Abstract
In this study, the non-polynomial spline function is used to find the numerical solution of the third order singularly perturbed boundary value problems. The convergence analysis is discussed and the method is shown to have second order convergence. The order of convergence is improved up to fourth order using the improved end conditions. Numerical results are given to describe the efficiency of the method and compared with the method developed by Akram (2012), which shows that the present method is better.
Highlights
The purpose of the study is to develop a new spline method for the solution of third order singularly perturbed boundary value problem
Theorem 1: The method given by Eq (7) and (14) for solving the boundary value problem (1) for sufficiently small h gives a fourth order convergent solution
Theorem 2: The method given by Eq (7) and (16) for solving the boundary value problem (2) for sufficiently small h gives a fourth order convergent solution
Summary
The purpose of the study is to develop a new spline method for the solution of third order singularly perturbed boundary value problem. Lie (2008) constructed a computational method for singularly perturbed two point BVP in the form of series in reproducing Kernel space. Rashidinia and Mahmoodi (2007) developed the classes of methods for the numerical solution of singularly perturbed two point BVP using non polynomial cubic spline and the method is second order as well as fourth order accurate. Khan et al (2006) used sextic spline to solve second order singularly perturbed BVP and the method is fifth order accurate. Akram (2012) presented a quartic spline solution of a third order singularly perturbed BVP and the method is second order accurate.
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