UNBOUNDED SOLUTION OF CHARACTERISTIC SINGULAR INTEGRAL EQUATION USING DIFFERENTIAL TRANSFORM METHOD for Innovative Research

: In this paper, The differential transform method is extended to solve the Cauchy type singular integral equation of the first kind. Unbounded solution of the Cauchy type singular Integral equation is discussed. Numerical results are shown to illustrate the efficiency and accuracy of the present solution.

where K(x, t) and f(x) are given real valued functions belonging to the Holder class and g(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 integral is considered as Cauchy principal value integral. Chakrabarti and Berge [4] 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. Kim [5] solved CSIE by using Gaussian quadrature and chose the zeros of Chebyshev polynomials of the first and second kinds as the collocation and abscissa points. Abdulkawi [6] 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 identical with the exact solution when the force function is a polynomial of degree one.
The characteristic CSIE is of the form It is known that the analytical solution of the equation (2) for unbounded case is given by the following expression [7]     2 1

DIFFERENTIAL TRANSFORM METHOD
The transformation of the kth derivative of a function in one variable is as follows: (5) and the inverse transformation is defined by The following theorems can be deduced from Eqs. (5) and (6) [8].
where a is a constant and G(k) is a differential transform of g(x).

THE SCHEME OF THE NUMERICAL SOLUTION
The numerical solution of Eq. (2) is derived using the following procedures: 1-The unknown function g(t) is written as: where  (x) is regular function, so Eq. (2) becomes 2-The following singular integrals are evaluated 3-The following condition is imposed to obtain the unique solution   Rewriting g(t) as follows Using (12) into (13) yields It is known that and [7]   Substituting (15) and (16) into (14) we get It is not difficult to see that D e c e m b e r 31, 2014 Taking the differential transform onto (21) and using theorems 1, 2 and 3 yields

and C(k) is defined by (19)
Proof : Using (12) Taking the differential transform onto (22) and using theorems 1, 2 and 3 we obtain The proof is completed 