New recurrence algorithms for the nonclassic Adomian polynomials

Computers and Mathematics with Applications

Volumn 62, Issue 8, 2011, Pages 2961-2977

  Duan, Jun Sheng ^a  

^a SHANGHAI INSTITUTE OF TECHNOLOGY   (China)

Author keywords

Adomian decomposition method; Adomian polynomials; MATHEMATICA; Nonlinear differential equations

Indexed keywords

ADOMIAN DECOMPOSITION METHOD; ADOMIAN POLYNOMIALS; ALGORITHMIC APPROACH; CLASS II; COMPUTER GENERATION; EXPLICIT EXPRESSIONS; MATHEMATICA; NON-LINEARITY; NONLINEAR DIFFERENTIAL EQUATION; NONLINEAR FUNCTIONAL EQUATIONS; RECURRENCE ALGORITHMS; RUNNING TIME; SUMMANDS;

ALGORITHMS; CONTROL NONLINEARITIES; DIFFERENTIAL EQUATIONS; NONLINEAR EQUATIONS;

POLYNOMIALS;

EID: 80053367331     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2011.07.074     Document Type: Article

References (65)

1. G. Adomian Stochastic Systems 1983 Academic New York
2. G. Adomian, and R. Rach Inversion of nonlinear stochastic operators J. Math. Anal. Appl. 91 1983 39 46
3. G. Adomian Nonlinear Stochastic Operator Equations 1986 Academic Orlando, FL
4. G. Adomian Nonlinear Stochastic Systems Theory and Applications to Physics 1989 Kluwer Academic Dordrecht
5. G. Adomian, R. Rach, and R. Meyers An efficient methodology for the physical sciences Kybernetes 20 1991 24 34
6. G. Adomian Solving Frontier Problems of Physics: The Decomposition Method 1994 Kluwer Academic Dordrecht
7. R. Rach A new definition of the Adomian polynomials Kybernetes 37 2008 910 955
8. A.M. Wazwaz Partial Differential Equations and Solitary Waves Theory 2009 Higher Education, Beijing, and Springer Berlin
9. A.M. Wazwaz Linear and Nonlinear Integral Equations: Methods and Applications 2011 Higher Education, Beijing, and Springer Berlin
10. R. Rach, and A. Baghdasarian On approximate solution of a nonlinear differential equation Appl. Math. Lett. 3 1990 101 102
11. R. Rach, A. Baghdasarian, and G. Adomian Differential equations with singular coefficients Appl. Math. Comput. 47 1992 179 184
12. H.L. Arora, and F.I. Abdelwahid Solution of non-integer order differential equations via the Adomian decomposition method Appl. Math. Lett. 6 1993 21 23
13. J. Duan, J. An, and M. Xu Solution of system of fractional differential equations by Adomian decomposition method Appl. Math. J. Chinese Univ. Ser. B 22 2007 7 12
14. S. Momani, and N. Shawagfeh Decomposition method for solving fractional Riccati differential equations Appl. Math. Comput. 182 2006 1083 1092 (Pubitemid 44838818)
15. M. Dehghan, and M. Tatari Finding approximate solutions for a class of third-order non-linear boundary value problems via the decomposition method of Adomian Int. J. Comput. Math. 87 2010 1256 1263
16. M. Dehghan The use of Adomian decomposition method for solving the one-dimensional parabolic equation with non-local boundary specifications Int. J. Comput. Math. 81 2004 25 34 (Pubitemid 41710456)
17. S.E. Serrano Engineering Uncertainty and Risk Analysis: A Balanced Approach to Probability, Statistics, Stochastic Processes, and Stochastic Differential Equations 2nd ed. 2011 HydroScience Ambler, PA
18. K. Srivastava, and S.E. Serrano Uncertainty analysis of linear and nonlinear groundwater flow in a heterogeneous aquifer J. Hydrol. Eng. 12 2007 306 318 (Pubitemid 46605602)
19. A.M.A. El-Sayed, S.H. Behiry, and W.E. Raslan Adomian's decomposition method for solving an intermediate fractional advectiondispersion equation Comput. Math. Appl. 59 2010 1759 1765
20. G. Adomian, R. Rach, and N.T. Shawagfeh On the analytic solution of the LaneEmden equation Found. Phys. Lett. 8 1995 161 181
21. A.M. Wazwaz A new method for solving singular initial value problems in the second order ordinary differential equations Appl. Math. Comput. 128 2002 45 57
22. A.M. Wazwaz The modified decomposition method for analytic treatment of differential equations Appl. Math. Comput. 173 2006 165 176
23. A.M. Wazwaz A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems Comput. Math. Appl. 41 2001 1237 1244 (Pubitemid 32366171)
24. R. Rach, and G. Adomian Multiple decompositions for computational convenience Appl. Math. Lett. 3 1990 97 99
25. A.M. Wazwaz Necessary conditions for the appearance of noise terms in decomposition series solutions Appl. Math. Comput. 81 1997 265 274
26. A.M. Wazwaz A reliable modification of Adomian decomposition method Appl. Math. Comput. 102 1999 77 86
27. G. Adomian, and R. Rach The noisy convergence phenomena in decomposition method solutions J. Comput. Appl. Math. 15 1986 379 381 (Pubitemid 16640498)
28. G. Adomian, and R. Rach Noise terms in decomposition solution series Comput. Math. Appl. 24 1992 61 64
29. A.M. Wazwaz, and S.M. El-Sayed A new modification of the Adomian decomposition method for linear and nonlinear operators Appl. Math. Comput. 122 2001 393 405 (Pubitemid 32645602)
30. K. Abbaoui, and Y. Cherruault Convergence of Adomian's method applied to differential equations Comput. Math. Appl. 28 1994 103 109
