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Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method. (English) Zbl 1070.65105

Summary: The time fractional heat-like and wave-like equations with variable coefficients are obtained by replacing the first order and second order time derivative by a fractional derivative of order \(\alpha\), \(0 < \alpha \leqslant 2\). The applications of the decomposition method are extended to derive analytical solutions in the form of a series with easily computed terms for these generalized fractional equation. Some examples are presented to show the efficiency and simplicity of the method.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35L70 Second-order nonlinear hyperbolic equations
26A33 Fractional derivatives and integrals
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[1] Adomian, G., A review of the decomposition method in applied mathematics, J. math. anal. appl., 135, 501-544, (1988) · Zbl 0671.34053
[2] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Publishers Boston · Zbl 0802.65122
[3] K. Al-Khaled, D. Kaya, M.A. Noor, Numerical comparison of methods for solving parabolic equations, Appl. Math. Comput., in press · Zbl 1061.65098
[4] Agrawal, O.P., Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear dynam., 29, 145-155, (2002) · Zbl 1009.65085
[5] Andrezei, H., Multi-dimensional solutions of space-time-fractional diffusion equations, Proc. R. soc. lond., ser. A, math. phys. eng. sci., 458, 2018, 429-450, (2002)
[6] Caputo, M., Linear models of dissipation whose Q is almost frequency independent. part II, J. roy. astral. soc., 13, 529-539, (1967)
[7] Cherrualt, Y., Convergence of adomian’s method, Kybernetes, 18, 31-38, (1989)
[8] Cherrualt, Y.; Adomian, G., Decomposition methods: a new proof of convergence, Math. comput. model., 18, 103-106, (1993) · Zbl 0805.65057
[9] M. Dehghan, Application of the Adomian decomposition method for two-dimensional parabolic equation subject to nonstandard boundary specification, Appl. Math. Comput., in press · Zbl 1054.65105
[10] Fujita, Y., Cauchy problems of fractional order and stable processes, Japan J. appl. math., 7, 3, 459-476, (1990) · Zbl 0718.35026
[11] Hilfer, R., Foundations of fractional dynamics, Fractals, 3, 3, 549-556, (1995) · Zbl 0870.58041
[12] Hilfer, R., Fractional diffusion based on Riemann-Liouville fractional derivative, J. phys. chem., 104, 3914-3917, (2000)
[13] Klafter, J.; Blumen, A.; Shlesinger, M.F., Fractal behavior in trapping and reaction: a random walk study, J. stat. phys., 36, 561-578, (1984) · Zbl 0587.60062
[14] A.Y. Luchko, R. Groreflo, The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series A08–98, fachbreich mathematik und informatik, Freic Universitat Berlin, 1998
[15] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (), 291-348 · Zbl 0917.73004
[16] Metzler, R.; Klafter, J., Boundary value problems fractional diffusion equations, Physica A, 278, 107-125, (2000)
[17] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), John Wiley and Sons, Inc. New York · Zbl 0789.26002
[18] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004
[19] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[20] Shawagfeh, N.T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. math. comput., 123, 133-140, (2001)
[21] Wazwaz, A.M., Blow-up for solutions of some linear wave equations with mixed nonlinear boundary conditions, Appl. math. comput., 131, 517-529, (2002)
[22] Wazwaz, A.M.; Goruis, A., Exacat solution for heat-like and wave-like equations with variable coefficient, Appl. math. comput., 149, 51-59, (2004)
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