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

Abstract

In the paper, we investigate a local boundary value problem with transmitting condition of the integral form for mixed parabolic-hyperbolic equation with non-characteristic line of type changing. Theorem on strong solvability of the considered problem has been proved and integral representation of the solution is obtained in a functional space. Using Lidskii Theorem on coincidences of matrix and spectral traces of nuclear operator and Gaal’s formula for evaluating traces of nuclear operator, which is represented as a product of two Hilbert-Schmidt operators, we prove the existence of eigenvalues of the considered problem.

Extension of the Tricomi problem for a loaded parabolic–hyperbolic equation with a characteristic line of change of type

“…We should note recent publications [5][6][7][8][15][16][17][18][19][20][21][22][23][24] solved the most important boundary value problems for the differential equations parabolic, hyperbolic, and parabolic-hyperbolic types with fractional derivative in the sense of Riemann-Liouville and Caputo.…”

Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition

“…In this regard, great attention is paid to partial differential equations of fractional order, which are a generalization of partial differential equations of integer order. At the same time, various statements are possible [6][7][8][12][13][14][15]. The paper [6] is devoted to the study of the solvability of boundary value problems with integral conditions for a class of quasi-linear pseudo-parabolic equations.…”

“…The works [4][5][6][7][12][13][14][15][20][21][22][23][24][25][26][27][28][29] are devoted to the study of the solvability of various classes of problems for a pseudo-parabolic equation with a fractional derivative.…”