The recovery of a parabolic equation from measurements at a single point

Abstract

By measuring the temperature at an arbitrary single point located inside an unknown object or on its boundary, we show how we can uniquely reconstruct all the coefficients appearing in a general parabolic equation which models its cooling. We also reconstruct the shape of the object. The proof hinges on the fact that we can detect infinitely many eigenfunctions whose Wronskian does not vanish. This allows us to evaluate these coefficients by solving a simple linear algebraic system. The geometry of the domain and its boundary are found by reconstructing the first eigenfunction.

Reconstructing a fractional integro‐differential equation

“…Local existence results in the case of Dirichlet boundary conditions are known; however, the existence results presented here are new for the Neumann and Robin conditions and do not rely on semigroup techniques 7 . As for the inverse problem, we use new reconstruction methods from single point observations found in other works 8‐11 . Taking into consideration the nature and wide applications of fractional integro‐differential equations in engineering, we shall make use of simple computational tools derived from integral transforms that are familiar to the engineering community, so to make the reconstruction of the sought parameters computationally possible in practice.…”

“…7 As for the inverse problem, we use new reconstruction methods from single point observations found in other works. [8][9][10][11] Taking into consideration the nature and wide applications of fractional integro-differential equations in engineering, we shall make use of simple computational tools derived from integral transforms that are familiar to the engineering community, so to make the reconstruction of the sought parameters computationally possible in practice.…”