Application of radial basis function to approximate functional integral equations

In the present paper, Radial Basis Function (RBF) interpolation is applied to approximate the numerical solution of both Fredlholm and Volterra functional integral equations. RBF interpolation is based on linear combinations of terms which include a single univariate function. Applying RBF in functional integral equation, a linear system ΨC = G will be obtain in which by defining coefficient vector C, target function will be approximiated. Finally, validity of the method is illustrated by some examples.


Introduction
Functional differential equations is one of the subjects which are interested by many researchers aspect of numerical or analytical solutions.In this study, Fredholm and Volterra functional integral equations of the second kind are investigated.The general concept of functional equations blongs to Arndt [9] and further details are investigated by some researchers [10,11,12].A new interpolation method for functional integral equations and functional integro-differential equations is introduced by Rashed [2].Among variety of the numerical methods, RBF method appears to be the best one in literature.RBF approximations are usually finite linear combinations of the translation of a radially symmetric basis function, φ(∥ .∥) where (∥ .∥) is the Euclidean norm.The set of RBFs, {ϕ i } m 1 is as follows: where ∥ .∥ denote the Euclidean norm and x i is the center of RBF.Gaussian (GA) ϕ (r) = exp(−σ r 2 ), multi quadric (MQ) ϕ (r) = √ (r 2 + σ 2 ) are some well-known functions that generate RBF.More functions are shown in Table 1 in which r = ∥x − x i ∥.The mentioned functions due to having the parameter c i have exponentially convergence [15].
RBFs are computationally means to approximate functions which are complicated or have many variables, by other simpler functions which are easier to understand and readily evaluated.One of the outstanding advantages of interpolation by RBF, unlike multivariable polynomial interpolation or splines [14], is applicability in scattered data aspect of existence and uniqueness results since there is little restrictions on dimension and also high accuracy or fast convergence to the target function.As another advantage of RBF there are not required to triangulations of the data points, while other numerical methods such as finite elements or multivariate spline methods need triangulations [14,13].This requirement cost computationally especially in more than two dimensions.In this paper, we consider RBF to approximate the solution of the problem as meshfree approximations.In this study, we consider two types of Fredholm and Volterra functional integral equations of the second kind as the following; • General form of Fredholm functional integral equations: • General form of Volterra functional integral equations: where A(x), h(x) and g(x) are analytical known functions.
To approximate the target function y(x) , we employ RBF interpolation in distinct grids from a definite domain.To this purpose, a linear component is considered as follows; where ϕ i (x) can be chosen from one of the basis functions which is mentioned in Table 1, according to type of the target function which is desired to approximate.For instance, consider Gaussian function, then we would have; and similarly it is true for y ] is a known function.In the next section, the application of RBF to functional integral equations will be described in details.Section 3 is devoted to some examples of both Fredholm and Volterra type equations with numerical solutions.Section 4, presents some concluding remarks.

Describtion of the method
The method is applied in n dimensional Euclidean space.To this purpose, consider m distinct points as (x 1 , x 2 , . . ., x m ) ∈ R d , in this space at which the function to be approximated is known and real scalers (g(x 1 ), g(x 2 ), . . ., g(x m )) which are given values at the points.We desire to construct a continuous function s : R n → R so that s(x j ) = g j for j = 1, 2, . . ., m.
Radial basis function method is based on continuous function such as ϕ : R + → R + and a norm ∥ .∥ in R d , then s can be as the following form: where c j are scalar parameters which should be chosen so that sapproximates g in point x j for j = 1, 2 . . ., m.Then funtions γ → ϕ (∥ γ − x j ∥) translates s into a vector space.According to interpolation conditions a linear system will be defined as ΨC = G, where Ψ ∈ R n×n is called a distance matrix or interpolation matrix, and given by and also The interpolation matrix is non-singular since it is a positive definite matrix, so we have the unique existence of the coefficients c j .

Convergence of the method
To analysis the convergence of RBF, let data points are on equispaced grids in R d and the spacing are denoted by h, when h → 0. Indeed, we have infinite uniform grids of spacing h.Considering s as an approxiamte of function g by RBF, uniform difference between s and g is closed to zero at the same rate as some power of h.In rest, we consider the convergenced of two basis functions, Gaussians and inverse quadratics, the methodology holds for other functions.Definition 2.1.( Native space [16]) The native space that is conditionally positive definite on Ω with respect to Punisolvent subset is defined by The space is equipped with a semi-inner product via where T P : C(Ω) → P and T P ( f The concept of convergence of RBF interpolation on a domain Ω ∈ R d is based on functions on native spaces N ϕ (Ω) .
For some basis functions such as Gaussians and inverse quadratics [16], which are strictly positive definite basis functions, we can construct the native space of a conditionally positive basis functions as the completion of the pre-Hilbert space.Consider the linear space Defining the inner product, F ϕ (Ω) becomes a pre-Hilbert space In RBF applications fill distance of h is considerd as follows; so for sufficiently small h N and data x j , the following relations are resulted for inverse quadratics and Gaussians; (2.9) Concept of convergence for RBF approximation is defined for functions over doamain Ω ∈ R d which are on native spaces N ϕ (Ω) .

