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Zbl 1071.65135
Al-Khaled, Kamel; Momani, Shaher
An approximate solution for a fractional diffusion-wave equation using the decomposition method. (English)
[J] Appl. Math. Comput. 165, No. 2, 473-483 (2005). ISSN 0096-3003

Summary: The partial differential equation of diffusion is generalized by replacing the first order time derivative by a fractional derivative of order $\alpha$, $0 < \alpha \leqslant 2$. An approximate solution based on the decomposition method is given for the generalized fractional diffusion (diffusion-wave) equation. The fractional derivative is described in the sense of {\it M. Caputo} [Linear models of dissipation whose $Q$ is almost frequency independent. II. J. Roy. Austral. Soc. 13, 529--539 (1967)]. A numerical example is given to show the application of the present technique. Results show the transition from a pure diffusion process $(\alpha = 1)$ to a pure wave process $(\alpha = 2)$.

MSC 2000:: *65M70 Spectral, collocation and related methods (IVP of PDE)
35K55 Nonlinear parabolic equations
35K05 Heat equation
26A33 Fractional derivatives and integrals (real functions)
35L05 Wave equation

Keywords: Diffusion-wave equation; Heat equation; Decomposition method; Fractional calculus; numerical examples; Adomian decomposition method; numerical example

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Zentralblatt MATH Berlin [Germany]