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Solution of the nonlinear fractional diffusion equation with absorbent term and external force. (English) Zbl 1221.35437

Summary: The article presents the approximate analytical solutions of general nonlinear diffusion equation with fractional time derivative in the presence of an absorbent term and a linear external force obtained with the help of powerful mathematical tool like Homotopy Perturbation Method. By using initial value, the approximate analytical solutions of the equation are derived. The fractional derivatives are described in the Caputo sense. Numerical results for different particular cases are presented graphically. The anomalous behavior of nonlinear diffusivity in the presence or absence of external force and reaction term are calculated numerically and presented graphically.

MSC:

35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35K20 Initial-boundary value problems for second-order parabolic equations
35K59 Quasilinear parabolic equations
45K05 Integro-partial differential equations
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[1] Wazwaz, A. M., Exact solutions to nonlinear diffusion equations obtained by the decomposition method, Appl. Math. Comput., 123, 109-122 (2001) · Zbl 1027.35019
[2] Laskin, N., Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268, 298-305 (2000) · Zbl 0948.81595
[3] Schot, A.; Lenzi, M. K.; Evangelista, L. R.; Malacarne, L. C.; Mendes, R. S.; Lenzi, E. K., Fractional diffusion equation with an absorbent term and a linear external force: exact solution, Phys. Lett. A, 366, 346-350 (2007)
[4] Zahran, M. A., On the derivation of fractional diffusion equation with an absorbent term and a linear external force, Appl. Math. Model., 33, 3088-3092 (2008)
[5] Bologna, M.; Tsallis, C.; Grigolini, P., Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: exact time-dependent solutions, Phys Rev E, 62, 2213-2218 (2000)
[6] Lenzi, E. K.; Mendes, G. A.; Mendes, R. S.; Silva, L. R.; Lucena, L. S., Exact solutions to nonlinear non autonomous space-fractional diffusion equations with absorption, Phys. Rev. E, 67, 051109 (2003)
[7] Lenzi, E. K.; Mendes, R. S.; Fa, K. S.; Moraes, L. S.; Silva, L. R.; Lucena, L. S., Nonlinear fractional diffusion equation: exact results, J. Math. Phys., 46, 083506 (2005) · Zbl 1110.60058
[8] Assis, P. C.; Silva, L. R.; Lenzi, E. K.; Malacarne, L. C.; Mendes, R. S., Nonlinear diffusion equation, Tsallis formalism and exact solutions, J. Math. Phys., 46, 123303 (2005) · Zbl 1111.82307
[9] Silva, A. T.; Lenzi, E. K.; Evangelista, L. R.; Lenzi, M. K.; Silva, LRda, Fractional nonlinear diffusion equation, solutions and anomalous diffusion, Physica A, 375, 65-71 (2007)
[10] Lenzi, E. K.; Lenzi, M. K.; Evangelista, L. R.; Malacarne, L. C.; Mendes, R. S., Solutions for a fractional nonlinear diffusion equation with external force and absorbent term, J. Stat. Mech., P02048 (2009)
[11] Das, S., A note on fractional diffusion equation, Chaos Soliton. Fract., 42, 2074-2079 (2009) · Zbl 1198.65137
[12] Das, S.; Gupta, P. K., An approximate analytical solution of the fractional diffusion equation with absorbent term and external force by homotopy perturbation method, Z. Naturforsch. A, 65a, 3, 182-190 (2010)
[13] He, J. H., Homotopy perturbation technique, Comput. Meth. Appl. Mech. Eng., 178, 257-262 (1999) · Zbl 0956.70017
[14] He, J. H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear Mech., 35, 37-43 (2000) · Zbl 1068.74618
[15] He, J. H., Periodic solutions and bifurcations of delay-differential equations, Phys. Lett. A, 347, 228-230 (2005) · Zbl 1195.34116
[16] He, J. H., Application of homotopy perturbation method to nonlinear wave equations, Chaos Soliton. Fract., 26, 695-700 (2005) · Zbl 1072.35502
[17] He, J. H., Limit cycle and bifurcation of nonlinear problems, Chaos Soliton. Fract., 26, 827-833 (2005) · Zbl 1093.34520
[18] He, J. H., Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simulat., 6, 207-208 (2005) · Zbl 1401.65085
[19] He, J. H., Some asymptotic methods for strongly nonlinear equations, Int. J. Modern Phys. B, 20, 1141-1199 (2006) · Zbl 1102.34039
[20] S. Das, P.K. Gupta. A mathematical model on fractional Lotka-Volterra equations. J. Theor. Biol., In Press doi:10.1016/j.jtbi.2011.01.034; S. Das, P.K. Gupta. A mathematical model on fractional Lotka-Volterra equations. J. Theor. Biol., In Press doi:10.1016/j.jtbi.2011.01.034 · Zbl 1405.92227
[21] He, J. H., Homotopy perturbation method for solving boundary value problems, Phys. Lett. A, 350, 87-88 (2006) · Zbl 1195.65207
[22] Das, S.; Gupta, P. K.; Rajeev, A fractional predator-prey model and its solution, Int. J. Nonlinear Sci. Numer. Simulat., 10, 4, 873-876 (2009)
[23] Das, S.; Gupta, P. K.; Barat, S., A note on fractional Schrodinger equation, Nonlinear Sci. Lett. A, 1, 1, 91-94 (2010)
[24] Das, S.; Vishal, K.; Gupta, P. K., Approximate approach to the Das model of fractional logistic population growth, Applic. Appl. Math., 05, 10, 1702-1708 (2010) · Zbl 1205.65323
[25] He, J. H., Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput., 135, 1, 73-79 (2003) · Zbl 1030.34013
[26] Momani, S.; Odibat, Z., Comparison between the HPM and the VIM for linear fractional partial differential equations, Comput. Math. Appl., 54, 910-919 (2007) · Zbl 1141.65398
[27] Odibat, Z.; Momani, S., Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Soliton. Fract., 36, 1, 167-174 (2008) · Zbl 1152.34311
[28] Odibat, Z.; Momani, S., Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A, 365, 5-6, 345-350 (2007) · Zbl 1203.65212
[29] Wang, Q., Homotopy perturbation method for fractional KdV-Burgers equation, Chaos Soliton. Fract., 35, 5, 843-850 (2008) · Zbl 1132.65118
[30] Podlubny, I., Fractional differential equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[31] Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer Verlag: Springer Verlag New York) · Zbl 1030.26004
[32] Abbaoui, K.; Cherruault, Y., New ideas for proving convergence of decomposition methods, Comput. Math. Appl., 29, 103-108 (1995) · Zbl 0832.47051
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