IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 4, April 2015. www.ijiset.com ISSN 2348 – 7968 501 Solving Singular Partial Integro-Differential Equations Using Taylor Series Hussam E. Hashim 1 and Tarig M. Elzaki 2 1 Mathematics Department, Taif University Taif, Saudi Arabia 2 Mathematics Department, Jeddah University Jeddah, Saudi Arabia Abstract The aim of this study is to introduce a new technique to solve linear singular partial integro-differential equations (PIDEs) of first and second-order by using Taylor's series and convert the proposed PIDE to an partial differential equation. Solving this partial differential equation and applying the iteration method an exact solution of the problem is obtained. Some examples are presented in detail to show the accuracy and efficiency of this technique. Keywords: Partial integro-differential equations, Taylor's series, singular point 1. Introduction The theory and application of partial integro- differential equations (PIDEs) play an important role in the mathematical modeling of many fields: physical phenomena, biological models, chemical kinetics and engineering sciences in which it is necessary to take into account the effect of the real world problems. The general form of linear PIDE is:                  , , , , , , , , , , , , , , 0 , , 3 , 2 0 , 1 d y b c x a dt ds t s f t s y x k y x u y x y x f y x a y x y f y x a y x x f y x a x a y b m j i j i j i j i i i i n i i i i                               (1) where c b a , , and d are constants.   t s f , is the unknown function and   t s y x k , , , is the kernel of the integral equation. The functions         y x u y x a y x a y x a j i i i , , , , , , , , 3 , 2 , 1  and   y x f , are usually assumed to be continuous on the intervals   c a, and   d b, . Equations of this form are usually difficult to solve analytically so it is required to obtain an efficient approximate or numerical methods. These methods including Single-term Wash series method for Volterra integro-differential equations has been proposed by collocation method [4], Brunner applied a collecation- Sepehrian and Razzaghi [1], piecewise polynomials [2, 3], the spline type method to Volterra-Hammerstein integral equation as well as integro-di¤erential equations [5], the homotopy perturbation method (HPM) [6, 7], Haar wavelets [8], the wavelet-Galerkin method [9], the Tau method [10], the sinc-collocation method [11], the combined Laplace transform-Adomian decomposition method [12] to determine exact and approximate solutions, variational iterations method (VIM) [13] and Taylor polynomials[14]. The present work is motivated by the desire to obtain an exact solutions to first and second-order linear singular partial integro-differential equations, where the integrand is singular in the sense that its integral is continuous at the singular point, i.e. its kernel        s y t x t s y x k    1 , , , is singular as x t  and y s  . 2. Solutions by Taylor’s Series We propose an exact solution for solving linear singular partial integro-differential equations. The advantage of this method is that we remove the singularity of the kernel of first- and second-order linear singular partial integro-differential equations at x t  and y s  by judiciously applying Taylor’s approximation and then transforming the given singular partial integro- differential equation into an partial differential equation.