Note on Using Radial Basis Functions Method for Solving Nonlinear Integral Equations

In this note, the solution of nonlinear integral equations was discussed using radial basis functions (RBFs) method. This method will represent the solution of nonlinear integral equation by interpolating the RBFs based on LegendreGauss-Lobatto (LGL) nodes and weights. Zeros of the shifted Legendre polynomials are used as the collocation points. The method is used to some examples to illustrate the accuracy and the implementation of the method.


Introduction
The theory and application of integral equation is an important subject within applied mathematics.Integral equations are used as mathematical models for many and varied physical situations, and integral equations also occur as reformulations of other mathematical problems such as partial differential equations and ordinary differential equations.For this result, we need to solve this kind of equations.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) [1], and some other methods [2,3,4].The RBF method in solving integral equations was initially proposed in 2006 [5,6,7,8].The related research attracted a lot of attention recently.In this paper we propose a point interpolation mesh less method for solving the nonlinear Ferdholm and Volterra equations.The method is based upon radial basis functions, using zeros of the shifted Legendre polynomial as the collocation points.Many different basis functions have been used to estimate the solution of linear and nonlinear integral equations, such as orthogonal bases, wavelets and hybrid [9].RBFs are powerful tools in multi-variable approximation [10] and are increasingly being used in the numerical solution of partial differential equations [11].The main advantages of the RBFs for interpolating multidimensional scattered data are highlighted in [12].In recent decades,meshless methods have been proved to treat scientific and engineering problems efficiently.Meshless method based on the collocation method has been dominated and very efficient.Over the last several decades RBFs have been found to be widely successful for the interpolation of scattered data.RBF methods are not tied to a grid and in turn belong to a category of methods called meshless methods.They apply only a cloud of points without any information about nodal connections.It is (conditionally) positive definite, rotationally and translationally invariant.The RBF approximation is an extremely powerful tool for representing smooth functions in non-trivial geometries, since the method is meshfree and can be spectrally accurate [13].
A nonlinear Ferdholm integral equation is defined as and nonlinear Volterra integral equation is defined as where y(x) is an unknown function, η is constant and given functions assumed to have nth derivatives [14].Though the different choices of the parameters lead to various problems, the method can afford to approximate the solution of them.The layout of the paper is as follows.In Section 2, the radial basis functions are introduced.Section 3, reviews the Legendre-Gauss-Lobatto integration process.Section 4, as the main part, presents the solution nonlinear integral equations by direct and indirect process of RBFs.Numerical illustrative examples are included in Section 5. A conclusion is drawn in the Section 6.
2 Review on the principles of RBF method

RBF methodology
The history of RBF approximations goes back to 1968, when multiquadric RBFs were first used by Hardy to represent topographical surfaces given sets of sparse scattered measurements [15].This method was popularized in 1982 by Richard Franke with his report on 32 of the most commonly used interpolation methods [16].Richard Franke, in 1982 compared scattered data interpolation methods, and concluded MQs and TPs were best.Franke conjectured interpolation matrix for MQs is invertible [16].He also conjectured the unconditional non-singularity of the interpolation matrix associated with the multiquadric radial function, but it was not until a few years later that Micchelli [17] was able to prove it as mentioned above.The main feature of the MQ method is that the interpolant is a linear combination of translations of a basis function which only depends on the Euclidean distance from its center.This basis function is therefore radially symmetric with respect to its center.That is how its name radial basis function comes about.The MQ method was generalized to other radial functions, such as the thin plate spline, the Gaussian, the cubic, etc.In the 1990s researchers became to pay attention to the RBF method again when Kansa [18] introduced a way to use it for solving parabolic, elliptic and (viscously damped) hyperbolic PDEs.Results [19] on the spectral convergence rate of MQ interpolation followed from Madych and Nelson in 1992.Definition 2.1.Let R + = {x ∈ R x ≥ 0} the non-negative half-line and let ϕ : R + −→ R be a continuous function with ϕ (0) ≥ 0. A radial basis functions on R d is a function of the form ϕ (||x − x i ||) where x, x i ∈ R d and r = ||x − x i ||, denotes the Euclidean distance between x and x , i s.If one chooses N points is called a radial basis functions as well [20].(SeeTable 1) Parameter c is a parameter for controlling the shape of functions which effects on the rate of convergency.The standard radial basis functions are divided into two major classes [21]: Class 1. Infinitely smooth RBFs [21]: In this class the basis function ϕ (r) heavily depends on the free shape parameter c e.g.Hardy multiquadric (MQ),Gaussian (GA), inverse multiquadric (IMQ) and inverse quadric (IQ)(See Table 1)).Class 2. Piecewise smooth RBFs [21]: The key advantage is that the basis functions of this category are shape parameter free, including Linear r, Cubic r 3 , Thin plate spline (r 2n logr, n = 1, 2, 3, ..., ) ect.The choice of shape parameter is an important task in approximating functions by RBFs and researchers always have concerned about selecting a good shape parameter.Optimal shape parameter values are found experimentally and these values are written for exact text problems.Theoretically, RBF methods are most accurate when the shape parameter is small.Many authors have investigated the shape parameter.The implementation of RBF methods involves solving a linear system that is extremely ill-conditioned when the parameters of the method are such that the best accuracy is theoretically realized.Thus, in applications, RBF methods are not able to produce as accurate of results as they are theoretically capable of.

