×

zbMATH — the first resource for mathematics

Application of modified decomposition method for the analytical solution of space fractional diffusion equation. (English) Zbl 1133.65119
Summary: Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical super diffusive problems in fluid flow, finance and others areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by a modified decomposition method. By using initial conditions, the explicit solutions of the equations are presented in the closed form. The decomposition series analytic solution of the problem is quickly obtained by observing the existence of the self-cancelling “noise” phenomenon. Two examples, the first one is the one-dimensional and the second one is the two-dimensional fractional diffusion equation, are presented to show the application of the present technique. The present method performs extremely well in terms of efficiency and simplicity.

MSC:
65R20 Numerical methods for integral equations
26A33 Fractional derivatives and integrals
45K05 Integro-partial differential equations
35K15 Initial value problems for second-order parabolic equations
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Metzler, R.; Barkai, E.; Klafter, J., Anomalous diffusion and relaxation close to thermal equilibrium: a fractional fokker – planck equation approach, Phys. rev. lett., 82, 18, 3563-3567, (1999)
[2] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, 339, 1-77, (2000) · Zbl 0984.82032
[3] Metzler, R.; Klafter, J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. phys. A, 37, R161-R208, (2004) · Zbl 1075.82018
[4] Gorenflo, R.; Mainardi, F.; Scalas, E.; Raberto, M., Fractional calculus and continuous time finance. III. the diffusion limit. mathematical finance (Konstanz, 2000), Trends math. birkhuser basel, 171-180, (2001) · Zbl 1138.91444
[5] Mainardi, F.; Raberto, M.; Gorenflo, R.; Scalas, E., Fractional calculus and continuous-time finance II: the waiting-time distribution, Physica A, 287, 468-481, (2000)
[6] Scalas, E.; Gorenflo, R.; Mainardi, F., Fractional calculus and continuous-time finance, Physica A, 284, 376-384, (2000)
[7] Raberto, M.; Scalas, E.; Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Physica A, 314, 749-755, (2002) · Zbl 1001.91033
[8] Benson, D.A.; Wheatcraft, S.; Meerschaert, M.M., Application of a fractional advection – dispersion equation, Water resour. res., 36, 1403-1412, (2000)
[9] Baeumer, B.; Meerschaert, M.M.; Benson, D.A.; Wheatcraft, S.W., Subordinated advection – dispersion equation for contaminant transport, Water resour. res., 37, 1543-1550, (2001)
[10] Benson, D.A.; Schumer, R.; Meerschaert, M.M.; Wheatcraft, S.W., Fractional dispersion, Lévy motions, and the MADE tracer tests, Transp. porous media, 42, 211-240, (2001)
[11] Schumer, R.; Benson, D.A.; Meerschaert, M.M.; Wheatcraft, S.W., Eulerian derivation of the fractional advection – dispersion equation, J. contam. hydrol., 48, 9-88, (2001)
[12] Schumer, R.; Benson, D.A.; Meerschaert, M.M.; Baeumer, B., Multiscaling fractional advection – dispersion equations and their solutions, Water resour. res., 39, 1022-1032, (2003)
[13] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (), 291-348 · Zbl 0917.73004
[14] Mainardi, F.; Pagnini, G., The weight functions as solutions of the time-fractional diffusion equation, Appl. math. comput., 141, 51-62, (2003) · Zbl 1053.35008
[15] Agrawal, O.P., Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear dynam., 29, 145-155, (2002) · Zbl 1009.65085
[16] Schneider, W.R.; Wyss, W., Fractional diffusion and wave equations, J. math. phys., 30, 134-144, (1989) · Zbl 0692.45004
[17] Meerschaert, M.M.; Scheffler, H.; Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J. comput. phys., 211, 249-261, (2006) · Zbl 1085.65080
[18] Tadjeran, C.; Meerschaert, M.M.; Scheffler, H., A second-order accurate numerical approximation for the fractional diffusion equation, J. comput. phys., 213, 205-213, (2006) · Zbl 1089.65089
[19] Adomian, G., Nonlinear stochastic systems theory and applications to physics, (1989), Kluwer Academic Publishers The Netherlands · Zbl 0659.93003
[20] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Publishers Boston · Zbl 0802.65122
