Mixed Galerkin - Implicit Differences Methods for Solving a Coupled Nonlinear Parabolic System with Variable Coefficients

The approximate solution of a coupled nonlinear parabolic system with variable coefficients (CNPSVC) is found by using the mixing Galerkin finite element method (GFEM) in the variable of space with implicit finite difference method (IFDM) in the variable of time, for this reason, the method will be denoted by MGIM. In this method and at any step of time the CNPSVC is transformed into couple Galerkin nonlinear algebraic system (CGNAS), which is solved by applying the predictor and the corrector techniques (PCT), these techniques transform the CGNAS into a coupled Galerkin linear algebraic system (CGLAS). Then the Cholesky decomposition (ChDe) is used to solve it. The existence and uniqueness of the solution are proven. The stability and the convergence of the method are studied. Some Illustrative examples are given to solve the proposed system, the results are given by tables and figures and we show the accuracy and effectiveness of the proposed method.