The loaded parabolic-hyperbolic equation and its relation to non-local problems

Abstract

In this present paper, unique solvability is proved for the boundary value problems for the loaded differential equations associated with non-local boundary value problems, for the classical partial differential equations.

Analog of the Darboux problem for a loaded integro-differential equation involving the Caputo fractional derivative

“…[4], Zikirov O. S. and Kholikov D. K. [5], Kh. Belakroum, A. Ashyralyev, A. Guezane-Lakoud [6], Khashimov A. R, Smetanová D. [7], as well as in [8,9].…”

“…We can cite by many examples as which emphasized the development of the theory of the loaded equation in the last several decades. As we know [1], mathematical models of nonlocal physics-biological fractal processes represented by loaded integro-differential equations or loaded differential equations especially with integro-differential operators [1,9].…”

“…Based on the above, the main aim of this paper is to formulate and investigate the unique solvability of a boundaryvalue problem type of Darboux (Cauchy-Goursat) for a third-order loaded partial differential equation of hyperbolic type, with the Caputo operators. For investigating this problem, we reduce the boundary-value problem for the loaded equation to a new non-local problem for the fractional order integro-differential equation [8,9]. Taking into account the unique solvability of non-local problems, we prove the solvability of the local boundary-value problem for the loaded equation.…”

Analogues of the Cauchy–Goursat problem for a loaded third-order hyperbolic type equation in an infinite three-dimensional domain

“…Local and nonlocal problems for second-order partial dierential equations of hyperbolic and mixed types with loaded terms in special domains were considered in [18]- [22]. In the study of these problems, the problem of studying integral and integro-dierential equations arises.…”

Analog of the Gellerstedt problem for a loaded equation of the third order

“…The field of partial differential equations of fractional order is experiencing a sustained growing interest in the recent years, see for instance the previous studies. () From the available bibliography, it is observed that the boundary‐value problems for a third‐order loaded differential equations containing mixed parabolic‐hyperbolic and elliptic‐parabolic operators in the principal part have been scarcely investigated . On the other hand, it is well known that in connection with the important problems of partial differential equations of mixed type and gas dynamics of transonic currents, the construction of mathematical models describing various processes developed in media with a fractal structure leads to differential equations of fractional order.…”