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An approximate solution for a fractional diffusion-wave equation using the decomposition method. (English) Zbl 1071.65135
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:
 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 (hyperbolic PDE)
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