The fundamental matrix properties of Dickson and Graph-Matching polynomials and their applications to functional integro-differential equations

This thesis aims to obtain the approximate solutions of the functional integro-differential equations and their some sub-classes, such as fractional and partial differential, integral equations arising in science and engineering by using two matrix-collocation methods based on Dickson and matching polynomials. These methods are made up of the matrix relations of the polynomials, collocation points and a fundamental matrix equation. After taking the matrix expansions of the functions at the collocation points in the equations, the methods are performed with solving them in the collocated matrix-system and so they can reach the solutions immediately. Furthermore, residual error analysis (or estimation), convergence analysis techniques based on residual function and first-second mean value theorems, are constructed to test their efficiency. The computer programs for each method are devised in accordance with the equations of general type to be considered. Hence, by precisely and sensitevely providing the approximations, the comparisons can be materialized at the scientific level. Numerical and graphical results are investigated and interpreted in tables and figures. Thanks to these methods, the numerical solutions of the equations, which are hardly solved analtically, are easily obtained and their consistencies are determined by the convergence analysis techniques. With the investigation of the results, it is observed that the methods are practical, simple and precise; also their programs are algorithmically in non-complex structure. Especially, some new problems derived from the general equation formulations can be directly considered in the future studies. One can conclude that the presented methods are widely applied to the nonliner partial integro-differential and nonlinear fractional partial differential equations.