Adomian Decomposition Method with Green’s Function for Solving Tenth-Order Boundary Value Problems

Waleed Al-Hayani

Department of Mathematics, College of Computer Science and Mathematics, University of Mosul, Mosul, Iraq.

DOI: 10.4236/am.2014.510136                

Abstract

In this paper, the Adomian decomposition method with Green’s function (Standard Adomian and Modified Technique) is applied to solve linear and nonlinear tenth-order boundary value problems with boundary conditions defined at any order derivatives. The numerical results obtained with a small amount of computation are compared with the exact solutions to show the efficiency of the method. The results show that the decomposition method is of high accuracy, more convenient and efficient for solving high-order boundary value problems.

Keywords

Adomian Decomposition Method, Adomian’s Polynomials, Tenth-Order Boundary Value Problems, Green’s Function

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Adomian, G. (1983) Stochastic Systems. Academic Press, New York.
[2]	Adomian, G. (1986) Nonlinear Stochastic Operator Equations. Academic Press, New York.
[3]	Adomian, G. (1989) Nonlinear Stochastic Systems Theory and Applications to Physics. Kluwer Academic Publishers, Dordrecht. http://dx.doi.org/10.1007/978-94-009-2569-4
[4]	Adomian, G. (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Dordrecht. http://dx.doi.org/10.1007/978-94-015-8289-6
[5]	Chandrasekhar, S. (1961) Hydrodynamic Hydromagnetic Stability. Clarendon Press, Oxford. (Reprinted: Dover Books, New York, 1981)
[6]	Agarwal, R.P. (1986) Boundary Value Problems for Higher-Order Differential Equations. World Scientific, Singapore. http://dx.doi.org/10.1142/0266
[7]	Siddiqi, S.S. and Twizell, E.H. (1998) Spline Solutions of Linear Tenth-Order Boundary-Value Problems. International Journal of Computer Mathematics, 68, 345-362. http://dx.doi.org/10.1080/00207169808804701
[8]	Wazwaz, A.W. (2000) Approximate Solutions to Boundary Value Problems of Higher Order by the Modified Decomposition Method. Computers and Mathematics with Applications, 40, 679-691. http://dx.doi.org/10.1016/S0898-1221(00)00187-5
[9]	Erturk, V.S. and Momani, S. (2007) A Reliable Algorithm for Solving Tenth-Order Boundary Value Problems. Numerical Algorithms, 44, 147-158. http://dx.doi.org/10.1007/s11075-007-9093-3
[10]	Siddiqi, S.S. and Akram, G. (2007) Solutions of Tenth-Order Boundary Value Problems using Eleventh Degree Spline. Applied Mathematics and Computation, 185, 115-127. http://dx.doi.org/10.1016/j.amc.2006.07.013
[11]	Siddiqi, S.S. and Akram, G. (2007) Solution of 10th-Order Boundary Value Problems Using Non-Polynomial Spline Technique. Applied Mathematics and Computation, 190, 641-651. http://dx.doi.org/10.1016/j.amc.2007.01.075
[12]	Siddiqi, S.S., Akram, G. and Zaheer, S. (2009) Solution of Tenth Order Boundary Value Problems Using Variational Iteration Technique. European Journal of Scientific Research, 30, 326-347.
[13]	Ravi Kanth, A.S.V. and Aruna, K. (2009) He’s Homotopy Perturbation Method for Solving Higher-Order Boundary Value Problems. Chaos, Solitons and Fractals, 41, 1905-1909. http://dx.doi.org/10.1016/j.chaos.2008.07.044
[14]	Stakgold, I. (1998) Green’s Functions and Boundary Value Problems. John Wiley & Sons, Inc., Hoboken.
[15]	Abbaoui, K. and Cherruault, Y. (1994) Convergence of Adomian’s Method Applied to Differential Equations. Mathematical and Computer Modelling, 28, 103-109.
[16]	Abbaoui, K. and Cherruault, Y. (1995) New Ideas for Proving Convergence of Decomposition Methods. Computers and Mathematics with Applications, 29, 103-108. http://dx.doi.org/10.1016/0898-1221(95)00022-Q
[17]	Abbaoui, K. and Cherruault, Y. (1994) Convergence of Adomian’s Method Applied to Nonlinear Equations. Mathematical and Computer Modelling, 20, 69-73. http://dx.doi.org/10.1016/0895-7177(94)00163-4
[18]	Cherruault, Y. and Adomian, G. (1993) Decomposition Methods: A New Proof of Convergence. Mathematical and Computer Modelling, 18, 103-106. http://dx.doi.org/10.1016/0895-7177(93)90233-O
[19]	Guellal, S. and Cherruault, Y. (1994) Practical Formula for Calculation of Adomian’s Polynomials and Application to the Convergence of the Decomposition Method. International Journal Bio-Medical Computing, 36, 223-228. http://dx.doi.org/10.1016/0020-7101(94)90057-4
[20]	Wazwaz, A.W. (1999) A Reliable Modification of Adomian Decomposition Method. Applied Mathematics and Computation, 102, 77-86. http://dx.doi.org/10.1016/S0096-3003(98)10024-3