B-Spline Based Numerical Algorithm for Singularly Perturbed Problem of Fourth Order

In the present paper, a numerical algorithm using fifth degree quintic B-spline for fourth order singular perturbation problem has been developed. The most of the numerical methods used for higher order singularly perturbed boundary value problems transform the problems into equivalent system of first and/or second order differential equations. However, in the present method, fifth degree B-spline is applied directly to the problem without transforming the problem into an equivalent system. The method uses values of fifth degree B-spline function and its derivatives up to the order four at nodal points. Resulting system of equations is solved to get the required quintic B-spline solution. Since perturbed problems contain boundary layers, the strategy of fitted mesh is used which assigns more mesh points in the boundary layer regions. The algorithm is tested on two problems to demonstrate the practical usefulness and superiority of the approach.