A cauchy problem for the sub-diffusion equation with the Prabhakar fractional derivative

In this work, we have investigated the Cauchy problem for the second order partial differential equation involving the Prabhakar fractional derivative of the Riemann-Liouville type in the time variable. This equation has been solved by converting it into a system of two first order differential equations. Therefore, the boundary and Cauchy problems for the first order equations have first been solved as auxiliary problems. We have presented the explicit solution of the Cauchy problem for the second order differential equation using these auxiliary problems. Then, it has been proven that the obtained solution satisfies the problem.

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