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
