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

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 a. 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.