Numerical solutions of partial functional integro differential equations based on Laguerre polynomials and their applications

In this thesis study, Laguerre collocation method is applied to obtain approximate solutions of partial differential equations, parabolic convection-diffusion problems, partial integro differential equations, partial functional integro differential equations and nonlinear partial functional integro differential equations. In the method, equations and given conditions with the collocation points are reduced to a system of algebraic equations with Laguerre coefficients by putting them in the form of matrices. This system of algebraic equations is solved by using the codes developed in Maple and MATLAB programs and approximate solutions are obtained. In addition, error analysis of the approximate solutions are used for estimation of errors by different norms; Laguerre polynomial solutions are improved and the errors are reduced with the help of residual function. The study consists of five parts. In the first part, the general development of the partial functional integro differential equations and solution methods and also the aim of the thesis work are given. In the second part, basic concepts about these equations, different methods used for their numerical solutions, resource summaries about Laguerre polynomials and error analysis are available. In the third part, Laguerre collocation method, matrix relations, method of solution and error analysis are explained. In the fourth section, numerical examples are given for each problem type and the results are shown with the help of tables and graphs. Finally, in the fifth section, conclusions and suggestions are included.