Non-polynomial septic spline method for singularly perturbed two point boundary value problems of order three

Abstract

This study introduces a non-polynomial septic spline method for solving singularly perturbed two point boundary value problems of order three. First, the given interval is discretized. Then, the spline coefficients are derived and the consistency relation is obtained by using continuity of second, fourth and fifth derivatives. Further, the obtained fifteen different systems of equations are reduced to a system of equations and boundary equations are developed in order to equate a system of linear equations. The convergence analysis of the obtained hepta-diagonal scheme is investigated. To validate the applicability of the method, two model examples are considered for different values of perturbation parameter $\varepsilon $ and different mesh size h. The proposed method approximates the exact solution very well when $\varepsilon \ll h $. Moreover, the present method is convergent and gives more accurate results than some existing numerical methods reported in the literature.

Keywords

• singular perturbation
• non-polynomial septic spline
• convergence