Solving nonlinear fractional partial differential equations using the homotopy analysis method

Numerical Methods for Partial Differential Equations

Volumn 26, Issue 2, 2010, Pages 448-479

  Dehghan, Mehdi ^a   Manafian, Jalil ^a   Saadatmandi, Abbas ^b  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

^b UNIVERSITY OF KASHAN   (Iran)

Author keywords

Analytical solution; Coupled Kdv and Boussinesq like B(m,n) equations; Fractional Kdv, K(2, 2), Burgers, BBM Burgers, cubic Boussinesq; Fractional partial differential equations (FPDEs); homotopy analysis method (HAM)

Indexed keywords

EID: 76649094637     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20460     Document Type: Article

References (76)

1. 1 M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press, New York, 1991.
2. 2 K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993.
3. 3 I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, New York, 1999.
4. 4 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives: Theory and applications, Yverdon, Gordon and Breach, 1993.
5. 5 B. J. West, M. Bolognab, and P. Grigolini, Physics of fractal operators, Springer, New York, 2003.
6. 6 S. Abbasbandy, An approximation solution of a nonlinear equation with Riemann–Liouville's fractional derivatives by He's variational iteration method, J Computat Appl Math 207 (2007), 53–58.
7. 7 M. Caputo, Linear models of dissipation whose Q is almost frequency independent, J R Astronomic Soc 13 (1967), 529–539.
8. 8 L. Debanth, Recents applications of fractional calculus to science and engineering, Int J Math Math Sci 54 (2003), 3413–3442.
9. 9 K. Diethelm, N. J. Ford, and A. D. Freed, A predictor‐corrector approach for the numerical solution of fractional differential equation, Nonlinear Dyn 29 (2002), 3–22.
10. 10 T. Hayat, S. Nadeem, and S. Asghar, Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model, Appl Math Comput 151 (2004), 153–161.
11. 11 H. Jafari and S. Seifi, Homotopy Analysis Method for solving linear and nonlinear fractional diffusion‐wave equation, Commun Nonlinear Sci Numer Simul 14 (2009), 2006–2012.
12. 12 H. Jafari and S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun Nonlinear Sci Numer Simul 14 (2009), 1962–1969.
13. 13 S. Kemple and H. Beyer, Global and causal solutions of fractional differential equations, Transform methods and special functions: Varna 96, Proceedings of 2nd international workshop (SCTP), Vol. 96, Singapore, 1997, pp. 210–216.
14. 14 A. A. Kilbas and J. J. Trujillo, Differential equations of fractional order: methods, results problems, Appl Anal 78 (2001), 153–192.
15. 15 S. V. Kiryakova, Multiple (multiindex) Mittag‐Leffler functions and relations to generalized fractional calculus, J Comput Appl Math 118 (2000), 441–452.
16. 16 S. Momani and N. T. Shawagfeh, Decomposition method for solving fractional Riccati differential equations, Appl Math Comput 182 (2006), 1083–1092.
17. 17 K. B. Oldham and J. Spanier, The fractional calculus, Academic Press, New York, 1974.
18. 18 G. Adomian, A review of the decomposition method in applied mathematics, J Math Anal Appl 135 (1988), 501–544.
19. 19 M. Dehghan and F. Shakeri, The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics, Physica Scripta 78 (2008), 1–11.
20. 20 M. Dehghan and M. Tatari, Solution of a semilinear parabolic equation with an unknown control function using the decomposition procedure of Adomian, Numer Methods Partial Differential Eq 23 (2007), 499–510.
21. 21 M. Tatari and M. Dehghan, Numerical solution of Laplace equation in a disk using the Adomian decomposition method, Physica Scripta 72 (2005), 345–348.
22. 22 M. Dehghan, A. Hamidi, and M. Shakourifar, The solution of coupled Burgers' equations using Adomian‐Pade technique, Appl Math Comput 189 (2007), 1034–1047.
23. 23 A. M. Wazwaz, The variational iteration method for rational solutions for K(2,2), Burgers, and cubic Boussinesq equations, J Comput Appl Math 207 (2007), 18–23.
24. 24 C. L. Peter, V. Muto, and S. Rionero, Solitary wave solutions to a system of Boussinesq‐like equations, Chaos, Solitons Fractals 2 (1992), 529–539.
