A numerical solution of doubly degenerate parabolic equations in a class of discontinuous functions

As is known, differentiability order of the solution of a nonlinear degenerate parabolic type equation is lower than the order of differentiability which is required by the solution of equation. This feature of the equation shows that there is no classical solution of the equation. Also, because of this feature implementation of the well-known methods in the literature raises difficulties to find the solution. In this thesis, a special auxiliary problem having some advantages over the main problem has been proposed for the solution of one dimensional nonlinear doubly degenerate parabolic equation. Using the auxiliary problem, the convergence of the numerical solution to the exact solution is proven. High-precision numerical algorithms were created in a class of discontinuous functions and these algorithms have been applied to some problems. The thesis contains four chapters. In the first chapter of the thesis the required background and the necessary information and some concepts to obtain the solutions of the doubly degenerate equations are included. In the second chapter, Cauchy problem, boundary value problem and first-kind initial-boundary value problem are defined separately for a parabolic equation with double nonlinearity and for each of these problems the weak solutions have been introduced. In the third chapter, using the proposed auxiliary problem and results obtained in the second chapter, the numerical solution of the nonlinear equation is obtained and the convergence of this to the exact solution is proven. Also, numerical experiments were carried out for demonstrating the effectiveness of the proposed method. In the last chapter, the asymptotic structure of the solution of the Burgers equation is investigated.