Numerical Solution of Freholm-Volterra Integral Equations by Using Scaling Function Interpolation Method

Wavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions are the key elements of wavelet methods. In this article, we use scaling function interpolation method to solve Volterra integral equations of the first kind, and Fredholm-Volterra integral equations. Moreover, we prove convergence theorem for the numerical solution of Volterra integral equations and Freholm-Volterra integral equations. We also present three examples of solving Volterra integral equation and one example of solving Fredholm-Volterra integral equation. Comparisons of the results with other methods are included in the examples.


Introduction
The study of finite-dimensional linear systems is well developed.As an infinite-dimensional counter part of finite-dimensional linear systems, one can view integral equations as extensions of linear systems of algebraic equations.An integral equation maybe interpreted as an analogue of a matrix equation which is easier to solve.There are many different ways to transform integral equations to linear systems.Many different methods have been used for solving Volterra integral equations and Freholm-Velterra integral equations numerically.
In this paper, we first recall the method of scaling function interpolation.Then we solve linear Volterra integral equation of the form: and Fredholm-Volterra integral equations of the form: is known, and the function   y t is to be determined.One of the motivations in this study arose from equations in theoretical physics.In fact, there are many applications in several disciplines as well.We will use scaling function interpolation method to solve integral equations.As a natural question, one would wonder any possible convergence properties and how this method would compare with other methods.We will prove two convergence theorems and present several examples.

Approximation
Wavelets and scaling functions are a useful tool in approximation methods of solutions of differential and integral equations [1].We first recall Multiresolution analysis (MRA) [2].We assume the scaling function and wavelet function , Ψ are sufficiently smooth and satisfy MRA with compact support and Ψ has N vanishing moments (defined below).The scaling function   for some coefficients   , p p Z   .By using this dilation and translation we defined a nested of sequence spaces   For the subspace is built by In fact, for each j we define the orthogonal subspace j W of j V in the subspace 1 j V  , the or thogonal basis of j W is denoted by and the wavelet function can be obtained by for some coefficients p  .Some interesting properties of scaling and wavelet functions make wavelet method more efficiently than other methods such as spline approximations in solving an equation.A lot of computational time and storage capacity can be saved since we do not require a huge number of arithmetic operations partly due to the following properties.
Vanishing moments: and in this case the wavelet is said to have a vanishing moment of order k.Semiorthogonality: .
The set of scaling functions   , n j  is orthogonal at the same level n, which means: .
Coiflet (of order L) has more symmetries and it is an orthogonal multiresolution wavelet system with,  d 0, 1,2, , where is the moment of scaling functions.
  k M

Scaling Function Interpolation
In MRA, any given function     2 f x L R  can be interpolated by using the basis functions in the subspace j V as follows: where the coefficientsv p  are evaluated by using the semiorthogonality of th g functions (12) such that e scalin Hence the Equation (16) becomes as follows: To approximate a given function f, one can use sampling values of f at certain points.It is proved in [3], namely, an interpolation theorem using coiflet, namely, if where the index set is , sup sup where C is a constant depending only on N, diameter of Ω and where In this section, Coiflet is used to solve linear int Equations (1) and ( 2), where we will explain the method in terms of matrix notation.

Solutions of Linear Integral Equation
In this subsection we will use the interpolatio By substituting Equation ( 21) into the Equation (1), we have the following system, To simplify the system, let The coefficients can be evaluated by subst  equivalent to the system aA  , then the system (23) is f , and the solution is This gives raise to coefficien a

Equation
olve the Fred use a similar algorithm as we use in 4.1.The unknown function can be approximated by using Equation (1) and one can have the system of linear equations; where a is the vector of unknowns as we introduce in Equation ( 21), with and the set of t sec-which one can be ly spaced.In the nex tion we will discuss the convergence for the method by deriving a convergence theorem of this numerical solution.choose equal

Error Analysis
ide with the convergence rate of , suppose that the functions In this section, we prov our method for the numerical solution of solving linear Volterra integral equations and Freholm-Volterra integral equation respectively.We will explain the necessary conditions for the convergence.
Theorem 5.1 In Equation (1) and the two functions If an approximate solution of the Equation (1) with coefficients obtained in (24), and the error at the point where c is a constant.n with the following equation.

Proof:
We begi x t e t t m e t t m e t t m e t   

 
Then, By (19), the unknown function   y t can be interpola the coiflet such that: ted by using , j p  If we add and subtract Equation (28) in we get the following inequality: But by (20), we have that; And since  By using the above results and the rthonomality of the scaling functions In Equation (2), suppose that the functions , and If an oximate solution of the Equation (2) with coefficients obtained in (24), and the rror at the point

y t y t t k x t y t y t t k x t y t y t t k x t y t y t t y t y t t y t y t t
Add and subtract Equation (28) for absolute value in the previous equation, we get the following equation.
We use the same idea in the proof of 5.1, and obtain the following error estimate.

Remark
Here we discuss only the case when the kernel function   , k x t any given is positive.We can generalize our method for And hence the result is concluded i ilar fashion.n a sim

erica
lve several linea ns using coiflet of order 5 1-3) are al the e r resu

Num l Examples
In the following examples, we will so r Volterra integral equations of the first kind and Fredholme-Volterra integral equatio and provide the absolute errors.The examples ( so shown in [4] and xample 4 is presented in [5].We will compare ou lts with others and show that our method has better approximations than other methods.

Consider the integral
and the exact solution is   .
are known functions and called kernels.The function   f x

egral 4 . 1 .
Linear Volterra Integral Equation n Formula then the system (23) n If we use the notation can be written in the form

y x x  e 4 .
The numerical results are presented in Tabl Absolute errors where β is a constant  