IMA Journal of Applied Mathematics (1990) 45, 225-231 Mathematical Analysis: An Inverse Problem Arising in Convective-Diffusive Flow W. R. SMITH Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario NIG 2W1, Canada G. C. WAKE Department of Mathematics and Statistics, Massey University, Palmerston North, New Zealand [Received 11 May 1989 and in revised form 12 October 1989] A particular system constructed and analysed for the combined convective flow and diffusive movement of concentrations through a packed column gives rise to a novel inverse problem analogous to 'the sideways heat equation', which may be ill-posed. It is possible to find an explicit formula for the solute concentration at the top of the column in terms of that at the bottom of the column. This is used in the simulation of biological experiments involving hormonal stimulation of perifused cells in elution chromatography. The formula obtained for the concentration at the top of the column involves an infinite series of time derivatives of increasing order of the concentration at the bottom of the column. While this presents possible complications from a computational perspective, the practical use of this formula is possible by use of splines tofitexperimental data. Provided a priori estimates can be made on the inlet data, such as bounded Lz-norm and rapid decay on the higher frequencies of its spectrum, the problem can be stabilized. 1. Introduction IN AN attempt to determine the mechanisms governing the release of hormones from secretory cells in response to a stimulus in a medium flowing past them and through a diffusion column, the following inverse problem arises. We presume that the layer of diffusion (here one dimensional for simplicity, but the same question arises with more than one space dimension) is accompanied by forced convection (here presumed constant, again for simplicity) and so the concentration C(x, t) (of hormone) satisfies the convection-diffusion equation (see Smith et ai, 1989) f + "f =D 0 (0<x<U>0). (1) Here, x is the distance measured down the column, t the time, u the flow velocity down the column, and D (also constant) the diffusion constant. Initially we have taken, for simplicity, C(x,0) = 0 (0«x*/) (2) 225 © Oxford Univeriity Pren 1990 at University of Guelph on March 2, 2016 http://imamat.oxfordjournals.org/ Downloaded from