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

Advances in Difference Equations

Ranjan Kumar Mohanty¹  Deepti Kaur² 

[1] Department of Applied Mathematics, Faculty of Mathematics and Computer Science, South Asian University, Akbar Bhawan, Chanakyapuri, India;Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, India

关键词: Euler-Bernoulli beam equation;    off-step nodal points;    quasi-variable mesh;    finite difference method;    successive tangential partial derivatives;    good Boussinesq equation;   

DOI  :  10.1186/s13662-016-1048-3

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

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.

【 授权许可】

CC BY