25. 25 P. Rosenau and J. M. Hyman, Compactons: Solitons with finite wavelengths, Phys Rev Lett 70 (1993), 564–567.
26. 26 T. J. Priestly and P. A. Clarkson, Symmetries of a generalized Boussinesq equation, IMS Technical Report, UKC/IMS/ 59, 1996.
27. 27 D. Kaya, Explicit solutions of generalized Boussinesq equations, J Appl Math 1 (2001), 29–37.
28. 28 A. M. Wazwaz, Necessary conditions for the appearance of noise terms in decomposition solution series, Appl Math Comput 81 (1997), 265–274.
29. 29 A. M. Wazwaz, A new method for solving singular initial value problems in the second order differential equations, Appl Math Comput 128 (2002), 47–57.
30. 30 K. Al‐Khaled and S. Momani, An approximate solution for a fractional diffusion‐wave equation using the decomposition method, Appl Math Comput 165 (2005), 473–483.
31. 31 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.
32. 32 N. T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Appl Math Comput 131 (2002), 241–259.
33. 33 L. Song and H. Zhang, Solving the fractional BBM‐Burgers equation using the homotopy analysis method, Chaos, Solitons Fractals 40 (2009), 1616–1622.
34. 34 H. Zhu and B. Xuan, Existence and convergence of solutions for the generalized BBM‐Burgers equation with dissipative term, Nonlinear Anal 28 (1997), 1835–1849.
35. 35 Y. T. Gao and B. Tian, Ion–acoustic shocks in space and laboratory dusty plasmas: Two‐dimensional and non‐traveling‐wave observable effects, Phys Plasma 8 (2001), 3146–3149.
36. 36 A. Osborne, The inverse scattering transform: Tools for the nonlinear fourier analysis and filtering of ocean surface waves, Chaos, Solitons Fractals 5 (1995), 2623–2637.
37. 37 Z. Yan, New families of solitons with compact support for Boussinesq–like B(m,n) equations with fully nonlinear dispersion, Chaos, Solitons Fractals 14 (2002), 1151–1158.
38. 38 E. Fermi, J. R. Pasta, and S. M. Ulam,In: E. Segre, editor, Journal of collected papers of Enrico Fermi, Vol. 2, University of Chicago, 1965.
39. 39 Y. Zhu, Q. Chang, and S. Wu, Exact solitary solutions with compact support for the nonlinear dispersive Boussinesq‐like B (m, n) equations, Chaos, Solitons Fractals 26 (2005), 407–413.
40. 40 M. Dehghan and M. Tatari, The use of He's variational iteration method for solving a Fokker‐Planck equation, Phys Scripts 74 (2006), 310–316.
41. 41 M. Dehghan and M. Tatari, Identifying an unknown function in a parabolic equation with overspecified data via He's variational iteration method, Chaos, Solitons Fractals 36 (2008), 157–166.
42. 42 M. Dehghan and F. Shakeri, Application of He's variational iteration method for solving the Cauchy reaction‐diffusion problem, J Comput Appl Math 214 (2008), 435–446.
43. 43 F. Shakeri and M. Dehghan, Numerical solution of the Klein‐Gordon equation via He's variational iteration method, Nonlinear Dyn 51 (2008), 89–97.
44. 44 M. Tatari and M. Dehghan, Solution of problems in calculus of variations via He's variational iteration method, Phys Lett A 362 (2007), 401–406.
45. 45 M. Tatari and M. Dehghan, On the convergence of Hes variational iteration method, J Comput Appl Math 207 (2007), 121–128.
46. 46 M. Tatari and M. Dehghan, He's variational iteration method for computing a control parameter in a semi‐linear inverse parabolic equation, Choas, Solitons and Fractals 33 (2007), 671–677.
47. 47 M. Dehghan and F. Shakeri, Numerical solution of a biological population model using He's variational iteration method, Comput Math Applic 54 (2007), 1197–1209.
48. 48 M. Dehghan and F. Shakeri, Solution of parabolic integro‐differential equations arising in heat conduction in materials with memory via He's variational iteration technique, Communications in Numerical Methods in Engineering, 2008 Sep 4; [Epub ahead of print (DOI: 10.1002/cnm.1166)].
49. 49 M. Dehghan and F. Shakeri, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy 13 (2008), 53–59.
50. 50 F. Shakeri and M. Dehghan, Solution of a model describing biological species living together using the variational iteration method, Mathematical and Computer Modelling 48 (2008), 685–699.
