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An analog of the Tricomi problem for a mixed type equation with a partial fractional derivative. (English) Zbl 1201.35148

An analog of Tricomi boundary value problem is for a special partial differential equation of mixed type is under consideration. It involves a diffusion equation with the Riemann-Liouville partial fractional derivative and a hyperbolic equation with two degenerate lines.

Uniqueness and existence of a solution of the considered problem are proved by using the properties of the Gauss hypergeometric function and of the generalized fractional integrals and derivatives with such a function in the kernel. Its explicit solution is established in terms of the new special function.

MSC:
 35M10 PDE of mixed type 35R11 Fractional partial differential equations 26A33 Fractional derivatives and integrals (real functions) 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$ 33E12 Mittag-Leffler functions and generalizations 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$