A class of quasi-variable mesh methods based on off-step discretization for the solution of non-linear fourth order ordinary differential equations with Dirichlet and Neumann boundary conditions

Abstract

We propose a class of second and third order techniques based on off-step discretizations for a general non-linear ordinary differential equation of order four, subject to the Dirichlet and Neumann boundary conditions. Our approach uses only three grid points and involves the construction of a quasi-variable mesh. This type of a mesh is framed using a mesh ratio parameter η> 0 whose value is chosen in accordance with the occurrence of boundary layer in the problem, and varies with the number of grid points taken. The third order technique reduces to a fourth order one when taken with η= 1. The stability and convergence analysis of the techniques are discussed over a model problem. Computational results obtained upon the application to seven linear as well as non-linear problems endorse the theoretically claimed accuracies. We also provide a comparison with the computational results using approaches of other authors, which shows that the proposed methods are better.