Abstract
This paper considers the inverse problem of recovering state-dependent source terms in a reaction–diffusion system from overposed data consisting of the values of the state variables either at a fixed finite time (census-type data) or a time trace of their values at a fixed point on the boundary of the spatial domain. We show both uniqueness results and the convergence of an iteration scheme designed to recover these sources. This leads to a reconstructive method and we shall demonstrate its effectiveness by several illustrative examples.
Highlights
Reaction diffusion equations have a rich history in the building of mathematical models for physical processes
They are descendants of nonlinear ordinary differential equations in time with an added spatial component making for a partial differential equation of parabolic type
As a matter of fact, in [16], we considered a scalar problem with only one unknown function f but the setting there is slighty more general than the one described in the introduction in the sense that a subdiffusion equation is considered
Summary
Reaction diffusion equations have a rich history in the building of mathematical models for physical processes They are descendants of nonlinear ordinary differential equations in time with an added spatial component making for a partial differential equation of parabolic type. The time trace data involves monitoring the population (or of chemical concentrations) at a fixed spatial point as a function of time. Both of these data measurements are quite standard in applications. In the case of a single equation using time trace data, uniqueness results and the convergence of reconstruction algorithms were shown in [4, 27, 25] for the recovery of the unknown term f (u). Both settings of final time data (3) and of time trace data (4) are considered
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