A practical review of the Adomian decomposition method: computer implementation aspects

Document Type : Research Article

Author

• A. Molabahrami

Department of Mathematics, Ilam University, PO Box 69315516, Ilam, Iran

10.22067/ijnao.v5i2.40124

Abstract

In this paper, a practical review of the Adomian decomposition method, to extend the procedure to handle the strongly nonlinear problems under the mixed conditions, is given and the convergence of the algorithm is proved.
For this respect, a new and simple way to generate the Adomian polynomials, for a general nonlinear function, is proposed. The proposed procedure, provides an explicit formula to calculate the Adomian polynomials of a nonlinear function. The efficiency of the approach will be shown by applying the procedure on several interesting integro-differential problems. The Mathematica programs generating the Adomian polynomials and Adomian solutions based on the procedures in this paper are designed.

Keywords

• Adomian decomposition method
• Adomian polynomials
• Nonlinear integro-differential problems
• Series solution
• Strongly nonlinear problems
• Explicit machine computation and programs

20.1001.1.24236977.2015.5.2.3.0

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References

1. Adomian, G. Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer Academic, Dordrecht, 1989.

2. Adomian, G. Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht, 1994.

3. Adomian, G. and Rach, R. On composite nonlinearities and the decomposition method, J. Math. Anal. Appl. 113 (1986) 504-509.

4. Cherruault, Y., Adomian, G., Abbaoui, K. and Rach R. Further remarks on convergence of decomposition method, International Journal of BioMedical Computing 38 (1995) 89-93.

5. Duan, J., Rach, R., Baleanu, D. and Wazwaz, A. A review of the Adomian decomposition method and its applications to fractional differential equations, Commun. Frac. Calc. 3 (2) (2012) 73-99.

6. Gabet, L. The Theoretical Foundation of the Adomian Method, Computers Math. Applic. 27 (1994) 41-52.

7. Ghorbani, A. Beyond Adomian polynomails: He polynomails, Chaos, Soliton and Fractals 39 (2009) 1486-1492.

8. Liao, S. J. Notes on the homotopy analysis method: Some deffnitions and theorems, Commun Nonlinear Sci Numer Simul 14 (2009) 983-997.

9. Molabahrami, A. Integral mean value method for solving a general nonlinear Fredholm integro-differential equation under the mixed conditions, Communications in Numerical Analysis 2013 (2013) 1-15. http://dx.doi.org/10.5899/2013/cna-00146.

10. Molabahrami, A. and Khani, F. The homotopy analysis method to solve the Burgers-Huxley equation, Nonlinear Anal-Real 10 (2009) 589-600.

11. Ngarhasta, N., Some, B., Abbaoui, K. and Cherruault, Y. New numerical study of adomian method applied to a diffusion model, Kybernetes 31(2002) 61-75.

12. Rach, R. A convenient computational form for the Adomian polynomials, J. Math. Anal. Appl. 102 (1984) 415-419.

13. Rach, R. A new deffnition of the Adomian polynomials, Kybernetes 37 (2008) 910-955.