High accuracy implicit variable mesh methods for numerical study of special types of fourth order non-linear parabolic equations

In this paper, we propose a new implicit high order two-level finite difference method of O(k2+khl+hl3) for the solution of special type of 1D fourth-order non-linear parabolic partial differential equation of the form uxxxx−2uxxt+utt=f(x,t,u,ux,uxx−ut,uxxx−uxt),a0 and a new three-level finite difference method of O(k2+hl3) for the solution of another type of 1D fourth-order non-linear parabolic partial differential equation of the form A(x,t,u,uxx)uxxxx+utt=g(x,t,u,ut,ux,uxx,uxxx),a0, both on a variable mesh, subject to suitable initial and Dirichlet boundary conditions, where k > 0 and hl > 0 are the mesh sizes in time and space coordinates, respectively. The third order variable mesh methods proposed are based on only three spatial grid points xl, xl ± 1, meaning that no fictitious points are required for incorporating the boundary conditions. We discuss how our formulation is able to handle linear singular problem. The proposed two-level finite difference method is shown to be unconditionally stable for a model linear problem. We applied the proposed methods to find the numerical solution of more complex fourth-order non-linear equations like second-order Benjamin–Ono equation and the good Boussinesq equation. It is evident from the numerical experiments that the numerical results agree well with the exact solutions, hence demonstrating the efficiency of the methods derived.