Numerical solutions of partial differential equations by using restrictive Taylor approximation method

This thesis is consists of four sections.In the first section, Taylor Approximations for time-dependent partial differential equation and Restrictive Taylor Approximation in which errors are totally removed at certain points by the help of this approximation has been given. The Taylor series relates the value of a differentiable function at any point to its first and higher order derivatives at a reference point, and consequently the first or higher order derivatives at the reference point can be obtained in terms of the sampled values of the functions. In the solution to approximate the exponential matrix function by using Restrictive Taylor Approximation has been showed.In the second section, with the help of Restrictive Taylor Approximations has been examined nümerical solution of the heat conduction equation. For this solution, stability analysis and truncation error has been examined. Effectiveness of Restrictive Taylor Approximation has been compared with other methods by considered problems.In the third section, Restrictive Taylor Approximations method has been applied to this equation to find the numerical solution of convection-diffusion equation. Gerschgorin theorem has been applied to examine the stability of obtained equation system, and truncation error for the solution has been examined. Effectiveness of Restrictive Taylor Approximation has been compared with Douglas method by considered problems.In the fourth section, with the help of Restrictive Taylor Approximations has been examined nümerical solution of Burgers? equation. For the solution, firstly non-linear Burgers? equation has been transformed to linear heat conduction by Hopf Cole transformation. For Burgers? equation, effectiveness of Restrictive Taylor Approximation has been compared with other methods by considered problems.Keywords: Burgers?equation, Convection-diffusion equation, Heat conduction equation, Restrictive Taylor Approximations.