On Boundary Value Problems for a Mixed Type Fractional Differential Equation with Caputo Operator

Abstract

This article is devoted to study the boundary value problems of the first and second kind with respect to the spatial variable for a mixed inhomogeneous differential equation of parabolic-hyperbolic type with a fractional Caputo operator in a rectangular domain. In the study of such boundary value problems, we abandoned the boundary value condition with respect to the first argument and instead it is used additional gluing condition. In this case, in the justification of the unique solvability of the problems, the conditions on the boundary domain are removed. This allowed us to weaken the criterion for the unique solvability of boundary value problems under consideration. The solution is constructed in the form of Fourier series with eigenfunctions corresponding to homogeneous spectral problems. Estimates for the convergence of Fourier series are obtained as a regular solution of this mixed equation.

On a Nonlocal Problem for Mixed-Type Equation with Partial Riemann-Liouville Fractional Derivative

“…The Gellerstedt-type problem, with nonlocal boundary and integral gluing conditions for the parabolic-hyperbolic-type equation, with nonlinear terms and Gerasimov-Caputo operator of differentiation, was studied in [8]. The work [9] is devoted to study the boundary value problems for a mixed type fractional differential equation with Caputo operator. A nonlocal boundary value problem for weak nonlinear partial differential equations of mixed type, with a fractional Hilfer operator, was solved in [10].…”