Analysis of a causal diffusion model and its backwards diffusion problem

In Kowar (2012) [25] a diffusion model was developed and analyzed that obeys causality, i.e. the speed of propagation of the concentration is finite. In this article we analyze the respective causal backwards diffusion problem. The motivation for this paper is that because real diffusion obeys causality, a causal diffusion model may contain smaller modeling errors than the noncausal standard model and thus an increase of resolution of inverse and ill-posed problems related to diffusion is possible. We derive an analytic representation of the Green function of causal diffusion in the k − t -domain (wave vector-time domain) that enables us to analyze the properties of the causal backwards diffusion problem. In particular, it is proven that this inverse problem is ill-posed, but not exponentially ill-posed. Furthermore, a theoretical and numerical comparison between the standard diffusion model and the causal diffusion model is performed. The paper is concluded with numerical simulations of the backwards diffusion problem via the Landweber method that confirm our theoretical results.