A class of quasi-variable mesh methods based on off-step discretization for the numerical solution of fourth-order quasi-linear parabolic partial differential equations

Ranjan Kumar Mohanty,Deepti Kaur

doi:10.1186/s13662-016-1048-3

Abstract

Numerical schemes based on off-step discretization are developed to solve two classes of fourth-order time-dependent partial differential equations subjected to appropriate initial and boundary conditions. The difference methods reported here are second-order accurate in time and second-order accurate in space and, for a nonuniform grid, second-order accurate in time and third-order accurate in space. In case of a uniform grid, the second scheme is of order two in time and four in space. The presented methods split the original problem to a coupled system of two second-order equations and involve only three spatial grid points of a compact stencil without discretizing the boundary conditions. The linear stability of the presented methods has been examined, and it is shown that the proposed two-level finite difference method is unconditionally stable for a linear model problem. The new developed methods are directly applicable to fourth-order parabolic partial differential equations with singular coefficients, which is the main highlight of our work. The methods are successfully tested on singular problems. The proposed method is applied to find numerical solutions of the Euler-Bernoulli beam equation and complex fourth-order nonlinear equations like the good Boussinesq equation. Comparison of the obtained results with those for some earlier known methods show the superiority of the present approach.

Highlights

In Section, we present and derive new quasi-variable mesh two-level off-step discretizations to solve the particular type of fourth-order partial differential equation (PDE) ( )
3 Two-level off-step discretization strategy and truncation error analysis we develop new quasi-variable mesh off-step finite difference methods for the differential equation ( ) with initial and boundary conditions given by ( a)-( c)
We propose finite difference approximations for the fourth-order timedependent parabolic PDEs ( ) and ( )

Summary

Introduction

Consider the fourth-order quasi-linear parabolic partial differential equation (PDE). Rashidinia and Mohammadi [ ] developed an approximation for finding the numerical solution of differential equation ( ) by replacing the time derivative by a finite difference approximation and the space derivative by sextic spline functions using off-step points to obtain three-level implicit methods of accuracies O(k + h ) and O(k + h ). An outline of the paper is as follows: In Section , we formulate and derive three-level quasi-variable mesh difference methods using off-step points for the solution of quasilinear fourth-order PDE ( ). The proposed difference method ( a)-( b) of order O(k + h ) for the uniform mesh when applied to this equation results in the following scheme written in the matrix form: Syj+ = ( S + T)yj – Syj– + w, where S = S S , S S.

Methods

Method

Findings

Conclusions