CUBIC SPLINE AND FINITE DIFFERENCE METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATION

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Published: 2021-08-10

Page: 750-792


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


How to Cite

WARE, A. N., & ASHINE, A. B. (2021). CUBIC SPLINE AND FINITE DIFFERENCE METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATION. Asian Journal of Advances in Research, 4(1), 750–792. Retrieved from https://mbimph.com/index.php/AJOAIR/article/view/2351

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References

Burden RL, Faiers JD. Numerical analysis. 8th edition. United States: Thomson Brooks/Cole; 2005.

Roberts SM, Shipman JS. Two-point boundary value problems; Shooting Method. United States: American Elsevier; 1972.

Nagle RK, Staff EB. Fundamentals of Differential equations and boundary value problems. 2ndEdition, United States: Addison-Wesley; 1996.

Zill DG, Cullen MR. Differential equations with boundary-value problems.7thEdn. United States: Brooks/Cole; 2009.

Fyfe DJ. The use of cubic splines in the solution of two-point boundary Value problems. The Computer Journal. 1968;2011;12(2):188-192.

Al-Said EA. Cubic spline method for solving two-point boundary value problems. Korean Computational and Applied Mathematics. 1998.5(3):669-680.

Khan A. Parametric cubic spline solution of two-point boundary value problems. Applied Mathematics and Computation. 2004;175(1): 175-182.

Fang Q, Tsuchiya T, Yamamota T. Finite difference, finite element and finite volume methods applied to two-point boundary value problems. J. Comput. Appl. Math. 2002;139:9-19.

Caglar HN, Caglar, Elfaiture K. B-spline interpolation Compared with finite difference, finite element and finite volume methods which applied to two-point boundary value problems. Applied Mathematics and Computation. 2006; 175:72-79.

Davis. B-splines and Geometric design. SIAM News. 1997;29(5).

Fan, Jiaguing, Yao, Qiwei. “Spline Methods”. Nonlinear time series: non-parametric and parametric methods.Springes. 2005; 247.ISBN 978-0-387-26142-3.

Jaud, Kenneth L. Numerical methods in economics MT press. 1998;225.

ISBN 978-0-262-10071-7.

Sky McKinley and Megan Levine, Cubic Spline Interpolation. Math 45: Linear Algebra.

Bickley WG. 'Piecewise cubic interpolation and two-point boundary problems. Computer Journal. 1968;11(2):206-208.

Balagurusamy E. Numerical methods, Tata Mc Grow –Hill Publishing Company, New Delhi. 1999; 302.

Richard L. Burden, Douglas Fairies J. Numerical Analysis (9th edition). International Edition. 1993; 147-157.

Rashidinia JR, Mohammadi, Jalilian R. Cubic spline method for two-point boundary value problems. Int. J. Eng. Sci, 2008;19(5-2):39-43.

Chang J, Yang Q, Zhao. Comparison of B-spline method and finite difference method to solve BVP of linear ODEs. J. Comput. 2011;6(10):2149-2155.

Salomon D. Curves and surfaces for computer graphics. United States: Springer; 2006.

Boor CD. A practical Guide to Splines. In Applied Mathematical Sciences 27(revised edition), United States: Springer-Verlag. 2001;87-107.

“Spline” Oxford English Dictionary (3rd edition.) Oxford University Press; September 2005.

Henrico P. Discrete variable methods in ordinary differential equations. Wiley, New York; 1962.

Manoj Kumar. A difference method for singular two-point boundary value problems. Applied Mathematics and Computation. 2003;879-884.