On Legendre polynomial approximation with the VIM or HAM for numerical treatment of nonlinear fractional differential equations

Journal of Computational and Applied Mathematics

Volumn 235, Issue 9, 2011, Pages 2956-2968

  Odibat, Zaid ^a  

^a AL BALQA APPLIED UNIVERSITY   (Jordan)

Author keywords

Caputo derivative; Fractional differential equations; Homotopy analysis method; Legendre polynomials; Variational iteration method

Indexed keywords

CAPUTO DERIVATIVES; FRACTIONAL DIFFERENTIAL EQUATIONS; HOMOTOPY ANALYSIS METHOD; LEGENDRE POLYNOMIALS; VARIATIONAL ITERATION METHOD;

APPROXIMATION ALGORITHMS; COMPUTATIONAL MECHANICS; DIFFERENTIAL EQUATIONS; DIFFERENTIATION (CALCULUS); ITERATIVE METHODS; NONLINEAR ANALYSIS; POLYNOMIAL APPROXIMATION;

NONLINEAR EQUATIONS;

EID: 79951512760     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2010.12.013     Document Type: Article

References (58)

1. S.G. Samko, A.A. Kilbas, and O.I. Marichev Fractional Integrals and Derivatives: Theory and Applications 1993 Gordon and Breach Yverdon
2. K.S. Miller, and B. Ross An Introduction to the Fractional Calculus and Fractional Differential Equations 1993 Wiley New York
3. I. Podlubny Fractional Differential Equations 1999 Academic Press New York
4. R. Hilfer Applications of Fractional Calculus in Physics 2000 World Scientific New Jersey
5. I. Petr A note on the fractional-order Chua's system Chaos Solitons Fractals 38 1 2008 140 147
6. S. Momani, and K. Al-Khaled Numerical solutions for systems of fractional differential equations by the decomposition method Appl. Math. Comput. 162 3 2005 1351 1365 (Pubitemid 40114500)
7. S. Momani Non-perturbative analytical solutions of the space- and time-fractional Burgers equations Chaos Solitons Fractals 28 4 2006 930 937 (Pubitemid 41607591)
8. S. Momani, and Z. Odibat Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method Appl. Math. Comput. 177 2006 488 494 (Pubitemid 43866876)
9. Z. Odibat, and S. Momani Approximate solutions for boundary value problems of time-fractional wave equation Appl. Math. Comput. 181 2006 1351 1358
10. Z. Odibat, and S. Momani Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order Chaos Solitons Fractals 36 1 2008 167 174 (Pubitemid 47576648)
11. S. Momani, and Z. Odibat Comparison between homotopy perturbation method and the variational iteration method for linear fractional partial differential equations Comput. Math. Appl. 54 78 2007 910 919 (Pubitemid 47488806)
12. S. Momani, and Z. Odibat Homotopy perturbation method for nonlinear partial differential equations of fractional order Phys. Lett. A 365 56 2007 345 350
13. M.S. Chowdhury, I. Hashim, and S. Momani The multistage homotopy perturbation method: a powerful scheme for handling the Lorenz system Chaos Solitons Fractals 40 4 2009 1929 1937
14. Z. Odibat Exact solitary solutions for variants of the KdV equations with fractional time derivatives Chaos Solitons Fractals 40 3 2009 1264 1270
15. V. Erturk, S. Momani, and Z. Odibat Application of generalized differential transform method to multi-order fractional differential equations Commun. Nonlinear Sci. Numer. Simul. 13 8 2008 1642 1654 (Pubitemid 351250728)
16. Z. Odibat, and S. Momani Generalized differential transform method for linear partial differential equations of fractional order Appl. Math. Lett. 21 2 2008 194 199 (Pubitemid 350181314)
17. S. Momani, Z. Odibat, and V. Erturk Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation Phys. Lett. A 370 56 2007 379 387
18. S. Momani, and Z. Odibat A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor's formula J. Comput. Appl. Math. 220 12 2008 85 95
19. Z. Odibat, and S. Momani Application of variational iteration method to nonlinear differential equations of fractional order I. J. Nonlinear Sci. Numer. Simul. 7 1 2006 27 34 (Pubitemid 43246579)
20. S. Momani, and Z. Odibat Numerical comparison of methods for solving linear differential equations of fractional order Chaos Solitons Fractals 31 5 2007 1248 1255 (Pubitemid 44345225)
21. S. Momani, and Z. Odibat Numerical approach to differential equations of fractional order J. Comput. Appl. Math. 207 1 2007 96 110 (Pubitemid 46935390)
22. Z. Odibat, and S. Momani Numerical methods for solving nonlinear partial differential equations of fractional order Appl. Math. Model. 32 1 2008 28 39 (Pubitemid 47355246)
23. Z. Odibat, and S. Momani The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics Comput. Math. Appl. 58 1112 2009 2199 2208
24. J. Cang, Y. Tan, H. Xu, and S.J. Liao Series solutions of non-linear Riccati differential equations with fractional order Chaos Solitons Fractals 40 1 2009 1 9
