Address: Vatutina st. 53, Vladikavkaz,
362025, RNO-A, Russia
Phone: (8672)23-00-54
E-mail: rio@smath.ru

DOI: 10.23671/VNC.2018.4.9167

A Numerical Method for the Solution of Fifth Order Boundary Value Problem in Ordinary Differential Equations

Pandey P. K.
Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 4.

Abstract: In this article we have proposed a technique for solving the fifth order boundary value problem as a coupled pair of boundary value problems. We have considered fifth order boundary value problem in ordinary differential equation for the development of the numerical technique. There are many techniques for the numerical solution of the problem considered in this article. Thus we considered the application of the finite difference method for the numerical solution of the problem. In this article we transformed fifth order differential problem into system of differential equations of lower order namely one and four. We discretized the system of differential equations into considered domain of the problem. Thus we got a system of algebraic equations. For the numerical solution of the problem, we have the system of algebraic equations. The solution of the algebraic equations is an approximate solution of the problem considered. Moreover we get numerical approximation of first and second derivative as a byproduct of the proposed method. We have shown that proposed method is convergent and order of accuracy of the proposed method is at lease quadratic. The numerical results obtained in computational experiment on the test problems approve the efficiency and accuracy of the method.

Keywords: boundary value problem, cubic order convergence, difference method, fifth order differential equation, odd order problems, odd-even order problems

For citation: Pandey P. K. A numerical method for the solution of fifth order
boundary value problem in ordinary differential equations //
Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], 2017, vol.
19, no. 1, pp. 50-57. DOI 10.23671/VNC.2018.4.9167

1. Sibley D. N. Viscoelastic Flows of PTT Fluids. Ph.D. Thesis. UK:
University of Bath, 2010.

2. Davies A. R., Karageorghis A., and Phillips T. N. Spectral
Galerkin methods for the primary two-point boundary-value problem in
modeling viscoelastic flows. Internat. J. Numer. Methods Eng., 1988.
vol. 26, no. 3, pp. 647-662.

3. Agarwal R. P. Boundary Value Problems for Higher Order
Differential Equations. Singapore: World Scientific, 1986.

4. Khan M. S. Finite difference solutions of fifth order boundary
value problems. Ph.D. thesis. UK: Brunel University London, 1994.

5. Caglar H. N., Caglar S. H., and Twizell E. H. The numerical
solution of fifth-order boundary value problems with sixth-degree
\(B\)-spline functions. Appl. Math. Letters, 1999, vol.12, no. 5,
pp. 25-30.

6. Lamnii A., Mraoui H., Sbibih D., and Tijini A. Sextic spline
solution of fifth order boundary value problems. Math. Computer
Simul. 2008. Vol. 77. P. 237-246.

7. Wazwaz A. M. The numerical solution of fifth-order boundary value
problems by the decomposition method. J. Comp. and Appl. Math. 2001.
vol. 136, no. 1-2, pp. 259-270.

8. Karageorghis A., Davies A. R., and Phillips T. N. Spectral
collocation methods for the primary two-point boundary-value problem
in modelling viscoelastic flows. Internat. J. Numer. Methods Eng.
1988, vol. 26, no. 4, pp. 805-813.

9. Viswanadham K. N. S. K., Krishnaa P. M., and Rao C. P. Numerical
Solution of Fifth Order Boundary Value Problems by Collocation
Method with Sixth Order \(B\)-Splines. Internat. J. Numer. Methods
Eng., 2010, vol. 8, no. 2, pp. 119-125.

10. Erturk V. S. Solving nonlinear fifth order boundary value
problems by differential transformation method. Selcuk J. Appl.
Math., 2007, vol. 8, no. 1, pp. 45-49.

11. Ali A., Mustafa I., and Adem K. A Comparison on Solutions of
Fifth-Order Boundary Value Problems. Appl. Math. Inf. Sci., 2016,
vol. 10, no. 2, pp. 755-764.

12. Pandey P. K. The Numerical Solution of Third Order Differential
Equation Containing the First Derivative. Neural, Parallel and Sci.
Comp., 2005, vol. 13, pp. 297-304.

13. Mohanty R. K. A fourth-order finite difference method for the
general one-dimensional nonlinear biharmonic problems of first kind.
J. Comp. Appl. Math., 2000, vol. 114, no. 2, pp. 275-290.

14. Mohanty R. K., Jain M. K. and Pandey P. K. Finite difference
methods of order two and four for 2-D nonlinear biharmonic problems
of first kind. Int. J. Comput. Math., 1996, vol. 61, pp. 155-163.

15. Varga R. S. Matrix Iterative Analysis, Second Revised and
Expanded Edition. Heidelberg: Springer-Verlag, 2000.

16. Horn R. A., Johnson C. R. Matrix Analysis. N. Y.: Cambridge
Univ. Press, 1990.

17. Gil M. I. Invertibility Conditions for Block Matrices and
Estimates for Norms of Inverse Matrices. Rocky Mountain J. of Math.
2003, vol. 33, no. 4, pp. 1323-1335.