The compact difference approach for the fourth‐order parabolic equations with the first Neumann boundary value conditions

Abstract

In this paper, the fourth‐order parabolic equation with the first Neumann boundary value conditions is concerned, where the values of the first and second spatial derivatives of the unknown function at the boundary are given. A compact difference scheme is established for this kind of problem by using the weighted average and order reduction methods. The difficulty lies in the challenges of handling the boundary conditions with high accuracy. The unique solvability, convergence and stability of the proposed compact difference scheme are proved by the energy method. Some novel techniques are introduced for the analysis. Then, the extension to a more general case with the reaction term is briefly explored. Finally, two numerical examples are numerically calculated to show the efficiency of the proposed numerical schemes.