Numerical solutions based on hermite polynomials of partial integro differential equations with two independent variables and their applications

In this study, Hermite collocation method is developed for approximate solution of partial differential equations, one dimensional parabolic convection-diffusion problems, two-dimensional integral equations, two-dimensional partial integro differential equations, one-dimensional delayed parabolic Volterra partial integro-differential equations and one-dimensional nonlinear partial integro-differential equations. The Hermite collocation method, using the matrix forms of the Hermite polynomials, provides the matrix form of each term in the equation and conditions under consideration and transforms the equation and conditions into a matrix form. The method reduces the solution of the given problem to the solution of a matrix equation corresponding to algebraic equations system with unknown Hermite coefficients. By solving this system of equations, Hermite polynomial solutions are obtained. In study, the method has been applied to examples of each type of equations in order to reveal to the correctness and validity of it. Absolute solutions have been compared to Hermite polynomial solutions through tables and figures. Also, the absolute error function and the residual error function have been tested by comparing with the help of tables and figures. The codes developed in MATLAB program have been used for the solution and analysis of the problems discussed in the study. Hermite collocation method is easy to program, gives fast and reliable results. This is also the biggest advantages of the proposed method.