31. J.S. Duan Recurrence triangle for Adomian polynomials Appl. Math. Comput. 216 2010 1235 1241
32. Y. Cherruault Convergence of Adomian's method Kybernetes 18 1989 31 38
33. L. Gabet The theoretical foundation of the Adomian method Comput. Math. Appl. 27 1994 41 52
34. N. Himoun, K. Abbaoui, and Y. Cherruault New results of convergence of Adomian's method Kybernetes 28 1999 423 429
35. A. Abdelrazec, and D. Pelinovsky Convergence of the Adomian decomposition method for initial-value problems Numer. Methods Partial Differential Equations 27 2011 749 766
36. G. Adomian, and K. Malakian A comparison of the iterative method and Picard's successive approximations for deterministic and stochastic differential equations Appl. Math. Comput. 8 1981 187 204
37. N. Bellomo, and D. Sarafyan On Adomian's decomposition method and some comparisons with Picard's iterative scheme J. Math. Anal. Appl. 123 1987 389 400
38. R. Rach On the Adomian (decomposition) method and comparisons with Picard's method J. Math. Anal. Appl. 128 1987 480 483
39. N. Bellomo, and R. Monaco A comparison between Adomian's decomposition method and perturbation techniques for nonlinear random differential equations J. Math. Anal. Appl. 110 1985 495 502
40. A.M. Wazwaz A comparison between Adomian decomposition method and Taylor series method in the series solutions Appl. Math. Comput. 97 1998 37 44
41. G. Adomian, and R. Rach Modified Adomian polynomials Math. Comput. Modelling 24 1996 39 46 (Pubitemid 27118829)
42. V. Daftardar-Gejji, and H. Jafari An iterative method for solving nonlinear functional equations J. Math. Anal. Appl. 316 2006 753 763 (Pubitemid 43033583)
43. S.H. Behiry, H. Hashish, I.L. El-Kalla, and A. Elsaid A new algorithm for the decomposition solution of nonlinear differential equations Comput. Math. Appl. 54 2007 459 466 (Pubitemid 46977636)
44. I.L. El-Kalla Error analysis of Adomian series solution to a class of nonlinear differential equations Appl. Math. E-Notes 7 2007 214 221
45. G.E. Andrews The Theory of Partitions Enc. Math. Appl. vol. 2 1976 Addison-Wesley Massachusetts
46. J.H. van Lint, and R.M. Wilson A Course in Combinatorics 2nd ed. 1992 Cambridge England
47. R. Rach A convenient computational form for the Adomian polynomials J. Math. Anal. Appl. 102 1984 415 419 (Pubitemid 14663530)
48. M. Azreg-Anou A developed new algorithm for evaluating Adomian polynomials CMES Comput. Model. Eng. Sci. 42 2009 1 18
49. R. Riganti On a class of nonlinear dynamical systems: the structure of a differential operator in the application of the decomposition method J. Math. Anal. Appl. 124 1987 189 199 (Pubitemid 17567080)
50. A.M. Wazwaz A new algorithm for calculating Adomian polynomials for nonlinear operators Appl. Math. Comput. 111 2000 53 69
51. F. Abdelwahid A mathematical model of Adomian polynomials Appl. Math. Comput. 141 2003 447 453
52. Y. Zhu, Q. Chang, and S. Wu A new algorithm for calculating Adomian polynomials Appl. Math. Comput. 169 2005 402 416 (Pubitemid 41491862)
53. J. Biazar, M. Ilie, and A. Khoshkenar An improvement to an alternate algorithm for computing Adomian polynomials in special cases Appl. Math. Comput. 173 2006 582 592 (Pubitemid 43208800)
54. J.S. Duan, and A.P. Guo Reduced polynomials and their generation in Adomian decomposition methods CMES Comput. Model. Eng. Sci. 60 2010 139 150
55. J.S. Duan An efficient algorithm for the multivariable Adomian polynomials Appl. Math. Comput. 217 2010 2456 2467
56. J.S. Duan Convenient analytic recurrence algorithms for the Adomian polynomials Appl. Math. Comput. 217 2011 6337 6348
57. G. Adomian, R. Rach, and R. Meyers Numerical algorithms and decomposition Comput. Math. Appl. 22 1991 57 61
58. G. Adomian, R.C. Rach, and R.E. Meyers Numerical integration, analytic continuation, and decomposition Appl. Math. Comput. 88 1997 95 116 (Pubitemid 127683557)
59. M.A. El-Tawil, A.A. Bahnasawi, and A. Abdel-Naby Solving Riccati differential equation using Adomian's decomposition method Appl. Math. Comput. 157 2004 503 514
60. M. Dehghan, A. Hamidi, and M. Shakourifar The solution of coupled Burgers' equations using AdomianPade technique Appl. Math. Comput. 189 2007 1034 1047 (Pubitemid 46813156)
61. Y.C. Jiao, Y. Yamamoto, C. Dang, and Y. Hao An aftertreatment technique for improving the accuracy of Adomian's decomposition method Comput. Math. Appl. 43 2002 783 798 (Pubitemid 34172747)
62. H. Chu, Y. Zhao, and Y. Liu A MAPLE package of new ADM-Padé approximate solution for nonlinear problems Appl. Math. Comput. 217 2011 7074 7091
63. R. Rach, and J.S. Duan Near-field and far-field approximations by the Adomian and asymptotic decomposition methods Appl. Math. Comput. 217 2011 5910 5922
64. R. Rach, G. Adomian, and R.E. Meyers A modified decomposition Comput. Math. Appl. 23 1992 17 23
65. J.S. Duan, and R. Rach New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods Appl. Math. Comput. 2011 10.1016/j.amc.2011.08.024