The use of Adomian decomposition method for solving problems in calculus of variations

Dehghan, Mehdi; Tatari, Mehdi

doi:https://doi.org/10.1155/MPE/2006/65379

Mathematical Problems in Engineering

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Volume 2006 | Article ID 065379 | https://doi.org/10.1155/MPE/2006/65379

The use of Adomian decomposition method for solving problems in calculus of variations

Mehdi Dehghan¹and Mehdi Tatari¹

Received16 Mar 2005

Revised09 Aug 2005

Accepted12 Sept 2005

Published07 Jun 2006

Abstract

In this paper, a numerical method is presented for finding the solution of some variational problems. The main objective is to find the solution of an ordinary differential equation which arises from the variational problem. This work is done using Adomian decomposition method which is a powerful tool for solving large amount of problems. In this approach, the solution is found in the form of a convergent power series with easily computed components. To show the efficiency of the method, numerical results are presented.

References

G. Adomian, “A new approach to the heat equation—an application of the decomposition method,” Journal of Mathematical Analysis and Applications, vol. 113, no. 1, pp. 202–209, 1986.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
G. Adomian, “Modification of the decomposition approach to the heat equation,” Journal of Mathematical Analysis and Applications, vol. 124, no. 1, pp. 290–291, 1987.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, vol. 46 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1989.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
G. Adomian, “Solving frontier problems modelled by nonlinear partial differential equations,” Computers & Mathematics with Applications, vol. 22, no. 8, pp. 91–94, 1991.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic, Dordrecht, 1994.
View at: Google Scholar | MathSciNet
G. Adomian and R. Rach, “Solution of nonlinear ordinary and partial differential equations of physics,” Journal of Mathematical and Physical Sciences, vol. 25, no. 5-6, pp. 703–718 (1993), 1991.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. Babolian, J. Biazar, and A. R. Vahidi, “The decomposition method applied to systems of Fredholm integral equations of the second kind,” Applied Mathematics and Computation, vol. 148, no. 2, pp. 443–452, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Y. Chang and M. L. Wang, “Shifted Legendre direct method for variational problems,” Journal of Optimization Theory and Applications, vol. 39, no. 2, pp. 299–307, 1983.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. F. Chen and C. H. Hsiao, “A Walsh series direct method for solving variational problems,” Journal of the Franklin Institute, vol. 300, no. 4, pp. 265–280, 1975.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Cherruault, “Convergence of Adomian's method,” Mathematical and Computer Modelling, vol. 14, pp. 83–86, 1990.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. Dehghan, “Application of the Adomian decomposition method for two-dimensional parabolic equation subject to nonstandard boundary specifications,” Applied Mathematics and Computation, vol. 157, no. 2, pp. 549–560, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Dehghan, “The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure,” International Journal of Computer Mathematics, vol. 81, no. 8, pp. 979–989, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Dehghan, “The use of Adomian decomposition method for solving the one-dimensional parabolic equation with non-local boundary specifications,” International Journal of Computer Mathematics, vol. 81, no. 1, pp. 25–34, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. Dyer and S. R. McReynolds, The Computation and Theory of Optimal Control, vol. 65 of Mathematics in Science and Engineering, Academic Press, New York, 1970.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
L. Elsgolts, Differential Equations and the Calculus of Variations, Mir, Moscow, 1977, translated from the Russian by G. Yankovsky.
View at: Google Scholar
I. M. Gelfand and S. V. Fomin, Calculus of Variations, revised English edition translated and edited by R. A. Silverman, Prentice-Hall, New Jersey, 1963.
View at: Google Scholar | MathSciNet
I. R. Horng and J. H. Chou, “Shifted Chebyshev direct method for solving variational problems,” International Journal of Systems Science, vol. 16, no. 7, pp. 855–861, 1985.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
C. Hwang and Y. P. Shih, “Laguerre series direct method for variational problems,” Journal of Optimization Theory and Applications, vol. 39, no. 1, pp. 143–149, 1983.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
N. Ngarhasta, B. Some, K. Abbaoui, and Y. Cherruault, “New numerical study of Adomian method applied to a diffusion model,” Kybernetes, vol. 31, no. 1, pp. 61–75, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Razzaghi and M. Razzaghi, “Fourier series direct method for variational problems,” International Journal of Control, vol. 48, no. 3, pp. 887–895, 1988.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. Razzaghi and M. Razzaghi, “Instabilities in the solution of a heat conduction problem using Taylor series and alternative approaches,” Journal of the Franklin Institute, vol. 326, no. 5, pp. 683–690, 1989.
View at: Publisher Site | Google Scholar
M. Tatari and M. Dehghan, “Numerical solution of Laplace equation in a disk using the Adomian decomposition method,” Physica Scripta, vol. 72, no. 5, p. 345, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
A.-M. Wazwaz, “A reliable modification of Adomian decomposition method,” Applied Mathematics and Computation, vol. 102, no. 1, pp. 77–86, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A.-M. Wazwaz, “A new algorithm for calculating Adomian polynomials for nonlinear operators,” Applied Mathematics and Computation, vol. 111, no. 1, pp. 53–69, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A.-M. Wazwaz, “Approximate solutions to boundary value problems of higher order by the modified decomposition method,” Computers & Mathematics with Applications, vol. 40, no. 6-7, pp. 679–691, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A.-M. Wazwaz, “The numerical solution of fifth-order boundary value problems by the decomposition method,” Journal of Computational and Applied Mathematics, vol. 136, no. 1-2, pp. 259–270, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A.-M. Wazwaz, “The numerical solution of sixth-order boundary value problems by the modified decomposition method,” Applied Mathematics and Computation, vol. 118, no. 2-3, pp. 311–325, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2006 Mehdi Dehghan and Mehdi Tatari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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