On a mixed problem with integral conditions for third-order equation

The solvability of mixed problems with integral conditions for third-order partial differential equations is an important area of research in the theory of differential equations and their applications. Such problems arise in various fields of physics, continuum mechanics, oscillation theory, and other processes. In this paper, we consider a non-local problem with an integral condition for a third-order partial differential equation with the thermal conductivity operator in the main part. The theorems on the existence and uniqueness of the solution of the studied non-local problem are proved here. Methods of the theory of differential equations, the Green function and the theory of integral equations are used to the prove the solvability of the problem. The problem under study is reduced to an equivalent second kind Volterra integral equation which is certainly solvable.