Decomposition-transform method for fractional differential equations

International Journal of Nonlinear Sciences and Numerical Simulation

Volumn 11, Issue , 2010, Pages 305-310

  Faraz, Naeem ^a   Khan, Yasir ^b   Sankar, D S ^c  

^a DONGHUA UNIVERSITY   (China)

^b ZHEJIANG UNIVERSITY   (China)

^c SCHOOL OF PHYSICS   (Malaysia)

Author keywords

Adomian decomposition method; Fractional partial differential equations; Laplace transforms method

Indexed keywords

EID: 84856015132     PISSN: 15651339     EISSN: None     Source Type: Journal    
DOI: 10.1515/ijnsns.2010.11.s1.305     Document Type: Article

References (40)

1. K. B. Oldham and J. Spanier, The Fractional Calculus. Academic Press, New York, 1974.
2. B. Ross (Ed.) Fractional Calculus and its Applications Lecture Notes in Mathematics #457. Springer Verlag, Berlin (1975) (Proc. Intr. Conf. held at Univ. of New Haven, USA. 1974.
3. K. Miller, B. Ross, An Introduction to the fractional Calculus and Fractional Differetial Equations, Wiley, New York, 1993.
4. M. Caputo, Elasticita e P Dissipazione. Zamchelli. Bologna (1969) (in Italian)
5. Yu. I. Babenko. Heat and Mass Transfer. Chimia, Leningrad (1986) (in Russian).
6. Ji Huan He, Approximate analytical solution for seepage flow with fractional derivative in porous media, Comput. Method. Appl. Meek Engeg, 167(1998), 57-68.
7. Z. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer Simul, 7(1) (2006), 27-34.
8. G. C. Wu, E. W. M. Lee, Fractional variational iteration method and its application, Physics Letters A, 2010, doi:10.1016/j.physleta.2010. 04.034.
9. H. Tari, D. D. Ganji, M. Rostamian, Approximate solutions of K (2, 2), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method, Int. J. Nonlin. Sci. Numer. Sim., 8(2) (2007), 203-210.
10. A. Arikoglu, I. Ozkol, Solution of fractional differential equations by using differential transform method, Chaos Solitons & Fractals, 34(5) (2007), 1473-1481.
11. F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput, 191(1) (2007), 12-20.
12. A. J. George, A. Chakrabarti, The Adomian method applied to some extraordinary differential equations, Appl. Math. Lett., 8(3) (1995), 91-97.
13. H. L. Arora, F. I. Abdelwahid, Solutions of non-integer order differential equations via the Adomian Decomposition Method, Appl. Math. Lett., 6(1) (1993), 21-23.
14. N. T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comp., 131(2002), 517-529.
15. I. Podlubny, Fractional Differential Equations, Academic press, San Diego, 1999.
16. R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals & Fractional order, in Continuum Mechanics, Springer, New York, (1997), 223-276.
17. G. Jumarie, Fourier's transform of fractional order via Mittag-Leffler function and modified Riemann-Liouville derivative, J. Appl Math. & informatics, 26(2008), 1101-1121.
18. Yasir Khan, An Effective Modification of the Laplace Decomposition Method for Nonlinear Equations, Internal. J. Nonlinear Sci. Numer. Simulation, 10(11-12) (2009), 1373-1376.
19. Yasir Khan, F. Austin, Application of the Laplace decomposition method to nonlinear homogeneous and non-homogenous Advection equations, Z. Naturforsch. A Accepted.
20. Esmail Hesameddini, Habibolla Latifizadeh, Reconstruction of Variational Iteration Algorithms using the Laplace Transform, Internal. J. Nonlinear Sci. Numer. Simulation, 10(11-12) (2009), 1377-1382.
21. M. Madani, M. Fathizadeh, Homotopy Perturbation Algorithm Using Laplace Transformation, Nonl. Sci. Lett. A, 1(3) (2010), 263-267.
22. A. M. Wazwaz, The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integrodifferential equations, Applied Mathematics and Computation, 216(2010), 1304-1309.
23. Francesco Mainardi, Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena, Chaos, Solitom & Fractah, 7(9) (1996), 146-1477.
24. F. Mainardi, The Fundamental Solutions for the Fractional Diffusion-Wave Equation, Appl. Math. Lett., 9(6) (1996), 23-28.
25. μ-Controllers, IEEE Transactions on Automatic Contro., 44(1) (1999), 208-214.
26. O. M. P. AGRAWAL, Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain, Nonlinear Dynamics, 29(2002), 145-155.
27. W. Mclean, V. Thomee, Numerical solution via Laplace transforms of a fractional order evolution equation, J. Integ. Eq. Appl., 22(2010), 57-94.
28. S. Hyder Ali Muttaqi Shah, Haitao Qi Starting solutions for a viscoelastic fluid with fractional Burgers model in an annular pipe, Nonlinear Analysis: Real World Applications, 11(2010), 547-554.
29. C. Xu, J. W. Feng, F. Austin, W. Q. Zhang, The synchronization of Rossler hyperchaotic system with a fractional order, Internat. J. Nonlinear Sci. Numer. Simulation, 10(11-12) (2009), 1517-1521.
30. G. Jumarie, Laplace's transform of fractional order via the Mittage-Leffler function and modified Riemann-Liouville derivative, Appl Math. Letters. 22(2009), 1659-1664.
31. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht, 1994.
32. G. Adomian, A review of the decomposition method and some recent results for nonlinear equation, Math. Comput. Model, 13(7) (1992), 17-43.
33. A. Wazwaz, A reliable modification of Adomian Decomposition Method, Appl. Math. Comp., 102(1) (1999), 77-86.
34. S. Momani, Z, Odibat, Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A., 355(4-5) (2006), 271-279.
35. Hossein Jafari, Varsha Daftardar-Gejji, Solving a system of nonlinear fractional differential equations using Adomian decomposition, J. Comput. Appl. Math, 196(2006), 644-651.
36. Shaher Momani, Zaid Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177(2006), 488-494.
37. S. Momani, Non-perturbative analytical solutions of the space- and time-fractional Burgers equations CHAOS SOLITONS & FRACTALS, 28(4) (2006), 930-937.
38. Kamel Al-Khaled, Shaher Momani, An approximate solution for a fractional diffusionwave equation using the decomposition method, Appl. Math. Comput., 165(2005), 473-483.
39. Zaid Odibat, Shaher Momani, Numerical methods for nonlinear partial differential equations of fractional order, Appl. Math. Modelling., 32(2008), 28-39.
40. S. Das, Solution of Fractional Vibration Equation by the Variational Iteration Method and Modified Decomposition Method, Internat. J. Nonlinear Sci. Numer. Simulation, 9(4) (2008), 361-366.