On the identification of a nonlinear term in a reaction–diffusion equation

Abstract

Abstract Reaction–diffusion equations are one of the most common partial differential equations used to model physical phenomena. They arise as the combination of two physical processes: a driving force that depends on the state variable u and a diffusive mechanism that spreads this effect over a spatial domain. The canonical form is . Application areas include chemical processes, heat flow models and population dynamics. As part of model building, assumptions are made about the form of , and these inevitably contain various physical constants. The direct or forward problem for such equations is now very well developed and understood, especially when the diffusive mechanism is governed by Brownian motion resulting in an equation of parabolic type. However, our interest lies in the inverse problem of recovering the reaction term not just at the level of determining a few parameters in a known functional form, but in recovering the complete functional form itself. To achieve this we set up the standard paradigm for the parabolic equation where u is subject to both given initial and boundary data; then we prescribe overposed data consisting of the solution at a later time T. For example, in the case of a population model this amounts to census data at a fixed time. Our approach will be two-fold. First we will transform the inverse problem into an equivalent nonlinear mapping from which we seek a fixed point. We will be able to prove important features of this map such as a self-mapping property and give conditions under which it is contractive. Second, we consider the direct map from f  through the partial differential operator to the overposed data. We will investigate Newton schemes for this case. Classical, Brownian motion diffusion is not the only version, and in recent decades various anomalous processes have been used to generalize this case. Amongst the most popular is one that replaces the standard time derivative by a subdiffusion process based on a fractional derivative of order . We will also include this model in our analysis. The final section of the paper will show numerical reconstructions that demonstrate the viability of the suggested approaches. This will also include the dependence of the inverse problem on both T and .