Recovery of a space-dependent vector source in thermoelastic systems

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 [GRAPHICS] and a vectorial hyperbolic equation for the displacement [GRAPHICS] . 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 [GRAPHICS] (with [GRAPHICS] componentwise strictly monotone increasing) is present in the hyperbolic equation. Despite the ill-posed nature of these inverse problems, a stable iterative algorithm is proposed to recover the unknown source in the case that [GRAPHICS] 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

Kernel reconstruction in a semilinear parabolic problem with integral overdetermination

A semilinear parabolic problem of second order with an unknown solely time-dependent convolution kernel is considered. The missing kernel is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is addressed. We design a numerical algorithm based on Rothe’s method, derive a priori estimates and prove convergence of iterates towards the exact solution

Nonlinear diffusion problem with dynamical boundary value

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