CUBIC SPLINE AND FINITE DIFFERENCE METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATION
ABE NURA WARE
Department of Mathematics, College of Natural and Computational Sciences, Arsi University, Asella, Ethiopia
AHMED BUSERI ASHINE
*
Department of Mathematics, College of Natural and Computational Sciences, Madda Walabu University, Bale Robe, Ethiopia.
*Author to whom correspondence should be addressed.
Abstract
The paper is an over view of the theory of cubic spline interpolation and finite difference method for solving boundary value problems of ordinary differential equation. It specially focuses on cubic spline functions to develop a numerical method for the solution of second-order two-point boundary value problems. The resulting system of equations has been solved by a tri-diagonal solver. The two-point boundary value problem of differential equations, which is part of differential equations with conditions imposed at different points, has been applied in mathematics, engineering and various field of science. The rapid increasing on its applications has led to formulating and upgraded of several existing methods and new approaches. To describe a numerical method for solving the boundary value problem of ordinary differential equation with second-order by using cubic spline function. First, the polynomial and non-polynomial spline basis functions are introduced, and then we use the correlation between polynomial and non-polynomial spline basis functions to approximate the solution. Finally, we obtain the numerical solution by solving tri-diagonal systems. The results are compared with finite difference method, and cubic B- spline through examples which show that the cubic spline method is efficient and feasible. The absolute errors in test examples are estimated, and the comparison of approximate values, exact values, and absolute errors at the nodal points are shown graphically. Further, we have shown that non-polynomial spline produces accurate results in comparison with the results obtained by the B-spline method and finite difference method.
Keywords: Cubic spline, Cubic B-spline interpolation method, Finite difference method, two-point boundary value problems, convergence analysis
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References
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