A galerkin-like numerical method for differential, functional differential and integro-differential equations and their systems

In its most general form, the interest of this thesis is to obtain numerical solutions of various problems encountered in natural sciences and engineering. The method used to obtain these numerical solutions makes use of Galerkin discretization, which is based on the integration of the product of the relevant equation with certain test functions. In this method, the approximate solution is sought in the form of a polynomial having a certain degree N, thus giving rise to the choice of the set of test functions as the classical polynomial basis {1,x,x^2,...,x^N}. The application of the method to various ordinary differential, delay differential and integro-differential equations has been separately described in detail, followed by the testing of the method on one or two example problems for each problem type. The results obtained from these tests have been compared within themselves and with those obtained by other numerical methods wherever possible. Then, some flaws of the method have been mentioned such as its non-applicability to partial differential equations and its failure to guarantee convergence by increasing the value of N in its original form. Some ways of overcoming these flaws have been suggested and these suggestions have been applied to example problems. As a result of these applications, the method has been observed to yield better results, constituting the original aspect of this study. Lastly, a technique has been considered which makes it possible to estimate the error of an obtained approximate solution by means of its residual and this technique has been used to improve the solution in question.