On the Volterra property of a boundary problem with integral gluing condition for a mixed parabolic-hyperbolic equation

Abstract

In the present work, we consider a boundary value problem with gluing conditions of an integral form for the parabolic-hyperbolic type equation. We prove that the considered problem has the Volterra property. The main tools used in the work are related to the method of the integral equations and functional analysis.

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

“…We multiply equation u xx − C D λ 0t u = 0 to the function u(x, t) and integrate along the domain Ω 0 . After the usage of the Green's formulas and considering designations (20), we obtain…”

“…Now we prove that τ + 1 (x) ≡ 0 at f (x, t) ≡ 0. For this aim in the equation u xx − C D λ 0t u = 0 we pass to the limit as t → +0 and considering designation (20)…”

On the Existence of Eigenvalues of a Boundary Value Problem with Transmitting Condition of the Integral Form for a Parabolic-Hyperbolic Equation

“…In the work [1], an analog problem to the generalized Tricomi problem, with integral gluing conditions for mixed parabolic-hyperbolic equation, was studied. Theorems on strong solvability and on the absence of eigenvalues were proved.…”

“…Theorems on strong solvability and on the absence of eigenvalues were proved. In [1] one can find historical information and notation on main scientific results on the related field.…”

“…In general, spectral theory for elliptic type equations is well-developed, while similar theory for the wave type and mixed type equations is still under development. Strong solvability and the Volterra property (the absence of eigenvalues) for the analog of the Tricomi problem for mixed parabolic-hyperbolic type equation with non-characteristic line of type-changing (in the domain with the deviation from characteristics) was studied in [1]. The existence of eigenvalues for a class of local boundary problems for the second and third order parabolic-hyperbolic type equations were investigated in [7,9].…”