Hybrid Functions Approach via Nonlinear Integral Equations with Symmetric and Nonsymmetrical Kernel in Two Dimensions

: The second kind of two-dimensional nonlinear integral equation (NIE) with symmetric and nonsymmetrical kernel is solved in the Banach space L 2 [ 0,1 ] × L 2 [ 0,1 ] . Here, the NIE’s existence and singular solution are described in this passage. Additionally, we use a numerical strategy that uses hybrid and block-pulse functions to obtain the approximate solution of the NIE in a two-dimensional problem. For this aim, the two-dimensional NIE will be reduced to a system of nonlinear algebraic equations (SNAEs). Then, the SNAEs can be solved numerically. This study focuses on showing the convergence analysis for the numerical approach and generating an estimate of the error. Examples are presented to prove the efﬁciency of the approach.


Introduction
Integral equations are used in many disciplines of applied mathematics to explore and solve problems. See [1][2][3][4][5][6] for more information on the topic of two-dimensional nonlinear integral equations, which have long been of growing interest in many fields, including medicine [7], biology [8], physics [9], geography and fuzzy control [10]. According to the references [11][12][13][14][15][16], many problems in engineering [17], applied mathematics and mathematical physics [18] can be reduced to two-dimensional nonlinear integral equations with a symmetric and nonsymmetrical kernel. The analytical solutions for these equations are typically difficult. Therefore, it is necessary to use numerical methods or semianalytical methods to obtain the solutions numerically. For example, Bernstein polynomial hybrids with functions of block-pulse form [19,20] and Legendre hybrids with functions of blockpulse form [21,22] have both recently been examined as computational approaches for solving two-dimensional nonlinear integral equations. Alhazmi et al.,in [23], used the Lerch polynomial method for solving mixed integral equations in position and time with a strongly symmetric singular kernel. In [24], Azeem et al. conducted research by using the fractional derivative as a treatment for a partial cancer stem cell model. In [25], Attia et al. presented numerical solutions for the fractional differential equations, while in [26], Liaqat et al. presented Shehu transform and the Adomian decomposition technique in a novel algorithm form to establish approximate and exact solutions to quantum mechanics models. Electrical engineering was originally introduced to block-pulse functions by Harmuth; also, other academics have discussed the topic [27].
where λ 1 and λ 2 are constant scalers having several physical meanings; the function ψ(x, y) is unknown in the Banach spaces L 2 [0, 1] × L 2 [0, 1]. The kernels Φ(x, τ; y, v) and G(x, τ; y, v) are continuous in the same space, and the known function 1]. In addition, the constant γ defines the kind of nonlinear integral equations.

Existence of a Unique Solution for the Integral Equation with Symmetric and Nonsymmetrical Kernel
The existence of a unique solution of problem (1) is discussed and proved in this section using the Banach fixed point theorem. For this, we write Equation (1) in the form of an integral operator: where We assume the following conditions: (i) The kernels Φ(x, τ; y, v) and G(x, τ; y, v) satisfy the following conditions: (iii) The function µ(x, y, ψ(x, y)) satisfies the following conditions: where M 1 and M 2 are constants. (iv) The function ν(x, y, ψ(x, y)) is bounded and satisfies the following: where, N 1 and N 2 are constants.
Theorem 1. Assume that the conditions (i)-(iv) are satisfied. Then, Equation (1) has a unique solution ψ(x, y) in the space is true.
To prove the theorem, the following two lemmas must be proven: Under the conditions (i), (ii), (iii-4) and (iv-6), the operator Vψ(x, y), defined by Equation (2), maps the space Proof. In light of Formulas (2) and (3), we obtain From conditions (i) and (ii), we obtain Given conditions (iii-4) and (iv-6), the above inequality takes on the following form: where max 0≤x≤1 |x| = 1, so that the last inequality becomes According to this inequality, the operator V maps the ball .

