On the Solution of Quadratic Nonlinear Integral Equation with Different Singular Kernels

All the previous authors discussed the quadratic equation only with continuous kernels by different methods. In this paper, we introduce a mixed nonlinear quadratic integral equation (MQNLIE) with singular kernel in a logarithmic form and Carleman type. An existence and uniqueness of MQNLIE are discussed. A quadrature method is applied to obtain a system of nonlinear integral equation (NLIE), and then the Toeplitz matrix method (TMM) and Nystrom method are used to have a nonlinear algebraic system (NLAS). 'e Newton–Raphson method is applied to solve the obtained NLAS. Some numerical examples are considered, and its estimated errors are computed, in each method, by using Maple 18 software.


Introduction
Integral equations of various types and kinds play an important role in several mathematical problems modelling. Analytical solutions of integral equations, however, neither exist nor simple to find, so several numerical methods have been developed for finding the solutions of integral equations. e quadratic equation provides an important tool for modelling many numerical phenomena, bio-mathematical problems, and process engineering. In genetics, it represents the reproduction equation, through which events affecting the cells can be predicted. Gripenberg [1] studied the existence and the uniqueness of a bounded continuous solution to the following integral equation of product type: which arises in the study of the spread of an infectious disease that does not induce permanent immunity. Abdou and Basseem [2] used Chebyshev polynomial in solving mixed integral equation in position and time using spectral relationships. Javidi and Golbabai [3] solved NLFIE by the modified homotopy perturbation method. Alipanah and Esmaeili [4] used radial basis function to find a solution of two-dimensional FIE. Bernstein's method is used to solve VIE by Maleknejad et al. [5]. e Toeplitz matrix method is used to solve NLIE of Hammerstein by Abdou et al. [6]. Orsi [7] used the product Nystrom method to get the solution of NVIE when its kernel takes a logarithmic form and Carleman function. e degenerate kernel method is discussed in three-dimensional NLIE by Basseem [8,9]. Guoqiang et al. [10] obtained numerically the solution of two-dimensional NVIE by collocation and iterated collocation methods. Brunner et al. [11] introduced a class of methods to obtain numerically the solution of Abel integral equation. Abdou and Raad [12] used the Adomian decomposition method for solving quadratic NLIE. e radial basis function method with collocation scheme for solving quadratic integral equation of Urysohn's type is described by Avazzadeh [13]. Assaria et al. used meshless methods for solving NLIE (see [14][15][16][17]).
In this paper, a new problem in a product type of mixed integral equation with singular kernel is considered. e existence and uniqueness of its solution are discussed. e quadratic method is applied to obtain a NLS of FIE, and then the Toeplitz matrix method or Nystrom method is used to obtain a NLAS which is solved numerically by the Newton-Raphson method.
Consider the QNLE where . c(x, t, φ(x, t))is a given nonlinear function of the unknown function φ(x, t). e constant λ has many physical meanings.

Existence and Uniqueness
In order to guarantee the existence of a unique solution of equation (2), assume (1) e discontinuous kernel of equation (2) (2) e positive kernel of time is continuous and satisfies max 0≤t,τ≤T (3) e given continuous function f(x, t) ∈ L 2 [−1, 1] × C[0, T]and its norm is defined as (4) e function c(x, t, φ(x, t))satisfies where D and E are constants. (ii) For any two functions φ 1 and φ 2 , c(x, t, φ(x, t)) satisfies Lipchitz condition which is

Integral Operator of MQNIE
Eq. (2) can be written in the operator form as where Γ and F are the Nemytskii operator generated by the functions c(x, t, φ 2 (x, t)) and f(x, t), respectively.

Theorem 1. e solution of equation (2) exists and is unique under the condition |λ| < (1/MC).
The proof of this theorem can be deduced after the following discussion.

Lemma 1.
e operator Y is bounded.
Proof. We assume two functions φ n and φ m satisfy equation (2), and then we get

Mathematical Problems in Engineering
Applying Cauchy-Schwarz inequality, we have

Quadratic Numerical Method (See [18])
To obtain a system of NLIE, divide the time interval [0, T]as Let t � t i , then equation (2) becomes Applying the quadrature rule, equation (14) reduces to where Using the notation we get which is the system of NLIE can be solved by two different methods, namely, Toeplitz matrix method and Nystrom method.

Algebraic System of NLIE. Consider
where and In order to guarantee the existence of a unique solution of an algebraic system of NIE, we assume the following conditions: Hence, formula (19) has a unique solution under condition λ < (1/CM * ).

Definition 1.
e estimate local errorR (1) is determined by the following relation:

Toeplitz Matrix Method (See [6])
We apply the TMM to have a nonlinear algebraic equation. For this, consider h � (1/N); therefore, Mathematical Problems in Engineering e functions A n (x) and B n (x) are arbitrary functions to be determined, and R is the error term. In order to obtain the values of two functions, assume φ(y) � 1, y . is yields a set of two equations in terms of two unknown functions. After ignoring the error term, equation (18) becomes Let x � mh, then using the following notation:

equation (25) becomes
where Equation (27) represents that the NLAS can be solved using the Newton-Raphson method.

Definition 2.
e Toeplitz matrix method is said to be convergent of order r in the interval [−1, 1], if and only if, for sufficient large N, there exists a constant D > 0 independent of N such that Definition 3. e estimate local error R (2) is determined by the following relation:

Existence and Uniqueness of NLAS.
In order to guarantee the existence of a unique solution of a NLAS, we assume the following conditions: e estimate local error R (T) is determined by the following relation: where R (T) ≤ R (1) + R (2) .

Nystrom Method (See [7])
Here, by using the product integration, we approximate the integral part of equation (18) Comparing equations (32) and (33), we deduce en, By substituting in equation (18), we get where equation (38) represents the NAS in which its existence and uniqueness can be easily shown as in the previous section.

Definition 5.
e Nystrom method is said to be convergent of order r in the interval [−1, 1], if and only if, for sufficient large N, there exists a constant K > 0 independent of N such that

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where f(x, t) is given by putting φ(x, t) � x 2 t 2 as an exact value with c � φ 2 ,V(t, τ) � (t − τ) 2 , and (1) e following table is selected among a large amount of data to compare between the exact solution and its numerical solution in the case of logarithmic kernel for both of the previous methods in some points in the region x ∈ e negative sign means that by increasing N, the error decreases (see Table 3).
where f(x, t)is given by setting ϕ(x, t) � xt as an exact value.

General Conclusion
From the above tables and our numerical results, we can deduce the following: (1) e estimated error increases by time, where its mean errors by using Toeplitz and Nystrom methods, when T � 0.02, are 1.876 × 10 −12 and 2.191 × 10 −12 , respectively, while, its mean errors when T � 0.8 are 8.565 × 10 −6 and 3.192 × 10 −5 , respectively. (2) e Toeplitz matrix method is comparatively better than the Nystrom method for different kernels (see Tables 1 and 2). (3) By increasing N, the error is extremely stable in both methods, but in the Toeplitz matrix method, the error almost decreases by increasing in N, where the convergence rate with +ve sign means the increasing of errors, while its −ve sign means the errors decreasing (see Table 3).  Table 2).

Data Availability
e authors confirm that the data supporting the findings of this study are included within the article.