×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[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: 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; 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) · Zbl 0999.60035
[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) · Zbl 0697.65051
[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; 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 \(A\); A.Y. Luchko, R. Groreflo, The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series \(A\)
[15] Mainardi, F., Fractional calculus: Some basic problems in continuum and statistical mechanics, (carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer-Verlag: Springer-Verlag New York), 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.: John Wiley and Sons, Inc. New York · Zbl 0789.26002
[18] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[19] Podlubny, I., Fractional Differential Equations (1999), Academic Press: 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) · Zbl 1027.35016
[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) · Zbl 1029.34003
[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)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.