Basic knowledge about RBFs approximation
For scattered data (x i , u(x i )) ∈ R d+1 , the approximation s(x) for a real function u(x) can be constructed by a linear combinations of translates of one function ϕ (||.||) of one real variable which is centered at The most attractive feature of the RBF methods is that the location of the centers can be chosen arbitrarily in the domain of interest.To determine the unknown coefficients λ j , j = 1, 2, ..., M., we can impose the interpolation conditions on s(x) i.e. s(x i ) = u i ; i = 1, 2, .., M.This gives the N × N linear system, This is summarized in a system of equations for the unknown coefficients λ j , where ] and u = u(x i ) are N × 1 matrices.The matrix A is called the interpolation matrix or the system matrix.Note that ϕ i (x j ) = ϕ (||x i − x j ||) therefore we have ϕ i (x j ) = ϕ j (x i ) consequently A = A T .The interpolant of u(x) is unique if and only if the matrix A is nonsingular.It has been discussed about sufficient conditions for ϕ (r) to guarantee nonsingularity of the A matrix i.e. there is a unique interpolant of the form Eq.(2.3), no matter how the distinct data points are scattered in any number of space dimensions.Micchelli [17] and Powell [22] have shown the existence of the interpolation.In the cases of inverse quadratic (IQ), inverse multiquadric (IMQ), hyperbolic secant (sech) and Gaussian (GA) the matrix A is positive definite and, for multiquadric (MQ), it has one positive eigenvalue and the remaining ones are all negative.In the piecewise smooth cases, a slight variation of the form of Eq.( 2.3) will again ensure nonsingularity [23,24,25].2l+d) for any y(x) satisfies N − y (α) ∥ ≤ ch 1−α where ϕ (x) is RBFs and the constant c depends on the RBFs, d is space dimension, l and α are nonnegative integer.It can be seen that not only RBFs itself but also its any order derivative has a good convergence.
Here, < ... > represents the usual L 2 [−1, 1] inner product and are{P i } i≥0 the well-known Legendre polynomials of order i which are orthogonal with respect to the weight function w(x) = 1 on the interval [−1, 1], and satisfy the following formulae: where x 1 = −1 < x 2 < ... < x N−1 < x N = 1 are Legendre-Guass-Lobatto nodes and w i Legendre-Gauss-Lobatto weights given in [28] where z = b−a 2 x + b+a 2 .It is well known that the integration in Eq.(3.6) is exact whenever f (x) is a polynomial of degree ≤ 2N + 1.

Description of Method
In this section we apply the results of the previous section to solve two the nonlinear integral equations.The first one is a fredholm integral equation which is solved by the RBF collocation method and the second one is a volterra integral equation which is solved by using RBFs.Also we approximate its corresponding integral by the Legendre-Gauss-Lobatto points and weights.

Fredholm integral equation
We consider the following integral equation of Fredholm type Suppose that the one dimensional approximation y(x) at an arbitrary point x by function ϕ (x), in the following form: where x j , s are known as centers.The unknown coefficients λ j are to be determined by the collocation method.Then, from substituting Eq.(4.10) into Eq.(4.9) we have, (4.11) We now collocate Eq.( 4.11) at points In above equation,we let t = b−a 2 x + b+a 2 .It reduces Eq.(4.12) to the following equation: for i = 1, ..., N. By using the Legendre-Gauss-Lobatto integration formula described in Eq.(3.6),we can approximate the integral in Eq.(4.13) and hence the above equation can be written as follows: where w i ,x i , i = 1, ..., N are weights and nodes of the integration rule respectively.Eq.(4.14) generates an N set of equations which can be solved by mathematical software for the unknowns vector λ .

