Direct and inverse problems for Barenblatt-Zheltov-Kochina type fractional equations

In the present work, we consider a Cauchy problem for a nonhomogeneous fractional-order equation of the Barenblatt-Zheltov-Kochina type with the Caputo derivative. A solution to the problem is constructed by means of the Fourier method. The continuity of the obtained solution is justified using the convergence properties of functional series. In addition, the uniqueness of the solution is proved. Throughout the analysis, essential use is made of the properties of the Mittag-Leffler function. Moreover, the inverse problem of determining the source function is also investigated, and the existence and uniqueness of its solution are rigorously established.

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