[21] Adomian, G., An analytical solution of the stochastic Navier-Stokes system, Foundations of physics, 21, 7, 831-843, (1991)
[22] Adomian, G.; Rach, R., Linear and nonlinear Schrödinger equations, Foundations of physics, 21, 983-991, (1991)
[23] Adomian, G., Solution of physical problems by decomposition, Comput. math. appl., 27, 9/10, 145-154, (1994) · Zbl 0803.35020
[24] Adomian, G., Solutions of nonlinear P.D.E, Appl. math. lett., 11, 3, 121-123, (1998) · Zbl 0933.65121
[25] Abbaoui, K.; Cherruault, Y., The decomposition method applied to the Cauchy problem, Kybernetes, 28, 103-108, (1999) · Zbl 0937.65074
[26] Kaya, D.; Yokus, A., A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations, Math. comput. simul., 60, 6, 507-512, (2002) · Zbl 1007.65078
[27] Wazwaz, A., A reliable modification of Adomian decomposition method, Appl. math. comput., 102, 1, 77-86, (1999) · Zbl 0928.65083
[28] Kaya, D.; El-Sayed, S.M., On a generalized fifth order KdV equations, Phys. lett. A, 310, 1, 44-51, (2003) · Zbl 1011.35114
[29] Kaya, D.; El-Sayed, S.M., An application of the decomposition method for the generalized KdV and RLW equations, Chaos solitons fractals, 17, 5, 869-877, (2003) · Zbl 1030.35139
[30] Kaya, D., An explicit and numerical solutions of some fifth-order KdV equation by decomposition method, Appl. math. comput., 144, 2-3, 353-363, (2003) · Zbl 1024.65096
[31] Kaya, D., A numerical simulation of solitary-wave solutions of the generalized regularized long-wave equation, Appl. math. comput., 149, 3, 833-841, (2004) · Zbl 1038.65101
[32] George, A.J.; Chakrabarti, A., The Adomian method applied to some extraordinary differential equations, Appl. math. lett., 8, 3, 91-97, (1995) · Zbl 0828.65081
[33] Arora, H.L.; Abdelwahid, F.I., Solutions of non-integer order differential equations via the Adomian decomposition method, Appl. math. lett., 6, 1, 21-23, (1993) · Zbl 0772.34009
[34] Shawagfeh, N.T., The decompostion method for fractional differential equations, J. frac. calc., 16, 27-33, (1999) · Zbl 0956.34004
[35] Shawagfeh, N.T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. math. comput., 131, 517-529, (2002) · Zbl 1029.34003
[36] Saha Ray, S.; Bera, R.K., Solution of an extraordinary differential equation by Adomian decomposition method, J. appl. math., 4, 331-338, (2004) · Zbl 1080.65069
[37] Saha Ray, S.; Bera, R.K., Analytical solution of a dynamic system containing fractional derivative of order 1/2 by Adomian decomposition method, Trans. ASME J. appl. mech., 72, 2, 290-295, (2005) · Zbl 1111.74611
[38] Saha Ray, S.; Bera, R.K., An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl. math. comput., 167, 1, 561-571, (2005) · Zbl 1082.65562
[39] Saha Ray, S.; Bera, R.K., Analytical solution of the bagley torvik equation by Adomian decomposition method, Appl. math. comput., 168, 1, 398-410, (2005) · Zbl 1109.65072
[40] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego, CA, USA · Zbl 0918.34010
[41] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York and London · Zbl 0428.26004
[42] Cherruault, Y., Convergence of adomian’s method, Kybernetes, 18, 31-38, (1989) · Zbl 0697.65051
[43] Abbaoui, K.; Cherruault, Y., Convergence of adomian’s method applied to differential equations, Comput.math. appl., 28, 5, 103-109, (1994) · Zbl 0809.65073
[44] Abbaoui, K.; Cherruault, Y., New ideas for proving convergence of decomposition methods, Comput. math. appl., 29, 103-108, (1995) · Zbl 0832.47051
[45] Himoun, N.; Abbaoui, K.; Cherruault, Y., New results of convergence of adomian’s method, Kybernetes, 28, 4-5, 423-429, (1999) · Zbl 0938.93019
[46] Adomian, G.; Rach, R., Noise terms in decomposition solution series, Comput. math. appl., 24, 11, 61-64, (1992) · Zbl 0777.35018
[47] Wazwaz, A.M., Necessary conditions for the appearance of noise terms in decomposition solution series, Appl. math. comput., 81, 2-3, 265-274, (1997) · Zbl 0882.65132
[48] Wazwaz, A.M., Partial differential equations: methods and applications, (2002), A.A. Balkema Publishers Lisse, The Netherlands · Zbl 0997.35083
[49] Wazwaz, A.M., The existence of noise terms for systems of inhomogeneous differential and integral equations, 146, 1, 81-92, (2003) · Zbl 1032.65114
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.