Volterra integral equation
The nonlinear Volterra integral equation take the following form: Let's approximate the function y(x) in terms of radial basis functions, ϕ (x), as follows Then, from substituting Eq.(4.16) into Eq.(4.15) we have, Substituting the collocation points {x i } N i=1 into Eq.(4.17), we obtain: In above equation,we let t = x i −a 2 x + x i +a 2 .It reduces Eq.(4.18) to the following equation: for i = 1, ..., N. Now, by applying Legendre-Gauss-Lobatto integration formula demonstrated in Eq.(3.6), we approximate the integral of Eq.(4.19) as follows where w i , x i , i = 1, ..., N are weights and nodes of the integration rule respectively.Again, we have a nonlinear system of equations that can be solved by mathematical software for the unknowns vector λ .

Numerical illustration
In this section,four examples are considered.The results of numerical experiments are compared with the exact solution in illustrative examples to confirm the accuracy and efficiency of the proposed method.We point out that the corresponding numerical solutions are obtained using Software Matlab.To study the convergence behavior of the RBFs method, we applied the following laws: 1.The L 2 error norm of the solution which is defined by 2. The L ∞ error norm of the solution which is defined by

The root mean square (RMS) is defined by
where x j , j = 1, ..., N are Legendre-Guass-Lobatto nodes.[30] and [31], which used Sinc collocation method, the errors in each row have been decreased, which is a good factor in the RBFs method.
Example 5.2.Next, we consider the following nonlinear Fredholm integral equation with the exact solution is y where 0 ≤ x ≤ 1 [31].
The error, L 2 -error,L ∞ -error and Root-Mean-Square(RMS)-error norms for N = 5, 10, 15 with MQ-RBF are illustrated in Table 3. From Table 3, it can be observed that the accuracy increases with the increase of number of collocation points.Also, we have compared it with the Sinc-collocation method [31] and Haar wavelet method [32].These results verify that our method is considerable accurate than the methods of [31,32].where the exact solution is y(x) = x 2 + 1 2 .The error, L 2 -error,L ∞ -error and Root-Mean-Square(RMS)-error norms for N = 5, 10, 15 with MQ-RBF are reported in Table 4 From Table 4, we find that the accuracy measured inL 2 ,L ∞ and RMS norm errors decreases as N increase.5 Also, in comparison with the results of [35], which used Legendre wavelets method, the errors in each row have been decreased, which is a good factor in the RBFs method.From this table , we find that the accuracy measured inL 2 ,L ∞ and RMS norm errors decreases as N increase.Exact solutions for nonlinear integral equations are not often available, so approximating these solutions is very important.Many authors have proposed different methods.In this article, we have applied meshless approach based on the RBF for numerical solution of nonlinear integral equations.The proposed method reduces an integral equation to a system of equations.Implementation of our method is easy and accurate, this has been verified by test examples.The numerical results given in the previous section demonstrate the efficiency and good accuracy of this scheme.

2 , 3 ,
....For convenience the solution we use RBFs with collocation nodes x i , 2 ≤ i ≤ N −1 which are the zeros of the Legendre polynomia P ′ (x),where P ′ (x) is derivative P(x) of on the interval [−1, 1].Also we approximate the integral of f (x) on [−1, 1] as

Table 1 :
Deffinition of some types of RBFs Name of RBF (Abbreviation)

Table 2 :
Errors in the solution of Example 5.1 with c = 2The error, L 2 -error,L ∞ -error and Root-Mean-Square(RMS)-error norms for N = 5, 10, 15 with MQ-RBF are reported in Table2.As this table illustrates,numerical results show simplicity and very good accuracy of the method.The errors decreases by increasing the number of collocation points N for MQ-RBFs.Accuracy of the present approximation is examined in the L 2 -error and L ∞ -error,RMS-error norms.The results of RMS-error are comparatively better than the L 2 -error and L ∞ -error norms.Also, in comparison with the results of

Table 3 :
Errors in the solution of Example 5.2 with c = 1.8