Lemma 2.
If the conditions (i), (iii-5) and (iv-7) are verified, then the operator Vψ(x, y) defined by Equation (2) is continuous in the space Proof. Suppose two functions, Ψ 1 (x, y) and Ψ 2 (x, y), satisfy Equation (2). Then, applying the properties of the norm, we obtain In view of the conditions (i), (iii-5) and (iv-7), the above inequality becomes This inequality shows that V is a continuous operator in L 2 [0, 1] × L 2 [0, 1]. Moreover, V is a contraction operator under the condition η < 1. The previous two Lemmas 1 and 2 show that the operator V defined by (2) is a contraction operator in the space L 2 [0, 1] × L 2 [0, 1]. Hence, from the Banach fixed point theorem, V has a unique fixed point which is, of course, the unique solution of Equation (1).

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 Equation (1) with respect to two-dimensional hybrid functions: where the finite series in Equation (10) can be written as where c m 1 n 1 m 2 n 2 , m 1 , m 2 = 1, 2, . . . , S, n 1 , n 2 = 0, 1, 2, . . . , K − 1, and S, K are the unknown hybrid coefficients. Substituting Equation (11) into Equation (1) yields Now, we discretize Equation (12) at the set of collocation nodes (x m , y n ) for m, n = 1, 2, . . . , SK as follows: where the integral operators in Equation (13) are approximated by 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], The integral over [0, x m ] must also be changed into the integral over [−1, 1], having the following formξ Then, Equation (13) is converted to The above equation can be expressed as follows using Gauss-Legendre quadrature: and w j , w i andw 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.

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. 1], then the function Ψ(x, y) converges uniformly to the infinite sum of the hybrid functions of Ψ(x, y) described by (10).

Theorem 3. The maximum absolute truncation error of the series solution (10) to nonlinear integral
Equation (1) is Using the orthogonality property of hybrid functions and taking relation (19) into consideration, we obtain

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: where Ψ(x, y) and Ψ S,K (x, y) are the exact solution and the approximate solution of the integral equations, respectively. Example 1. Consider the following two-dimensional nonlinear integral equation with a symmetric and nonsymmetrical kernel: 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 5, we present the absolute error |Ψ(x, y) − Ψ S,K (x, y)|, using the introduced numerical method with S = 3 and K = 2, 3, 6, 7 in the interval [0, 1). Table 6 shows the maximum absolute errors of the given method.  Moreover, in Figures 9-12, we show a comparison between the exact solution and the approximate solution using the presented numerical technique with different values of K = 2, 3, 6, 7 with S = 3 in the interval [0, 1).

Conclusions and Remarks
The following can be deduced from the above analysis and discussion: 1.
Under some conditions, Equation (1) has a unique solution Ψ(x, y) in the space

2.
After applying the proposed method, a two-dimensional integral equation with a symmetric and nonsymmetrical kernel of the second kind tends to result in an algebraic system of nonlinear equations. 3.
A nonlinear system of algebraic equations has a solution.

4.
Three illustrative examples are provided to evaluate and validate the effectiveness and dependability of the proposed method. Tables and Figures are used to show the numerical results. For example, Figures 1, 5 and 9 contained the numerical solution of Examples 1, 2 and 3, respectively, for different values of x, y and K. Figures 1-12 formed the absolute errors of each example with different values of x and y.

5.
In Example 1, absolute errors in four cases of K are presented in Table 1 and Figures 1-4. The error increases through x and y. When we take the Max. value error in Figure 1, it is (6.2103 × 10 −2 ) at S = 2, K = 2. Also, the Min. error value in Figure 4 is (1.36524 × 10 −6 ) at S = 2, K = 8 (see Table 2). 6.
In Example 2, from Table 4 at S = 2, K = 3, the error is as high as possible at point x = y = 0.9, and its value is (6.32541 × 10 −2 ). Likewise, the error begins to decrease, and when the value of x = y = 0, its value is (1.05214 × 10 −4 ).

7.
In Example 3, the error decreases as S and K increase, where the maximum value of the error at x = y = 0.9 for S = 3, K = 2 is (5.01478 × 10 −3 ), while for S = 3, K = 7, the minimum value of the error is (5.31231 × 10 −8 ). 8.
In general, the error obtained by the proposed method decreases when the number of (K) increases.