Recovery of a space-dependent vector source in thermoelastic systems

Abstract

In this contribution, an inverse problem of determining a space-dependent vector source in a thermoelastic system of type-I, type-II and type-III is studied using information from a supplementary measurement at a fixed time. These thermoelastic systems consist of two equations that are coupled: a parabolic equation for the temperature θ and a vectorial hyperbolic equation for the displacement u. In this latter one, the source is unknown, but solely space-dependent. A spacewise dependent additional measurement at the final time ensures that the inverse problem corresponding with each type of thermoelasticity has a unique solution when a damping term g (∂tu) (with g componentwise strictly monotone increasing) is present in the hyperbolic equation. Despite of the ill-posed of these inverse problems, a stable iterative algorithm is proposed to recover the unknown source in the case that g is also linear. This method is based on a sequence of well-posed direct problems, which are numerically solved at each iteration step by step using the finite element method. The instability of the inverse source problem is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results support the theoretically obtained results.

Identification of an unknown spatial load distribution in a vibrating beam or plate from the final state

“…The obtained results are in accordance with the numerical experiments performed for the heat conduction equation in [6,18] and for thermoelasticity in [35,36]:…”

“…Moreover, a Landweber-Fridman type iterative regularization method is developed to obtain an approximation of the unknown load source. This approach is already successfully applied for the heat conduction equation [6,18] and for thermoelastic systems [35,36]. Note that the coefficients in problem (2.1) are also time-dependent, which is not the case in the other papers referenced before.…”

“…This all gives rise to the procedure presented below for the reconstruction of the solution u and the source term f in problem (1.1)-(1.2). The procedure is similar to the one presented in [6,17,18,36] and reads as follows: (vi) Suppose that the algorithm is stopped afterk iterations. Denote the corresponding solution by vk, fk .…”

A parabolic inverse source problem with a dynamical boundary condition

“…solution at the final time), cf. [11,[19][20][21][22][23][24][25][26][27]. For solely time-dependent source a supplementary time-dependent measurement is needed, cf.…”

An inverse backward problem for degenerate parabolic equations

“…In particular, there are many researchers devoted to the study of inverse coefficient problems for nondegenerate parabolic equations and obtained many fruitful results, such as the existence, uniqueness, stability, numerical reconstruction, and so forth (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). In [22], an inverse problem of reconstructing a space-dependent vector source in a thermoelastic system of three types is considered, where the additional information is measured at a fixed time. The uniqueness of solution is proved and a well-posed algorithm of the Landweber-type is designed to obtain numerical solutions.…”