51. 51 M. Dehghan and A. Saadatmandi, Variational iteration method for solving the wave equation subject to an integral conservation condition, Chaos, Solitons Fractals 41 (2009), 1448–1453.
52. 52 S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University, 1992.
53. 53 S. J. Liao, Series solutions of unsteady boundary‐layer flows over a stretching flat plate, Studies Appl Math 117 (2006), 2529–2539.
54. 54 S. J. Liao, An approximate solution technique which does not depend upon small parameters: a special example, Int J Nonlinear Mech 30 (1995), 371–380.
55. 55 S. J. Liao, An approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics, Int J Nonlinear Mech 32 (1997), 815–822.
56. 56 S. J. Liao, An explicit, totally analytic approximation of Blasius viscous flow problems, Int J Nonlinear Mech 34 (1999), 759–778.
57. 57 S. J. Liao, Beyond perturbation: introduction to the homotopy analysis method, Chapman and Hall, CRC Press, Boca Rator, 2003.
58. 58 S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl Math Comput 147 (2004), 499–513.
59. 59 S. J. Liao and A. Campo, Analytic solutions of the temperature distribution in Blasius viscous flow problems, J Fluid Mech 453 (2002), 411–425.
60. 60 S. J. Liao, On the analytic solution of magnetohydrodynamic flows of non‐Newtonian fluids over a stretching sheet, J Fluid Mech 488 (2003), 189–212.
61. 61 S. J. Liao, An analytic approximate approach for free oscillations of self‐excited systems, Int J Nonlinear Mech 39 (2004), 271–280.
62. 62 S. Abbasbandy, Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method, Chem Eng J 136 (2008), 144–150.
63. 63 T. Hayat and M. Sajid, Homotopy analysis of MHD boundary layer flow of an upper‐convected Maxwell fluid, Int J Eng Sci 45 (2007), 393–401.
64. 64 L. Song and H. Zhang, Application of homotopy analysis method to fractional KdV‐Burgers‐Kuramoto equation, Phys Lett A 367 (2007), 88–94.
65. 65 H. Xu and C. Jie, Analysis of a time fractional wave‐like equation with the homotopy analysis method, Phys Lett A 372 (2008), 1250–1255.
66. 66 M. Dehghan and F. Shakeri, Use of He's homotopy perturbation method for solving a partial differential equation arising in modeling of flow in porous media, Journal of Porous Media 11 (2008), 765–778.
67. 67 F. Shakeri and M. Dehghan, Solution of the delay differential equations via homotopy perturbation method, Mathematical and Computer Modelling 48 (2008), 486–498.
68. 68 M. Dehghan and F. Shakeri, Solution of an integro‐differential equation arising in oscillating magnetic fields using He's homotopy perturbation method, Progress in Electromagnetic Research, PIER 78 (2008), 361–376.
69. 69 A. Saadatmandi, M. Dehghan, and A. Eftekhari, Application of He's homotopy perturbation method for non‐linear system of second‐order boundary value problems, Nonlinear Analysis: Real World Applications 10 (2009), 1912–1922.
70. 70 M. Dehghan and F. Shakeri, The numerical solution of the second Painleve equation, Numer Methods Partial Differential Eq, 2009 Jan 23; [Epub ahead of print (DOI: 10.1002/num.20416)].
71. 71 M. S. H. Chowdhury, I. Hashim, and O. Abdulaziz, Comparison of homotopy analysis method and homotopy perturbation method for purely nonlinear fin–type problems, Commun Nonlinear Sci Numer Simul 14 (2009), 371–378.
72. 72 M. Sajid and T. Hayat, Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations, Nonlinear Anal Real World Appl 9 (2008), 2296–2301.
73. 73 H. Lu and M. Wang, Exact soliton solutions of some nonlinear physical models, Phys Lett A 255 (1999), 249–252.
74. 74 S.A. El‐Wakil, E.M. Abulwafa, and M.A. Abdou, An improved variational iteration method for solving coupled KdV and Boussinesqlike B(m,n) equations, Chaos, Solitons Fractals 39 (2009), 1324–1334.
75. 75 S. Momani, Analytical approximate solution for fractional heat‐like and wave‐like equations with variable coefficients using the decomposition method, Appl Math Comput 165 (2005), 459–472.
76. 76 D. Kaya and I.E. Inan, Exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation, Appl Math Comput 151 (2004), 775–787.