Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the Rach-Adomian-Meyers modified decomposition method

Applied Mathematics and Computation

Volumn 218, Issue 17, 2012, Pages 8370-8392

  Duan, Jun Sheng ^a   Chaolu, Temuer ^b   Rach, Randolph ^c  

^a SHANGHAI INSTITUTE OF TECHNOLOGY   (China)

^b SHANGHAI MARITIME UNIVERSITY   (China)

^c NONE   (United States)

Author keywords

Adomian decomposition method; Adomian polynomials; Adomian Rach theorem; Fractional differential equations; Modified decomposition method; Power series

Indexed keywords

CONTROL NONLINEARITIES; INITIAL VALUE PROBLEMS; NONLINEAR EQUATIONS;

ADOMIAN; ADOMIAN DECOMPOSITION METHOD; ADOMIAN POLYNOMIALS; FRACTIONAL DIFFERENTIAL EQUATIONS; MODIFIED DECOMPOSITION METHOD; POWER SERIES;

ORDINARY DIFFERENTIAL EQUATIONS;

EID: 84859427726     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2012.01.063     Document Type: Article

References (95)

1. K.B. Oldham, and J. Spanier The Fractional Calculus 1974 Academic New York
2. K.S. Miller, and B. Ross An Introduction to the Fractional Calculus and Fractional Differential Equations 1993 Wiley New York
3. R. Gorenflo, and F. Mainardi Fractional calculus: integral and differential equations of fractional order A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics 1997 Springer-Verlag Wien/New York 223 276
4. F. Mainardi Fractional calculus: some basic problems in continuum and statistical mechanics A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics 1997 Springer-Verlag Wien/New York 291 348
5. I. Podlubny Fractional Differential Equations 1999 Academic San Diego
6. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo Theory and Applications of Fractional Differential Equations 2006 Elsevier Amsterdam
7. F. Mainardi Fractional Calculus and Waves in Linear Viscoelasticity 2010 Imperial College London
8. G. Jumarie A Fokker-Planck equation of fractional order with respect to time J. Math. Phys. 33 1992 3536 3541
9. R. Metzler, W.G. Glockle, and T.F. Nonnenmacher Fractional model equation for anomalous diffusion Physica A 211 1994 13 24
10. J.H. He Approximate analytical solution for seepage flow with fractional derivatives in porous media Comput. Methods Appl. Mech. Engrg. 167 1998 57 68
11. M.Y. Xu, and W.C. Tan Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion Sci. China Ser. A 44 2001 1387 1399
12. J.S. Duan, and M.Y. Xu Concentration distribution of fractional anomalous diffusion caused by an instantaneous point source Appl. Math. Mech. (English Ed.) 24 2003 1302 1308
13. J.S. Duan, and T. Chaolu Fractal circuit model deriving arbitrary-order derivative and integral Math. Practice Theory 39 2009 54 59 (in Chinese with English abstract)
14. J.S. Duan Time- and space-fractional partial differential equations J. Math. Phys. 46 2005 13504 13511
15. J.S. Duan, and M.Y. Xu The solution of semiboundless mixed problem of fractional diffusion equation Appl. Math. J. Chinese Univ. Ser. A 18 2003 259 266 (in Chinese with English abstract)
16. J.S. Duan, and M.Y. Xu The problem for fractional diffusion-wave equations on finite interval and Laplace transform Appl. Math. J. Chinese Univ. Ser. A 19 2004 165 171 (in Chinese with English abstract)
17. J.S. Duan, T. Chaolu, and J. Sun Solution for system of linear fractional differential equations with constant coefficients J. Math. (Wuhan) 29 2009 599 603
18. W. Wyss The fractional diffusion equation J. Math. Phys. 27 1986 2782 2785
19. J.S. Duan, and T. Chaolu Scale-invariant solution for fractional anomalous diffusion equation Ann. Differential Eqs. 22 2006 21 26
20. J.S. Duan, A.P. Guo, and W.Z. Yun Similarity method to solve fractional diffusion model in fractal media J. Biomath. 25 2010 218 224
21. 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
22. A.J. George, and A. Chakrabarti The Adomian method applied to some extraordinary differential equations Appl. Math. Lett. 8 1995 391 397
23. N.T. Shawagfeh Analytical approximate solutions for nonlinear fractional differential equations Appl. Math. Comput. 131 2002 517 529
24. V. Daftardar-Gejji, and H. Jafari Adomian decomposition: a tool for solving a system of fractional differential equations J. Math. Anal. Appl. 301 2005 508 518
25. S.S. Ray, and R.K. Bera An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method Appl. Math. Comput. 167 2005 561 571
26. S. Momani, and N. Shawagfeh Decomposition method for solving fractional Riccati differential equations Appl. Math. Comput. 182 2006 1083 1092
