A hybrid algorithm for solving inverse boundary problems with respect to intermediate masses on a beam

The inverse problem of determining the weight of three intermediate masses on a uniform beam from the known three natural frequencies has been solved. The performed numerical analysis allows restoring the value of only the second mass in a unique way. The inverse problem of determining the weight of three intermediate masses is solved uniquely except in the case when the first and the third masses are located geometrically symmetric relative to the middle of the beam. The hybrid algorithm for the unique solving inverse problem of determining the weight of three intermediate masses has been developed. The first three natural frequencies of the beam are calculated numerically by using the Maple computer package. Analytical relations between the masses are found.

Inverse problem of determining the coefficient and kernel in an integro - differential equation of parabolic type

This article is concerned with the study of the unique solvability of inverse boundary value problem for integro-differential heat equation. To study the solvability of the inverse problem, we first reduce the considered problem to an auxiliary system with trivial data and prove its equivalence (in a certain sense) to the original problem. Then using the Banach fixed point principle, the existence and uniqueness of a solution to this system is shown.

A New Approach to Solve Inverse Boundary Design of a Radiative Enclosure with Specular-Diffuse Surfaces

The maintenance of uniform temperature distribution affects the efficiency in the most industrial applications. In the current study, a novel strategy has been developed for inverse radiative boundary design problems in radiant enclosures. This study presents the Backward Monte Carlo method to investigate the inverse boundary design of an enclosure composed of specular and diffuse surfaces. A new optimized Monte Carlo method is proposed to determine the temperature distribution of heaters to achieve desirable prescribed uniform heat flux on the design surfaces. The proposed approach is highly efficient and simple to implement with appropriate results. The evaluated heat fluxes on design surfaces and temperature distribution of heaters are compared with the case where the reradiating walls are assumed to be perfectly diffuse. In the proposed approach, for a specific range of specularity, the absorptivity of the reradiating surfaces does not affect the temperature distribution of heaters. Compared to the diffuse walls, the specular walls have more uniform temperature distribution and heat flux of heaters. This finding will provide insight into solar furnaces design to enhance temperature uniformity, making specular surfaces suitable in many industrial applications.

Recovery of unknown coefficients in a two-dimensional hyperbolic equation with additional conditions

In this paper, a nonlocal inverse boundary value problem for a two-dimensional hyperbolic equation with overdetermination conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence (in a certain sense) to the original problem. Then using the Fourier method, solving an equivalent problem is reduced to solving a system of integral equations and by the contraction mappings principle the existence and uniqueness theorem for auxiliary problem is proved. Further, on the basis of the equivalency of these problems the uniquely existence theorem for the classical solution of the considered inverse problem is proved and some considerations on the numerical solution for this inverse problem are presented with the examples.