Bounded Solution of Cauchy Type Singular Integral Equation of The First Kind Using Differential Transform Method

In this paper, an efficient approximate solution for solving the Cauchy type singular integral equation of the first kind is presented. Bounded solution of the Cauchy type singular Integral equation is discussed. Two type of kernel, separable and convolution, are considered. The differential transform method is used in the solution. New theorems for transformation of Cauchy singular integrals are given with proofs. Approximate results areshown to illustrate the efficiency and accuracy of the approximate solution.


INTRODUCTION
The Cauchy type singular integral equation (CSIE)           11 11 , , 1 where K (x, t) and f(x) are given real valued functions belonging to the Holder class and (t) is to be determined, occurs in varieties of mixed boundary value problems of mathematical physics, isotropic elastic bodies involving cracks and other related problems [1][2][3]. The singular integral in CSIE (1) is considered as Cauchy principal value integral. Dezhborda et al. [5] investigated the numerical solution of various cases of CSIE (1) using reproducing kernel Hilbert space (RKHS) method. Chakrabarti and Berge [6] have proposed an approximate method to solve CSIE (1) using polynomial approximation of degree n and collocation points chosen to be the zeros of Chebyshev polynomial of the first kind for all cases. They showed that the approximate method is exact when the force function f(t) is linear. Abdulkawi [1] discussed the numerical solution of CSIE (1) for tow cases, unbounded and bounded. He approximated the unknown function by weighted Chebyshev polynomials of the first and second kind, respectively, and used Lagrange -Chebyshev interpolation to approximate the regular kernel. Eshkuvatov et al. [7] discussed approximate solution of CSIE (1) when K (x, t) = 0 for four cases. They used weighted Chebyshev polynomials of the first, second, third and fourth kinds. They showed that the numerical solution is ident ical with the exact solution when the force function is a polynomial of degree one. Abdulkawi and Akran [2] discussed the numerical solution of CSIE (1) when K (x, t) = 0 for bounded case. They used the Differential Transform Method in the solution. Dardery and Allan [10] discussed approximate solution of CSIE (1) by means of Chebyshev polynomials of first, second, third and fourth kind to obtain systems of linear algebraic equations which are solved numerically.
In this paper, we focus our discussion on approximate solution of equation for two types of the kernel, Separable and Convolution.
It is known that the analytical solution of the equation (2) for bounded case is given by the following expression [11]. The transformation of the kth derivative of a function in one variable is as follows: The following theorems can be deduced from Eqs. (5) and (6) [4].
The following Lemma and Theorems are used in this paper   2 .

 
and C(k ) is defined by Lemma 1.

THE SCHEME OF THE APPROXIMATE SOLUTION
The approximate solution is derived using the following procedures 1-The unknown function g (t) is approximated as: The proof is complete.
Using Theorem 2 and 5 into (23) we obtain The proof is complete.

SOLUTION OF THE SYSTEM FOR SEPARABLE KERNEL
Taking the differential transform for two sides of Eq.(7) and using Theorems 6 and 7 yields

SOLUTION OF THE SYSTEM FOR CONVOLUTION KERNEL
Taking the differential transform for two sides of Eq.(7) and using Theorems 9 and 6 yields  , then the differential transform of Using the system (24) with N=2 , we have