27. S. Momani A numerical scheme for the solution of multi-order fractional differential equations Appl. Math. Comput. 182 2006 761 770
28. J.S. Duan, J.Y. An, and M.Y. Xu Solution of system of fractional differential equations by Adomian decomposition method Appl. Math. J. Chinese Univ. Ser. B 22 2007 7 12
29. A.M.A. El-Sayed, S.H. Behiry, and W.E. Raslan Adomian's decomposition method for solving an intermediate fractional advection-dispersion equation Comput. Math. Appl. 59 2010 1759 1765
30. G.C. Wu Adomian decomposition method for non-smooth initial value problems Math. Comput. Modelling 54 2011 2104 2108
31. J.S. Duan, J. Sun, and T. Chaolu Nonlinear fractional differential equation combining Duffing equation and van der Pol equation J. Math. (Wuhan) 31 2011 7 10
32. Q. Wang Homotopy perturbation method for fractional KdV equation Appl. Math. Comput. 190 2007 1795 1802
33. L. Song, and H. Zhang Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation Phys. Lett. A 367 2007 88 94
34. H. Xu, S.J. Liao, and X.C. You Analysis of nonlinear fractional partial differential equations with the homotopy analysis method Commun. Nonlinear Sci. Numer. Simulat. 14 2009 1152 1156
35. A. Saadatmandi, and M. Dehghan A new operational matrix for solving fractional-order differential equations Comput. Math. Appl. 59 2010 1326 1336
36. J.S. Duan, Z.H. Liu, F.K. Zhang, and T. Chaolu Analytic solution and numerical solution to endolymph equation using fractional derivative Ann. Differential Eqs. 24 2008 9 12
37. K. Diethelm, N.J. Ford, and A.D. Freed Detailed error analysis for a fractional Adams method Numer. Algorithms 36 2004 31 52
38. R. Gorenflo Fractional calculus: some numerical methods A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics 1997 Springer-Verlag Wien/New York 277 290
39. C. Li, and Y. Wang Numerical algorithm based on Adomian decomposition for fractional differential equations Comput. Math. Appl. 57 2009 1672 1681
40. G.C. Wu A fractional characteristic method for solving fractional partial differential equations Appl. Math. Lett. 24 2011 1046 1050
41. A.M. Wazwaz, and S.A. Khuri A reliable technique for solving the weakly singular second-kind volterra-type integral equations Appl. Math. Comput. 80 1996 287 299
42. G. Adomian, and R. Rach Inversion of nonlinear stochastic operators J. Math. Anal. Appl. 91 1983 39 46
43. G. Adomian Stochastic Systems 1983 Academic New York
44. G. Adomian Nonlinear Stochastic Operator Equations 1986 Academic Orlando
45. G. Adomian Nonlinear Stochastic Systems Theory and Applications to Physics 1989 Kluwer Academic, Dordrecht
46. G. Adomian, R. Rach, and R. Meyers An efficient methodology for the physical sciences Kybernetes 20 1991 24 34
47. G. Adomian Solving Frontier Problems of Physics: The Decomposition Method 1994 Kluwer Academic, Dordrecht
48. N. Bellomo, and R. Riganti Nonlinear Stochastic System Analysis in Physics and Mechanics 1987 World Scientific Singapore and River Edge, NJ
49. A.M. Wazwaz A First Course in Integral Equations 1997 World Scientific Singapore and River Edge, NJ
50. A.M. Wazwaz Partial Differential Equations: Methods and Applications 2002 Balkema Lisse
51. A.M. Wazwaz Partial Differential Equations and Solitary Waves Theory, Higher Education, Beijing 2009 Springer Berlin
52. A.M. Wazwaz Linear and Nonlinear Integral Equations: Methods and Applications, Higher Education, Beijing 2011 Springer Berlin
53. K. Scott, and Y.P. Sun Approximate analytical solutions for models of three-dimensional electrodes by Adomian's decomposition method C.G. Vayenas, R.E. White, M.E. Gamboa-Adelco, Modern Aspects of Electrochemistry 41 2007 Springer New York 222 304
54. S.E. Serrano Hydrology for Engineers, Geologists, and Environmental Professionals: An Integrated Treatment of Surface, Subsurface, and Contaminant Hydrology second revised ed. 2010 HydroScience Ambler, PA
55. S.E. Serrano Engineering Uncertainty and Risk Analysis: A Balanced Approach to Probability, Statistics, Stochastic Modeling, and Stochastic Differential Equations second revised ed. 2011 HydroScience Ambler, PA
56. R. Rach A convenient computational form for the Adomian polynomials J. Math. Anal. Appl. 102 1984 415 419
57. G. Adomian, and R. Rach On composite nonlinearities and the decomposition method J. Math. Anal. Appl. 113 1986 504 509
58. A.M. Wazwaz A new algorithm for calculating Adomian polynomials for nonlinear operators Appl. Math. Comput. 111 2000 53 69
