 10^{- 13}$	1.32654 $\times 10^{- 16}$
(0.1,0.1)	2.51405 $\times 10^{- 7}$	2.36524 $\times 10^{- 8}$	1.36524 $\times 10^{- 10}$	6.32514 $\times 10^{- 13}$
(0.2,0.2)	5.62103 $\times 10^{- 6}$	2.36985 $\times 10^{- 7}$	5.36214 $\times 10^{- 9}$	8.22551 $\times 10^{- 12}$
(0.3,0.3)	2.02154 $\times 10^{- 4}$	3.58412 $\times 10^{- 5}$	8.32541 $\times 10^{- 8}$	6.32165 $\times 10^{- 10}$
(0.4,0.4)	4.58721 $\times 10^{- 4}$	3.65413 $\times 10^{- 4}$	2.21345 $\times 10^{- 7}$	1.32114 $\times 10^{- 9}$
(0.5,0.5)	7.36212 $\times 10^{- 4}$	2.23651 $\times 10^{- 4}$	3.65221 $\times 10^{- 7}$	2.36985 $\times 10^{- 8}$
(0.6,0.6)	1.36521 $\times 10^{- 3}$	1.65214 $\times 10^{- 4}$	7.32651 $\times 10^{- 7}$	2.92541 $\times 10^{- 8}$
(0.7,0.7)	5.26512 $\times 10^{- 3}$	1.36524 $\times 10^{- 3}$	6.32541 $\times 10^{- 6}$	6.32548 $\times 10^{- 8}$
(0.8,0.8)	5.62514 $\times 10^{- 2}$	4.36210 $\times 10^{- 3}$	8.36251 $\times 10^{- 6}$	7.32614 $\times 10^{- 8}$
(0.9,0.9)	5.65214 $\times 10^{- 2}$	6.25489 $\times 10^{- 3}$	5.32658 $\times 10^{- 5}$	1.36524 $\times 10^{- 6}$

Table 2. The maximum error

R_{m a x} (x, y)

for different values of

K = 2, 4, 6, 8

and

S = 2

for Eq (16).

Table 2. The maximum error

R_{m a x} (x, y)

for different values of

K = 2, 4, 6, 8

and

S = 2

for Eq (16).

	$S = 2, K = 2$	$S = 2, K = 4$	$S = 2, K = 6$	$S = 2, K = 8$
$R_{m a x}$	6.2103 $\times 10^{- 2}$	6.53210 $\times 10^{- 3}$	5.32658 $\times 10^{- 5}$	1.36524 $\times 10^{- 6}$

Table 3. Absolute error of solution of Eq. (17) by using present method with

S = 2

and

K = 3, 5, 7, 9

Table 3. Absolute error of solution of Eq. (17) by using present method with

S = 2

and

K = 3, 5, 7, 9

$(x_{i}, t_{i})$	$S = 2, K = 3$	$S = 2, K = 5$	$S = 2, K = 7$	$S = 2, K = 9$
(0,0)	3.20514 $\times 10^{- 5}$	5.32641 $\times 10^{- 6}$	6.32141 $\times 10^{- 9}$	2.36541 $\times 10^{- 11}$
(0.1,0.1)	3.25481 $\times 10^{- 4}$	9.32541 $\times 10^{- 5}$	5.32187 $\times 10^{- 7}$	3.65874 $\times 10^{- 8}$
(0.2,0.2)	3.32541 $\times 10^{- 3}$	3.21554 $\times 10^{- 4}$	2.36414 $\times 10^{- 6}$	7.36584 $\times 10^{- 8}$
(0.3,0.3)	4.32641 $\times 10^{- 3}$	5.32654 $\times 10^{- 4}$	5.32684 $\times 10^{- 6}$	3.36241 $\times 10^{- 7}$
(0.4,0.4)	5.36854 $\times 10^{- 3}$	6.36524 $\times 10^{- 4}$	8.32546 $\times 10^{- 6}$	6.32584 $\times 10^{- 7}$
(0.5,0.5)	6.93154 $\times 10^{- 3}$	7.1.365 $\times 10^{- 4}$	6.32541 $\times 10^{- 5}$	8.65241 $\times 10^{- 7}$
(0.6,0.6)	1.32511 $\times 10^{- 2}$	3.21547 $\times 10^{- 3}$	9.99215 $\times 10^{- 5}$	4.32516 $\times 10^{- 6}$
(0.7,0.7)	4.32658 $\times 10^{- 2}$	4.36561 $\times 10^{- 3}$	1.32154 $\times 10^{- 4}$	8.69854 $\times 10^{- 6}$
(0.8,0.8)	5.32666 $\times 10^{- 2}$	5.76524 $\times 10^{- 3}$	2.34541 $\times 10^{- 4}$	4.36215 $\times 10^{- 5}$
(0.9,0.9)	6.32541 $\times 10^{- 2}$	7.96525 $\times 10^{- 3}$	3.25456 $\times 10^{- 4}$	1.05214 $\times 10^{- 4}$

Table 4. The maximum error

R_{m a x} (x, y)

for different values of

K = 3, 5, 7, 9

and

S = 2

for Eq (17).

Table 4. The maximum error

R_{m a x} (x, y)

for different values of

K = 3, 5, 7, 9

and

S = 2

for Eq (17).

	$S = 2, K = 3$	$S = 2, K = 5$	$S = 2, K = 7$	$S = 2, K = 9$
$R_{m a x}$	6.32541 $\times 10^{- 2}$	7.96525 $\times 10^{- 3}$	3.25456 $\times 10^{- 4}$	1.05214 $\times 10^{- 4}$

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Hybrid Functions Approach to Solving Nonlinear Integral Equations in Two Dimensions

Abstract

1. Introduction

2. Existence of a unique solution for the integral equation

3. Method of solution for the main problem

4. Convergence analysis

5. Application and numerical results

6. Conclusions and Remarks

References

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