The fundamental solution of a diffusion-wave equation of fractional order

We construct a fundamental solution of a diffusion-wave equation with Dzhrbashyan-Nersesyan fractional differentiation operator with respect to the time variable. We prove reduction formulae and solve the problem of sign-determinacy for the fundamental solution. A general representation for solutions is constructed. We give a solution of the Cauchy problem and prove the uniqueness theorem in the class of functions satisfying an analogue of Tychonoff's condition. It is shown that our fundamental solution yields the corresponding solutions for the diffusion and wave equations when the order of the fractional derivative is equal to 1 or tends to 2. The corresponding results for equations with Riemann-Liouville and Caputo derivatives are obtained as particular cases of our assertions.

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