25. I. Hashim, O. Abdulaziz, and S. Momani Homotopy analysis method for fractional IVPs Commun. Nonlinear Sci. Numer. Simul. 14 3 2009 674 684
26. M. Zurigat, S. Momani, Z. Odibat, and A. Alawneh The homotopy analysis method for handling systems of fractional differential equations Appl. Math. Model. 34 1 2010 24 35
27. Z. Odibat, S. Momani, and H. Xu A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations Appl. Math. Model. 34 3 2010 593 600
28. C. Lubich Fractional linear multistep methods for AbelVolterra integral equations of the second kind Math. Comp. 45 172 1985 463 469
29. K. Diethelm, N. Ford, and A. Freed A predictor-corrector approach for the numerical solution of fractional differential equations Nonlinear Dynam. 29 2002 3 22 (Pubitemid 34945390)
30. K. Diethelm, N. Ford, and A. Freed Detailed error analysis for a fractional Adams method Numer. Algorithms 36 2004 31 52 (Pubitemid 39072748)
31. J.H. He Approximate solution of non linear differential equations with convolution product nonlinearities Comput. Methods Appl. Mech. Engrg. 167 1998 69 73
32. J.H. He Approximate analytical solution for seepage flow with fractional derivatives in porous media Comput. Methods Appl. Mech. Engrg. 167 1998 57 68
33. J.H. He Variational iteration method for autonomous ordinary differential systems Appl. Math. Comput. 114 2000 115 123
34. J.H. He Variational principles for some nonlinear partial differential equations with variable coefficients Chaos Solitons Fractals 19 4 2004 847 851
35. J.H. He Variational iteration method-some recent results and new interpretations J. Comput. Appl. Math. 207 1 2007 3 17
36. M.A. Abdou, and A.A. Soliman New applications of variational iteration method Physica D 211 12 2005 1 8 (Pubitemid 41491773)
37. A.M. Wazwaz A comparison between the variational iteration method and Adomian decomposition method J. Comput. Appl. Math. 207 1 2007 129 136
38. T.A. Abassy, M.A. El-Tawil, and H. El Zoheiry Solving nonlinear partial differential equations using the modified variational iteration Pad technique J. Comput. Appl. Math. 207 1 2007 73 91 (Pubitemid 46935388)
39. M. Tatari, and M. Dehghan On the convergence of He's variational iteration method J. Comput. Appl. Math. 207 1 2007 121 128 (Pubitemid 46950910)
40. A.M. Wazwaz The variational iteration method: a powerful scheme for handling linear and nonlinear diffusion equations Comput. Math. Appl. 54 78 2007 933 939
41. Z. Odibat Reliable approaches of variational iteration method for nonlinear operators Math. Comput. Model. 48 12 2008 222 231 (Pubitemid 351672229)
42. M. Tatari, and M. Dehghan Improvement of He's variational iteration method for solving systems of differential equations Comput. Math. Appl. 58 1112 2009 2160 2166
43. Z. Odibat A study on the convergence of variational iteration method Math. Comput. Model. 51 910 2010 1181 1192
44. S.J. Liao A kind of approximate solution technique which does not depend upon small parameters: a special example Internat. J. Non-Linear Mech. 30 1995 371 380
45. S.J. Liao Beyond Perturbation: Introduction to the Homotopy Analysis Method 2003 Chapman & Hall/CRC Press Boca Raton
46. S.J. Liao On the homotopy analysis method for nonlinear problems Appl. Math. Comput. 147 2004 499 513
47. S.J. Liao Notes on the homotopy analysis method: Some definitions and theorems Commun. Nonlinear Sci. Numer. Simul. 14 4 2009 983 997
48. T. Hayat, and M. Sajid On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder Phys. Lett. A 361 45 2007 316 322 (Pubitemid 44894740)
49. L. Song, and H.Q. Zhang Application of homotopy analysis method to fractional KdVBurgersKuramoto equation Phys. Lett. A 367 12 2007 88 94
50. 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 23 2008 144 150 (Pubitemid 351173865)
51. S. Abbasbandy, E. Magyari, and E. Shivanian The homotopy analysis method for multiple solutions of nonlinear boundary value problems Commun. Nonlinear Sci. Numer. Simul. 14 910 2009 3530 3536
52. I. Hashim, O. Abdulaziz, and S. Momani Homotopy analysis method for fractional IVPs Commun. Nonlinear Sci. Numer. Simul. 14 3 2009 674 684
53. S.J. Liao Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method Nonlinear Anal. RWA 10 4 2009 2455 2470
54. Y.M. Chen, and J.K. Liu A study of homotopy analysis method for limit cycle of van der Pol equation Commun. Nonlinear Sci. Numer. Simul. 14 5 2009 1816 1821
55. E. Kreyszig Advanced Engineering Mathematics 1999 Wiley New York
56. W. Gautschi Orthogonal Polynomials: Computation and Approximation 2004 Oxford University Press Oxford
57. M.M. Hosseini Adomian decomposition method with Chebyshev polynomials Appl. Math. Comput. 175 2 2006 1685 1693
58. W.-C. Tien, and C.-K. Chen Adomian decomposition method by Legendre polynomials Chaos Solitons Fractals 39 5 2009 2093 2101