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We consider one class of direct and inverse stationary diffusion problems describing by the Poisson equation. The problems is considered in a model domain, chosen as a half disk. Dirichlet classical boundary conditions are set on the arc of the circle. New nonlocal boundary conditions are set on the bottom base. The first condition means an equality of flows through opposite radii, and the second condition is the proportionality of distribution densities on these radii with a variable coefficient of proportionality. An inverse problem on the solution definition to the Poisson equation and its right-hand part depending only on an angular variable are considered. The considered direct and inverse problems are well-posed. [ABSTRACT FROM AUTHOR]

We consider one class of inverse problems for a one-dimensional heat equation with involution and with periodic type boundary conditions with respect to a space variable. This problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. The inverse problem consists in the restoration (simultaneously with the solution) of the unknown right-hand side of the equation, which depends only on the spatial variable. The conditions for overdetermination are initial and final states. Problems for a classical heat equation, for an equation with fractional derivatives with respect to a time variable, and for a degenerate equation are considered. Existence and uniqueness results for the given problem are obtained via the method of separation of variables. [ABSTRACT FROM AUTHOR]

We consider a class of problems modeling the process of determining the temperature and density of sources in the process of fractional heat conductivity with respect to given initial and finite temperatures. Their mathematical statement leads to inverse problems for a fractional heat equation in which along with solving the equation we have to find an unknown right-hand side depending only on a spatial variable. The fractional Riemann-Liouville derivative is used. Problems with general boundary conditions with respect to a spatial variable that are not strongly regular are investigated. The problem is considered independently of whether a corresponding spectral problem for an operator of multiple differentiation with not strongly regular boundary conditions has the basis property of root functions. The existence and uniqueness of classical solutions to the problem are proved. [ABSTRACT FROM AUTHOR]

We consider the problem of modeling the process of determining the distribution function of temperature and time varying structure of homogeneous bar of a given law changes in the average temperature. So there is an inverse problem for the heat equation in which together with finding the solution of the equation it is required to find unknown coefficient depending only on the time variable. The specific features of the considered problems is that the system of eigenfunctions of the multiple differentiation operator subject to boundary conditions of the initial problem does not have the basis property. In this paper, it is proved that this inverse problem has a unique generalized solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider one family of problems simulating the determination of target components and density of sources from given values of the initial and final states. The mathematical statement of these problems leads to the inverse problem for the diffusion equation, where it is required to find not only a solution of the problem, but also its right-hand side that depends only on a spatial variable. One of specific features of the considered problems is that the system of eigenfunctions of the multiple differentiation operator subject to boundary conditions of the initial problem does not have the basis property. The other specific feature of the considered problems is that an unknown function is simultaneously present both in the right- hand side of the equation and in conditions of the initial and final redefinition. We prove the unique existence of a generalized solution to the mentioned problem. [ABSTRACT FROM AUTHOR]

In the current paper, the problem of bending vibrations of a beam in which the binding on the right end is unknown and not available for visual inspection is studied. The main objective is to study an inverse problem: find additional unknown boundary conditions by additional spectral data, i.e., the conditions of fixing the right end of the rod. In this work, unlike many other works, as such additional conditions we choose the first natural frequencies (eigenvalues) of two new problems corresponding to the problem of bending vibrations of a beam with loads of different weights at the central point. [ABSTRACT FROM AUTHOR]