NUMERICAL SOLUTION FOR VOLTERRA-FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND BY USING PROJECTION METHODS

Abstract

This seminar was concerned within known function approximation by projection methods with help of projection operators for Volterra Fredholm integral equations of the second. The unknown function of the problem was approximated by Lagrange polynomials for the case of collocation methods and orthogonal Legendre polynomials for Galerkin methods. This is because in order to use the exact function the unknown function must found this is not possible always or may be very difficult. The necessary preliminary concepts, examples and theorems presented to make the concept of projection methods more clear. Many methods are available for approximating unknown function for those VFIEs to the desired precision in numerical approximation This Seminar is concerned with two popular projection methods (CM and GM).Under CM Lagrange polynomials were derived Properties of these trial functions were used to reduce the VFIEs to some algebraic system the speed of convergency of this method analyzed for some examples and numerical results were compared and showed that as the number of Lagrange basis increases error would decrease, but in GM the trial function are orthogonal Legendre polynomials these Legendre polynomials used to reduce VFIEs to algebraic system. For solving the mentioned algebraic system Gaussian elimination were applied to determine the corresponding coefficients (weight functions). This provide great advantage in computing VFIEs numerically by projection methods by comparisons of both methods we obtained that CM more speed of convergency and easy to compute Most computations were performed using MATLAB codes. The method is computationally attractive and applications are demonstrated through illustrative examples. As it shown in the reported tables of examples, compare the error of the methods, we can find that the presented method gets better approximate solution