Implementation of RBF to functional linear integral equations of the second kind
As it is mentioned in itroduction section, functional linear integral equations of the second kind are as the following form (2.10) If u(x) = b, it is Fredholm type and if u(x) = x, it is a Volterra functional integral equation.In the sequel, RBF interpolation has employd to approximate y(x) as the solution of functional linear integral equations of the second kind (2.10).To this end, linear component of functions ϕ j is replased in y(x) as the following form: and

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The matrix form for Eq.(2.16) is ΨC = G, where G = (g 1 , g 2 , . . ., g N ) T and C = (c 1 , c 2 . . ., c N ) T .So we have the following relation: where ψ i, j = ϕ i j + A j φi j + λ φ i j . (2.18) Finally, the functional integral equation is approximated by system of N linear equations.Also, the method is extended to treat the functional equations of advanced type (λ = 0 in (1.1) or (1.2)).
we investigate the performance and the ability of the present method by giving a test problem.Therefore, in order to illustrate the efficiency and the accuracy of the proposed method along with the TR method In next section, we investigate the performance and robustness of the method by giving some examples.

Numerical examples
To illustrate the efficiency of the proposed technique, we consider the following examples.The results have been provided by Matlab.
Example 3.1.Consider the following integral equation of the second kind: where y(x) = x 2 , is the exact solution.
• Case 1.For Fredholm integral equation of the second kind we have u(x) = b, g(x) = x 3 + x 2 + 1 3 x − 1 4 so integral equation is as the following form; According to Eq. (19) we have the following relations; We would obtain C from ΨC = G which was mentioned in previous section.Therefor, y(x) ≈ ∑ 20 i=0 c i ϕ i (x) is defined.In Figure 1, accuracy of the method and obtained values of approximation of y and the exact solution in interpolation grids for defined interval are shown.Similarly, we have the same methodology for Volterra functional integral equation, by some changes in ψ j (x i ) = ∫ x i 0 (x i − t)ϕ j (t)dt and g.The accuracy of the method for Volterra functional integral equation, is shown in Figure 2. As it is obvious, both figures 1 and 2 are the same since the solutions of Fredholm and Volterra integral equations are the same y(x) = x 2 , and the solution is approximated with the same accuracy in this example.In other problem the accuracy would different as it is shown in next example.Also, Table 1 shows the absolute errors for two types Fredholm and Volterra integral eqautions in various number of points.
We would obtain C from ΨC = G which was mentioned in previous section.Therefor, y(x) ≈ ∑ 20 i=0 c i ϕ i (x) is defined.In Figure3, accuracy of the method and obtained values of approximation of y and the exact solution in interpolation grids for defined interval are shown.Similarly, we have the same methodology for Volterra functional integral equation, by some changes in ψ j (x i ) = ∫ x i 0 (x i − t)ϕ j (t)dt and g.The accuracy of the method for Volterra functional integral equation, is shown in Figure 4. Also, Table 3 shows the absolute errors for two types Fredholm and Volterra integral eqautions in various number of points.

Conclusion
So far different approaches are suggested to functional integral equations, however, they were expensive computationally.In this study, we suggested RBF interpolation to approximate solution of functional integral equations.We also employed two examples to illustrate the qualification of the proposed method and the results showed that by applying RBF, approximate has high accuracy even in few distinct point of domains such as N = 3 or N = 5.Other advantages of this method can be low cost of computation and setting up the equations.

Theorem 2 . 1 .
Suppose that g is an analytical function in an open region that includes the strip |Im(z)| ≤ 1 2ε in its interior.Then for all node distributions {x j }} j, j = 1, . . ., N with fill distance h N = O(1/N) on the unit interval, the error in the inverse quadratic RBF interpolation is

Table 2 : 6 ,Figure 1 :
Figure 1: Left figure shows the exact and RBF approximatation for Fredholm functional integral equation of the second kind and accuracy of the method is shown in right figure.

Figure 2 :
Figure 2: Left figure shows the exact and RBF approximatation for Volterra functional integral equation of the second kind and Error of the proposed method on Volterra integral equation for Example 1, is shown in right figure.

Table 3 : 6 ,Figure 3 :
Figure 3: Left figure shows the exact and RBF approximatation for Volterra functional integral equation of the second kind and Error of the proposed method on Volterra integral equation for Example 2, is shown in right figure.

Figure 4 :
Figure 4: Left figure shows the exact and RBF approximatation for Volterra functional integral equation of the second kind and Error of the proposed method on Volterra integral equation for Example 2, is shown in right figure.

Table 1 :
Well-known functions that generate RBF