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Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion-wave equations. (English) Zbl 1231.65179
Summary: The Laplace decomposition method is employed to obtain approximate analytical solutions of the linear and nonlinear fractional diffusion-wave equations. This method is a combined form of the Laplace transform method and the Adomian decomposition method. The proposed scheme finds the solutions without any discretization or restrictive assumptions and is free from round-off errors and therefore, reduces the numerical computations to a great extent. The fractional derivative described here is in the Caputo sense. Some illustrative examples are presented and the results show that the solutions obtained by using this technique have close agreement with series solutions obtained with the help of the Adomian decomposition method.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35R11 Fractional partial differential equations
35K55 Nonlinear parabolic equations
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[1] Jafari, H.; Momani, S., Solving fractional diffusion and wave equations by modified homotopy perturbation method, Phys. lett. A, 370, 388-396, (2007) · Zbl 1209.65111
[2] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[3] Momani, S., Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method, Appl. math. comput., 149, 51-59, (2004)
[4] Weitzner, H.; Zaslavsky, G.M., Some applications of fractional equations, Commun. nonlinear sci. numer. simulat., 8, 273-281, (2003) · Zbl 1041.35073
[5] Schneider, W.R.; Wyss, W., Fractional diffusion and wave equations, J. math. phys., 30, 1, 134-144, (1998) · Zbl 0692.45004
[6] Agrawal, O.P., Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear dyn., 29, 145-155, (2002) · Zbl 1009.65085
[7] Ginoa, M.; Cerbelli, S.; Roman, H.E., Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A, 191, 449-453, (1992)
[8] Khuri, S.A., A Laplace decomposition algorithm applied to class of nonlinear differential equations, J. math. appl., 141-155, (2001) · Zbl 0996.65068
[9] Khuri, S.A., A new approach to bratu‘s problem, Appl. math. comput., 147, 131-136, (2004) · Zbl 1032.65084
[10] Yusufoglu (Aghadjanov), Elcin, Numerical solution of Duffing equation by the Laplace decomposition algorithm, Appl. math. comput., 177, 572-580, (2006) · Zbl 1096.65067
[11] Hosseinzadeh, H; Jafari, H.; Roohani, M., Application of Laplace decomposition method for solving klein – gordon equation, World appl. sci. J., 8, 7, 809-813, (2010)
[12] Khan, Majid; Hussain, M.; Jafari, Hossein; Khan, Yasir, Application of Laplace decomposition method to solve nonlinear coupled partial differential equations, World appl. sci. J., 9, 13-19, (2010)
[13] Khan, Majid; Asif Gondal, Muhammad, Applications of Laplace decomposition to solve nonlinear partial differential equations, J. adv. res. sci. comput., 2, 52-62, (2010)
[14] Khan, Majid; Hussain, Mazhar, Application of Laplace decomposition method on semi-infinite domain, Numer. algor., 56, 211-218, (2011) · Zbl 1428.35366
[15] Khan, M.; Gondal, M.A., Restrictions and improvements of Laplace decomposition method, J. adv. res. sci. comput., 3, 8-14, (2011)
[16] Jafari, H.; Daftardar-Gejji, V., Solving linear and nonlinear fractional diffution- and wave equations by Adomian decomposition, Appl. math. comput., 180, 488-497, (2006) · Zbl 1102.65135
[17] Luchko, Yu.; Gorenflo, R., An operational method for solving fractional differential equations with the Caputo derivatives, Acta math. vietnamica, 24, 207233, (1999)
[18] Moustafa, O.L., On the Cauchy problem for some fractional order partial differential equations, Chaos solitons fractals, 18, 135-140, (2003) · Zbl 1059.35034
[19] Samko, G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003
[20] Mainardi, F., On the initial value problem for the fractional diffusion-wave equation, (), 246-251
[21] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equation, (1993), Wiley-Intersciensce New York, Modelling · Zbl 0789.26002
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