Wavelet Interpolation Method for Solving Singular Integral Equations

Numerical solutions of singular Fredholm integral equations of the second kind are solved by using Coiflet interpolation method. Error analysis of the method is obtained and examples are presented. It turns out that our method provides better accuracy than other methods.


Introduction
In the early 1900s, Ivar Fredholm solved the integral equations named after him, , where the function   g x and continuous kernel   , k x t are given, and the unknown function y(x) is to be determined.A numerical method of solving this equation has been shown in [1].In this study, we discuss the numerical solution of singular Fredholm integral equation of the second kind which is defined as follows: where the functions   f x and are given, the numerical solution for Equation ( 1) is to provide an approximation for the unknown function .In fact, Equation ( 1) is known as an Abel's integral equation which is defined by Niels Henrik Abel.There are many approaches to find a numerical solution of the Abel's equation [2], such as Gauss-Jacobi quadrature rule which was proposed by Fettis (1964), orthogonal polynomials expansion by Kosarev (1973), the Chebyshev polynomials of the first kind by Piessens and Verbaeten (1973) and Piessens (2000), etc.Recently, K. Maleknejad, M. Nosrati and E. Najafi solved the equation by using wavelet Galerkin method [3].Here we used Coiflets to find a numerical solution of Equation (1).

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The Coiflets are discussed in the next section briefly.In Section 3, we solve Abel's Equation (1) by using Coiflets.The error analysis is discussed in Section 4. Finally, we apply our method for two singular equations in the examples and compare our method with other method [3].We obtain numerical solutions which have achieved better accuracy.

Coiflets and Wavelet Interpolation
In the context of wavelet theory, we usually deal with wavelets and scaling functions [4] which means that for any function   j f x V  it can be expressed as: If the basis functions of a subspace are orthogonal at the same level, then a given function   j f x V  can be expressed as follows: , , If the nested sequence of the subspaces  has the following properties then it is called a multiresolution analysis (MRA): The wavelet function is constructed in the orthogonal complement of each subspace j V in , and can be obtained from the following equation:
The orthogonally of j W on j V means that any member of j V is orthogonal to the members of j W , that is, In fact, scaling function and wavelet have the following properties: In what follows, we will recall a scaling function interpolation theorem and the definition of Coiflets.As an application, we will use Coiflets and this interpolation formula to find numerical solutions of singular integral equations.
Definition 2.1.The Coifman wavelet system (Coiflet) of order L is an orthogonal multiresolution wavelet system with  d 0,for 1,2, , Lin and Zhou proved the following interpolation theorem in R 2 and : , sup sup and sup denotes the support of the function.
In addition the moments where is a constant depending only on and diameter of

Solving Singular Fredholm Integral Equation Using Coiflets
This section provides a method of finding numerical solution of Equation ( 1).In what follows, we assume that satisfies Lipschitz condition.The unknown function in Equation (1) can be expressed in term of scaling functions in the subspacev, where the function is approximated by To find the numerical solution we need to determinate the unknowns p a in Equation (3).By substituting Equation (3) in (1) we have the following equation, By providing sufficient collocation points in for Equation (4) we will have a linear system of linear equations with unknown [0,1] p a .In fact, the linear system can be written as the following matrix equation, is obtained from the left hand side of Equation ( 4).Subsequently, we substitute the solutions of a p into Equation (3), and obtain an approximate solution of the integral equation.

Error Analysis
The integral Equation ( 1) can be rewritten as follows [3]. where Then the integral Equation ( 1) is equivalent to the following equation, The next theorem shows the convergence rate of our method for solving Equation (1).Without loss of generality, we suppose that the integral equation is defined on the interval   0,1 .Theorem 4.1.In Equation ( 1), suppose that the function k satisfies the Lipchitz condition.Moreover,   is an approximate solution of the unknown function in Equation ( 1) with coefficients obtained in Section 3. Then for some constant c.Proof: We prove in two cases, one at singularities (case 1) and the other at the points x t  (case 2).Case 1.In Equation (1) when x t  , the function satisfies the Lipchitz condition and the function is continuous, then Equation ( 1) is equivalent to Equation ( 6), then the function which gives us the exact solution.
Case 2. In this case we don't have singularities, and Equation ( 1) is equivalent to Equation ( 6) and  .Subtracting Equation ( 7) from ( 6) and applying the norm, we have The unknown function can be interpolated using Coiflet such that If we add and subtract Equation ( 9) to ( 8), then Equation (8) becomes: and by using Equation (2), for some constant c which is absorbed from the above inequality.

Numerical Examples
In the following examples we are solving singular Fredholm integral equation of the second kind by using Coiflet of order 5 and calculate errors between the exact and numerical solutions at level j = −10.and the exact solution is .tions, Vol. 13, No. 4, 1997, pp. 303-320.x

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E. Lin and X. Zhou, "Coiflet Interpolation and Approximation Solutions of Elliptic Partial Differential Equations," Numerical Methods for Partial Differential Equa-