A class of numerical methods for the solution of fourth-order nonlinear ordinary differential equations on a graded mesh with boundary conditions of first kind

Abstract

We propose difference schemes of order 2 and 3 on a graded mesh for fourth-order nonlinear ordinary differential equations (ODEs). The values of u and (du/dx) are prescribed at two boundary points. Our methods involve only 3-grid points. A graded mesh is generated by introducing a mesh ratio parameter σ. Numerical values of (du/dx) are obtained as a by-product of the proposed methods. The stability and convergence analysis are discussed in brief. The modified version of the method works well for singular problems. We have tested our methods on seven benchmark problems to demonstrate the usefulness of the proposed methods.

Keywords:

• Fourth order nonlinear boundary value problems
• boundary conditions of first kind
• graded mesh
• nonlinear biharmonic equation
• fourth-order singular boundary value problems

Acknowledgments

The authors thank the reviewers for their valuable suggestions, which substantially improved the standard of the article.

Disclosure statement

The authors declare that they have no competing interests. All authors drafted the manuscript, and they read and approved the final version.

Funding

The second author is supported by ‘SAARC Silver Jubilee Scholarship’, grant No. SAU/(S/ship)/003/2013-14.