Numerical solutions of nonlinear parabolic equations with Robin condition: Galerkin approach

Abstract

In this paper, classical solutions of nonlinear parabolic partial differential equations with the Robin boundary condition are approximated using the Galerkin finite element method (GFEM) which is associated with the combination of the Picard iterative scheme and α-family of approximation. The uniqueness, convergence, and structural stability analysis of solutions are studied. It is proven that the iterative scheme of the numerical method is stable. To ensure the efficiency and accuracy of the method, the comparative study between the exact and approximate solutions both numerically and graphically are given by solving two nonlinear parabolic problems. A reliable error estimation also opens possibilities of acceptance of the method. The results confirmed the consistency of the method and ensured the convergence of solutions.