A Numerical Approach for Solving Quadratic Integral Equations of Urysohn’s Type using Radial Basis Function

Introduction Quadratic integral equations provide an important tool for modeling the numerous problems in engineering and science. These equations appear in the modeling of radiative transfer, kinetic theory of gases, traffic theory, neutron transport and in many other phenomena [2-7]. So, it is clear that solving this class of integral equations can be used to describe many events in the real world. Recently, many different types of research have been focusing on the effective properties of quadratic integral equations such as existence, uniqueness, monotonic solutions and positive solutions of this class of equations [8-13]. There are a few numerical and analytical methods to estimate the solution of the quadratic integral equations such as Picard and Adomian decomposition method (ADM) [14], and some other methods [15]. In this study, the radial basis functions method with the collocation scheme for solving quadratic integral equations of Urysohn’s type is described. The use of radial basis functions for solving the Fredholm integral equation was offered by Makroglou [1] and Alipanah and Dehghan [16] facilitated this method with the quadrature integration technique. Also, this method is compared with the method via orthogonal polynomials [17]. We utilize the method for solving the quadratic integral equations. A nonlinear Fredholm quadratic integral equation of Urysohn’s type can be considered as the following general form


Introduction
Quadratic integral equations provide an important tool for modeling the numerous problems in engineering and science. These equations appear in the modeling of radiative transfer, kinetic theory of gases, traffic theory, neutron transport and in many other phenomena [2][3][4][5][6][7]. So, it is clear that solving this class of integral equations can be used to describe many events in the real world. Recently, many different types of research have been focusing on the effective properties of quadratic integral equations such as existence, uniqueness, monotonic solutions and positive solutions of this class of equations [8][9][10][11][12][13]. There are a few numerical and analytical methods to estimate the solution of the quadratic integral equations such as Picard and Adomian decomposition method (ADM) [14], and some other methods [15].
In this study, the radial basis functions method with the collocation scheme for solving quadratic integral equations of Urysohn's type is described. The use of radial basis functions for solving the Fredholm integral equation was offered by Makroglou [1] and Alipanah and Dehghan [16] facilitated this method with the quadrature integration technique. Also, this method is compared with the method via orthogonal polynomials [17]. We utilize the method for solving the quadratic integral equations. The presented paper is organized in the following way. In Basic definitions section, we review some basic definitions relevant to the radial basis functions and quadrature integration rules which were applied in solving the process. In Description of method section, we describe the method of solving quadratic integral equations by using radial basis functions in details. Some illustrative examples are presented in Numerical examples section. Numerical results confirm the efficiency and high accuracy of the method. Finally, Conclusion Section concludes this paper with the brief summary and more discussion of the numerical results.

Basic Definitions
In this section, we review some required tools and definitions. Firstly, we introduce radial basis functions as the effective tools for approximation of the given functions. These basis functions approximate the continuous function with exponential rate of convergency [18,19]. Also, we remind the quadrature formulae for numerical integration.

Radial basis functions
when the interpolation conditions are imposed as follows: The unknown coefficient j λ is determined by solving the following The interpolant of ( ) f x is unique if and only if the matrix A is nonsingular.

Description of Method
In 1992, Makroglou [1] proposed the use of radial basis functions for solving the Fredholm integral equation. Alipanah and Dehghan [16] facilitated this method with the quadrature integration technique. In this section, we develop the method for solving the quadratic integral equations.

Fredholm integral equation
Consider the following Urysohn Fredholm quadratic integral equation which is mentioned in Eq.(1). Let ( ) x ϕ be a radial basis function and we approximate ( ) u t with the following interpolant =0 ( ); (P P) = ( ).
To obtain , = 0,1,..., , j c j n as unknowns in above equation, we collocate the points , = 0,1,..., , i t i n such as follows By applying quadrature integration formula described in Eq.(11), we can rewrite Eq.(15) in the following form , , = 0,1,..., , where j w and j τ , = 0,1,..., j n , are weights and nodes of Legendre-Gauss-Lobatto integration rule. This is a nonlinear system of equation that can be solved by the Newton's iteration method to obtain the unknown vector T C .

Volterra integral equation
Consider the following quadratic integral equation of Volterra type which is mentioned in Eq. (2). Similarly, we substitute Eq.(13) in Eq. (17) and collocate the points , = 0,1,..., i t i n . So we have .
In above equation, we let = ( ) = Obviously, for arbitrary interval [ , ] a b ,we have Now, by applying Legendre-Gauss-Lobatto integration formula demonstrated in Eq.(9), we approximate the integral of Eq.(19) as follows ( , ( ), 2 ( ( ))), where j w and j τ , = 0,1,..., j n , are weights and nodes of the integration rule. Again, we have a nonlinear system of equations that can be solved by the Newton's iteration method to obtain the unknown vector T C .

Numerical Examples
In    Table 1. We solve the problem with multi quadratic (MQ), inverse quadratic (IQ) and Gaussian (GA) radial basis function. The further investigations will be described in the conclusion section.  Table 2. The further investigations will be described in the conclusion section.        Table 3. The further investigations will be described in the conclusion section.   Figure 4. Also, the rms values for different N are reported in Table 4. function is shown in Figure 5. Also, the rms values for different N are reported in Table 4.  Figure 6. Also, the rms values for different N are reported in Table 4.         Table 4.

Conclusion
The radial basis functions and the collocation method provide the efficient method to solve the general type of linear and nonlinear quadratic integral equations. Moreover, the Urysohn's type of Fredholm and Volterra integral equation can be solved by this method. The proposed method reduces an integral equation to a system of equations. The obtained results showed that this approach can be flexible to solve many different problems effectively.
There are some notable points in numerical results which are considerable. Different radial basis functions can be applicable.  Table 2 and 3 is related to = 1, 2 T and 5 . An increase in T causes decrease in the accuracy. Moreover, there are reasonable relationships between rms and absolute error, see Table 4 and Figures 4-7. It will be considerable for cases that exact solution is not given.
Since RBF method uses the norm properties, the presented method can be useful for two dimensional quadratic integral equations. Moreover, the proposed method can be extended to solve the more general types of problem such as quadratic integro-differential equations and systems included quadratic integral equations.