Integral equations are one of the most useful mathematical tools in the both pure and applied analysis . An integral equation is an equation in which the unknown function appears under an integral. Eric Ivar Fredholm was a swedish mathematician whose work on integral equations and operator theory foreshadowed the theory of Hilbert space . Fredholm introduced and analyzed a normal; MARGIN: 0cm 0cm 0pt" Vito Volterra was a physicist and a mathematician whose stature in the mathematical world was compared to that of David Hilbert . His work on integral and integro-differential equations and " functions of functions " led to the development of functional analysis . The type with integration over a fixed interval is called a Fredholm equation , while if the upper limit is x , a variable , it is a Volterra equation . The other fundamental division of these equations is into first and second kinds . If unknown function just appears under an integral , we have first kind of integral equations otherwise the integral equation is second kind . Many problems in engineering and mechanics can be transformed into integral equations . For example , it is usually required to solve Fredholm integral equations in the calculation of plasma physics. Integral equations occur in a variety of applications , often being obtained from a differential equation . Many physics problems that are usually solved by differential equations methods , can be solved more effectively by integral equation methods . Two kinds of partial differential equations are presented in this thesis that can be reformulated as 2D Volterra integral equation; Cauchy and Darboux problems. In recent years , a number of algorithms for the fast numerical solution of integral equations have been developed . In this thesis is considered numerical solution methods for two dimensionals Fredholm integral equation of the second kind with smooth kernel and a method for finding an approximate solution of a 0cm 0cm 10pt" I presented the properties of two dimensional shifted Legendre functions . The operational matrices of integration and product together with the collocation points are utilized to reduce the solution of the integral equation to the solution of a system of non-linear algebraic equations . The advantage of the orthogonal system , proposed in the present work , is that the Legendre bivariate polynomials provide an accurate approximation of the problem solution with the reduce number of basis functions . On the other hand , the computations can be handled in a simple way , making use of the recurrence formulae for Legendre polynomials and the operationals matrix techniques . Some results concerning the error analysis are obtained . The application of the method to the solution of certain partial differential equations also are considered . Numerical examples are given to illustrate the efficiency and accuracy of algorithm .