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Mapping between solutions of fractional diffusion-wave equations. (English) Zbl 1033.35161
Summary: We deal with a partial differential equation of fractional order where the time derivative of order $\beta \in \left(0;2\right]$ is defined in the Caputo sense and the space derivative of order $\alpha \in \left(0;2\right]$ is given as a pseudo-differential operator with the Fourier symbol $-{|\kappa |}^{\alpha }$, $\kappa \in ℝ$. This equation contains as particular cases the diffusion and the wave equations and it has already appeared both in mathematical papers and in applications. The main results of the paper consist in giving a mapping in the form of a linear integral operator between solutions of the equation with different parameters $\alpha$, $\beta$ and in presenting an explicit formula for the Green function of the Cauchy problem for the fractional diffusion-wave equation.
##### MSC:
 35S15 Boundary value problems for pseudodifferential operators 26A33 Fractional derivatives and integrals (real functions) 33C45 Orthogonal polynomials and functions of hypergeometric type 44A10 Laplace transform 45K05 Integro-partial differential equations