On representation of solution of the diffusion equation with Dzhrbashyan-Nersesyan operators

The paper investigates a parabolic partial differential equation with fractional differentiation with respect to one of two independent variables associated with time. Such equations are usually referred to the class of fractional diffusion equations. The fractional differentiation operator is a linear combination of two Dzhrbashyan-Nersesyan operators. The main result of the work is a theorem on the general representation of regular solutions of the equation under study in an infinite strip. A fundamental solution is constructed in terms of the Wright function and its main properties are studied. In particular, formulas for fractional differentiation are proved, the asymptotic behavior is investigated, and estimates are obtained for the fundamental solution and its derivatives for large and small values of the self-similar variable, and its positiveness is proved. To construct a general solution, the Green's function method adapted to equations containing Dzhrbashyan-Nersesyan operators is used. Particular cases of the equation under consideration include equations with Riemann-Liouville and Gerasimov-Caputo derivatives. Therefore, the results obtained remain valid for equations with these fractional differentiation operators and their combinations.