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a b s t r a c tIn this contribution, the reconstruction of a solely time-dependent convolution kernel is studied in an inverse problem arising in the theory of heat conduction for materials with memory. The missing kernel is recovered from a measurement of the average of temperature. The existence, uniqueness and regularity of a weak solution is addressed. More specific, a new numerical algorithm based on Rothe's method is designed. The convergence of iterates to the exact solution is shown.

In this contribution, we investigate an initial-boundary value problem for a fractional diffusion equation of distributed order where the coefficients of the elliptic operator are dependent on spatial and time variables. We consider a homogeneous Dirichlet boundary condition. Using a classical variational approach, we establish the existence of a unique weak solution to the problem in C [0, T ], H 1 0 (Ω) * ∩ L ∞ (0, T ), H 1 0 (Ω) if the initial data belongs to H 1 0 (Ω). The same result is also valid for the fractional diffusion equation with Caputo derivative.

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.

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