A boundary problem with integral gluing condition for a parabolic-hyperbolic equation involving the Caputo fractional derivative

In the present work we investigate the Tricomi problem with integral gluing condition for parabolic-hyperbolic equation with the Caputo fractional order derivative. Using the method of energy integrals we prove the uniqueness of the solution for considered problem. The existence will be proved using methods of ordinary differential equations, Fredholm integral equations and solution will be represented in an explicit form.Comment: 7 page

Inverse source problems for degenerate time-fractional PDE

In this paper, we investigate two inverse source problems for degenerate time-fractional partial differential equation in rectangular domains. The first problem involves a space-degenerate partial differential equation and the second one involves a time-degenerate partial differential equation. Solutions to both problem are expressed in series expansions. For the first problem, we obtained solutions in the form of Fourier-Legendre series. Convergence and uniqueness of solutions have been discussed. Solutions to the second problem are expressed in the form of Fourier-Sine series and they involve a generalized Mittag- Leffler type function. Moreover, we have established a new estimate for this generalized Mittag-Leffler type function. The obtained results are illustrated by providing example solutions using certain given data at the initial and final time.Comment: 12 pages, 8 figure

Non-local boundary value problem for a mixed-type equation involving the bi-ordinal Hilfer fractional differential operators

In this paper, we consider a nonlocal boundary-value problem for a mixed-type equation involving the bi-ordinal Hilfer fractional derivative in a rectangular domain. The main target of this work is to analyze the uniqueness and the existence of the solution of the considered problem by means of eigenfunctions. Moreover, we construct the solution of the ordinary fractional differential equation with the right-sided bi-ordinal Hilfer derivative by the method of reduction to the Volterra integral equation. Then, we present sufficient conditions for given data in order to show the existence of the solution.Comment: 13 page

Initial, inner and inner-boundary problems for a fractional differential equation

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann-Liouville derivatives is less understood. In this paper we propose new type initial, inner and inner-boundary value problems for fractional differential equations with the Riemann-Liouville derivatives. The results on the existence and uniqueness are proved, and conditions on the solvability are found. The well-posedness of the new type initial, inner and inner-boundary conditions are also discussed. Moreover, we give explicit formulas for the solutions. As an application fractional partial differential equations for general positive operators are studied.Comment: 18 pages