Recovering Heat Source from Fourth-Order Inverse Problems by Weighted Gradient Collocation

The weighted gradient reproducing kernel collocation method is introduced to recover the heat source described by Poisson’s equation. As it is commonly known that there is no unique solution to the inverse heat source problem, the weak solution based on a priori assumptions is considered herein. In view of the fourth-order partial differential equation (PDE) in the mathematical model, the high-order gradient reproducing kernel approximation is introduced to efficiently untangle the problem without calculating the high-order derivatives of reproducing kernel shape functions. The weights of the weighted collocation method for high-order inverse analysis are first determined. In the benchmark analysis, the unclear illustration in the literature is clarified, and the correct interpretation of numerical results is given particularly. Two mathematical formulations with four examples are provided to demonstrate the viability of the method, including the extreme cases of the limited accessible boundary.

Inverse heat source problem for a coupled hyperbolic-parabolic system

International audience Dans ce papier, on a prouvé une estimation de stabilité de type Höldérienne pour un problème inverse de détermination du terme source de l'équation de la chaleur à l'aide d'une inégalité de Carleman pour un système d'équations hyperbolique-parabolique couplé. ABSTRACT. In this paper we consider a coupled system of mixed hyperbolic-parabolic type which describes the Biot consolidation model in poro-elasticity. Using a local Carleman estimate for a coupled hyperbolic-parabolic system, we prove the uniqueness and a Hölder stability in determining the heat source by a single measurement of solution over ω × (0, T), where T > 0 is a sufficiently large time and a suitable subbdomain ω ⊂ Ω such that ∂ω ⊃ ∂Ω. MOTS-CLÉS : Problème inverse, estimation de Carleman, système couplet