59. R. Rach A new definition of the Adomian polynomials Kybernetes 37 2008 910 955
60. F. Abdelwahid A mathematical model of Adomian polynomials Appl. Math. Comput. 141 2003 447 453
61. 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
62. Y. Zhu, Q. Chang, and S. Wu A new algorithm for calculating Adomian polynomials Appl. Math. Comput. 169 2005 402 416
63. M. Azreg-Aïnou A developed new algorithm for evaluating Adomian polynomials CMES-Comput. Model. Eng. Sci. 42 2009 1 18
64. J.S. Duan Recurrence triangle for Adomian polynomials Appl. Math. Comput. 216 2010 1235 1241
65. J.S. Duan An efficient algorithm for the multivariable Adomian polynomials Appl. Math. Comput. 217 2010 2456 2467
66. J.S. Duan Convenient analytic recurrence algorithms for the Adomian polynomials Appl. Math. Comput. 217 2011 6337 6348
67. 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
68. J.S. Duan, A.P. Guo, Symbolic implementation of a new, fast algorithm for the multivariable adomian polynomials, in: Proceedings 2011 World Congress on Engineering and Technology, vol. 1, IEEE, Beijing, 2011, pp. 72-74.
69. G. Adomian, and R. Rach Modified Adomian polynomials Math. Comput. Modelling 24 1996 39 46
70. J.S. Duan New recurrence algorithms for the nonclassic Adomian polynomials Comput. Math. Appl. 62 2011 2961 2977
71. J.S. Duan New ideas for decomposing nonlinearities in differential equations Appl. Math. Comput. 218 2011 1774 1784
72. R. Rach, G. Adomian, and R.E. Meyers A modified decomposition Comput. Math. Appl. 23 1992 17 23
73. G. Adomian, and R. Rach Transformation of series Appl. Math. Lett. 4 1991 69 71
74. G. Adomian, and R. Rach Nonlinear transformation of series-part II Comput. Math. Appl. 23 1992 79 83
75. G. Adomian, and R. Rach Inhomogeneous nonlinear partial differential equations with variable coefficients Appl. Math. Lett. 5 1992 11 12
76. G. Adomian, and R. Rach Modified decomposition solution of nonlinear partial differential equations Appl. Math. Lett. 5 1992 29 30
77. G. Adomian, and R. Rach Solution of nonlinear partial differential equations in one, two, three, and four dimensions World Sci. Ser. Appl. Anal. 2 1993 1 13
78. G. Adomian, and R. Rach Modified decomposition solution of linear and nonlinear boundary-value problems Nonlinear Anal. 23 1994 615 619
79. G. Adomian, R. Rach, and N.T. Shawagfeh On the analytic solution of the Lane-Emden equation Found. Phys. Lett. 8 1995 161 181
80. J.C. Hsu, H.Y. Lai, and C.K. Chen An innovative eigenvalue problem solver for free vibration of uniform Timoshenko beams by using the Adomian modified decomposition method J. Sound Vibration 325 2009 451 470
81. G. Adomian, R.C. Rach, and R.E. Meyers Numerical integration, analytic continuation, and decomposition Appl. Math. Comput. 88 1997 95 116
82. 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. 218 2011 2810 2828
83. J.J. Trujillo, M. Rivero, and B. Bonilla On a Riemann-Liouville generalized Taylor's formula J. Math. Anal. Appl. 231 1999 255 265
84. Z.M. Odibat, and N.T. Shawagfeh Generalized Taylor's formula Appl. Math. Comput. 186 2007 286 293
85. G. Jumarie Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results Comput. Math. Appl. 51 2006
86. G. Adomian, and R. Rach Generalization of Adomian polynomials to functions of several variables Comput. Math. Appl. 24 1992 11 24
87. K. Abbaoui, Y. Cherruault, and V. Seng Practical formulae for the calculus of multivariable Adomian polynomials Math. Comput. Modelling 22 1995 89 93
88. I.O.A. Arikoglu Solution of fractional differential equations by using differential transform method Chaos Solitons Fractals 34 2007 1473 1481
89. 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
90. J.W. Burgmeier, and M.R. Scott A method for obtaining bounds on eigenvalues and eigenfunctions by solving non-homogeneous integral equations Computing 10 1972 9 22
91. J.S. Duan, and R. Rach On the domain of convergence of the Adomian series solution J. Comput. Appl. Math. 2012 unpublished results
92. J.S. Duan, and R. Rach A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations Appl. Math. Comput. 218 2011 4090 4118
93. F. Mainardi, and R. Gorenflo On Mittag-Leffler-type functions in fractional evolution processes J. Comput. Appl. Math. 118 2000 283 299
94. G. Adomian, and R. Rach Anharmonic oscillator systems J. Math. Anal. Appl. 91 1983 229 236
95. A.M. Wazwaz A reliable modification of Adomian decomposition method Appl. Math. Comput. 102 1999 77 86