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

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

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 problemThe research of the first author is supported by the grant of the Committee of Sciences, Ministry of Education and Science of the Republic of Kazakhstan to the Institute of Information and Computational Technologies, project AP05131026. Second author is supported by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDER and by Xunta de Galicia, project ED431C 2019/02 (Spain)S