Boundary Value Problems with Integral Gluing Conditions for Fractional‐Order Mixed‐Type Equation

Abstract

Analogs of the Tricomi and the Gellerstedt problems with integral gluing conditions for mixed parabolic-hyperbolic equation with parameter have been considered. The considered mixed-type equation consists of fractional diffusion and telegraph equation. The Tricomi problem is equivalently reduced to the second-kind Volterra integral equation, which is uniquely solvable. The uniqueness of the Gellerstedt problem is proven by energy integrals' method and the existence by reducing it to the ordinary differential equations. The method of Green functions and properties of integral-differential operators have been used.

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

“…In the present work we use integral gluing condition with kernel, which has a more general form than the kernel used in [11]. When we prove the uniqueness of the solution we must put some restrictions to the kernel (see theorem 1), but for the existence of solution we don't need these conditions (see theorem 2).…”

“…We substitute (11) into the integral I and consider τ 1 (0) = 0, τ 1 (1) = 0 (which are deduced from conditions (2), (3) in the homogeneous case), we have…”

Solvability of a Non-local Problem with Integral Transmitting Condition for Mixed Type Equation with Caputo Fractional Derivative

“…We as well would like to note results on local and non-local problems for parabolic-hyperbolic type equations with fractional order derivatives. Precisely, in [8] the Tricomi and Gellerstedt problem for parabolic-hyperbolic equation with the Riemann-Liouvill fractional operator in the hyperbolic part were under discussion and unique solvability of these problems were proved. In [9] authors consider the same equation, but with two lines of type-changing in a domain with deviation from the characteristics.…”

Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations

“…Omitting many papers on direct boundary problems for PDEs involving fractional differential operators, we note some works [1][2][3], where time-fractional parabolichyperbolic type equations were investigated. Precisely, main boundary problems for mixed parabolic-hyperbolic equations with the Riemann-Liouville fractional differential operator in parabolic part, were objects of investigations.…”

“…There exist many types of gluing conditions such as continuous, discontinuous, integral form and etc. For instance, in the works [2,3] gluing conditions of integral form were in use, but in the work [5] authors consider boundary problems with continuous gluing conditions, i.e values of seeking function and its derivative from the both parabolic and hyperbolic parts of mixed domain are equal on the line of type changing. Depending on which gluing conditions are used, solvability conditions to given data vary.…”