In this paper, we give an integral representation of the solution of the Dirichlet problem for the Laplace and Helmholtz equations in the form of a generalized convolution.

In this paper, we construct a lateral boundary condition for an elliptic-parabolic equation in a finite domain. Theorem on existence and uniqueness of a solution of the considered problem is proved by method of theory potential.

In this paper, the solution to the Dirichlet problem for the wave equation on the star graph is constructed. To begin, we solve the boundary value problem on the interval (on one edge of the graph). We use the generalized functions method to obtain the wave equation with a singular right-hand side. The solution to the Dirichlet problem is determined through the convolution of the fundamental solution with the singular right-hand side of the wave equation. Thus, the solution found on the interval is determined by the initial functions, boundary functions, and their derivatives (the unknown boundary functions). A resolving system of two linear algebraic equations in the space of the Fourier transform in time is constructed to determine the unknown boundary functions. Following inverse Fourier transforms, the solution to the Dirichlet problem of the wave equation on the interval is constructed. After determining all the solutions on all edges and taking the continuity condition and Kirchhoff joint condition into account, we obtain the solution to the wave equation on the star graph.

In this paper we consider a model case of the problem of heat (or mass) diffusion in a homogeneous body with a special initial state. The peculiarity of this initial state is its local inhomogeneity. That is, there is a closed domain Ω inside a body such that the initial state is constant out of the domain. Mathematical modeling leads to the problem for a homogeneous multi-dimensional diffusion equation. We construct the boundary conditions on the boundary of the domain Ω, which can be characterized as ”transparent” boundary conditions. We separately consider a special case, that is, a model of redistribution of heat (or mass) in a uniform linear rod, the side surface of which is insulated in the absence of (internal and external) sources of heat (or mass) and of locally inhomogeneous initial state.

A spectral decomposition of the Green’s function of the Holmgren problem in a cylindrical domain is used to obtain Bitsadze–Samarskii boundary conditions for a regular elliptic-parabolic volume...