# The Effect of Combining the Adomian Decomposition and Homotopy Perturbation Methods on Solving Fifth-Order Boundary Value Problems

S. S. Hosseini , A. Aminataei, F. Kiany, M. Alizadeh, M. Zahraei

#### Abstract

This paper considers fifth-order boundary value problems with two-point boundary conditions. For the investigation, first, the Adomian decomposition method (ADM) was applied to the equation and then the homotopy perturbation method (HPM) was used to continue the solution. In addition, this combined method was applied to solve two linear and nonlinear experiments. Then the numerical results obtained by other methods were compared. Furthermore, the convergence of the methods was analyzed. In most of the articles, the HPM is used for equations with initial conditions and the ADM is used for equations with initial and boundary conditions. In this study, a combination of two methods (the ADM and HPM) was used to solve equations with boundary conditions. The results of this combined method are remarkable.

Keywords: homotopy perturbation method, Adomian decomposition method, combination, convergence.

https://doi.org/10.55463/issn.1674-2974.49.8.28

PDF

#### References

CAGLAR H. N., CAGLAR S. H., and TWIZELL E. E. The numerical solution of fifth order boundary-value problems with sixth degree B-spline functions. Applied Mathematics Letters, 1999, 12: 25-30. https://doi.org/10.1016/S0893-659(99)00052-X

DAVIES A. R., KARAGEOGHIS A., and PHILLIPS T. N. Spectral Glarkien methods for the primary two-point boundary-value problems in modeling viscelastic flows. International Journal for Numerical Methods in Engineering, 1988, 26: 647-662. https://doi.org/10.1002/nme.1620260309

FYFE D. J. Linear dependence relations connecting equal interval N th degree splines and their derivatives, Journal of the Institute of Mathematics and its Applications, 1971, 7: 398-406. https://doi.org/10.1093/imamat/7.3.398

SIDDIQI S. S., and IFTIKHAR M. Variational iteration method for the solution of seventh order boundary value problems using He’s polynomials. Journal of the Association of Arab Universities for Basic and Applied Sciences, 2015, 18: 60-65. https://doi.org/10.1016/j.jaubas.2014.03.001

AMINATAEI A., and HOSSEINI S. S. Comparison of Adomian decomposition and double decomposition methods for boundary-value problems. European Journal of Scientific Research, 2005, 2: 48-56. https://doi.org/10.1016/j.amc.2006.08.011

NOOR M. A., and MOHYUD-DIN S. T. An efficient algorithm for solving fifth-order boundary value problems. Mathematical and Computer Modeling, 2007, 45: 954-964. https://doi.org/10.1016/j.mcm.2006.09.004

HOSSEINI S. S., AMINATAEI A., KIANY F., ALIZADEH M., and ZAHRAEI M. Improvement of the homotopy perturbation method for solving differential equations with boundary conditions. International Journal of Applied Mathematics & Statistics, 2019, 58: 89-95. http://www.ceser.in/ceserp/index.php/ijamas/article/view/6162

AKRAM G., and ASLAM I. A. Solution of fourth order three-point boundary value problem using ADM and RKM. Journal of the Association of Arab Universities for Basic and Applied Sciences, 2016, 20: 61-67. https://doi.org/10.1016/j.jaubas.2014.08.001

LIAO S. J. An approximate solution technique not depending on small parameter: a special example, International Journal of Non-Linear Mechanics, 1995, 30: 371-380. https://doi.org/10.1016/0020-7462(94)00054-E

LIAO S. J. Boundary element method for general nonlinear differential operators. Engineering Analysis with Boundary Elements, 1997, 20: 91-99. https://doi.org/10.1016/S0955-7997(97)00043-X

NAYFEH A. H. Introduction to Perturbation Technique. John Wiley and Sons, New York, 1981.

NAYFEH A. H. Problems in Perturbation. John Wiley and Sons, New York, 1985. https://www.amazon.com/Problems-Perturbation-Ali-H-Nayfeh/dp/0471822922

KHAN Y., and WU Q. Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Computers and Mathematics with Applications, 2011, 61: 1963-1967. https://doi.org/10.1016/j.camwa.2010.08.022

WAZWAZ A. M. The numerical solution of fifth-order boundary-value problems by Adomian decomposition, Journal of Computational and Applied Mathematics, 2001, 136: 259-270. https:/doi.org/10.1016/S0377-0427(00)00618-X

KHAN M. S. Finite-difference solutions of fifth-order boundary-value problems, Ph.D. Thesis, Brunel University, England, 1994.

BIAZAR J., and AMINIKHAH H. Study of convergence of homotopy perturbation method for systems of partial differential equations. Computers and Mathematics with Applications, 2009, 58: 2221-2230. https://doi.org/10.1016/j.camwa.2009.03.030

### Refbacks

• There are currently no refbacks.