Identification of unknown diffusion and convection coefficients in ion transport problems from flux data: an analytical approach

Abstract

This article presents an analytical approach for identification problems related to ion transport problems. In the first part of the study, relationship between the flux \({\varphi_L := (D(x)u_x(0, t)_{x=0}}\) and the current response \({{\mathcal I}(t)}\) is analyzed for various models. It is shown that in pure diffusive linear model case the flux is proportional to the classical Cottrelian \({{\mathcal I}_C(t)}\). Similar relationship is derived in the case of nonlinear model including diffusion and migration. These results suggest acceptability of the flux data as a measured output data in ion transport problems, instead of nonlocal additional condition in the form an integral of concentration function. In pure diffusive and diffusive-convective linear models cases, explicit analytical formulas between inputs (diffusion or/and convection coefficients) and output (measured flux data) are derived. The proposed analytical approach permits one to determine the unknown diffusion coefficient from a single flux data given at a fixed time t1 > 0, and unknown convection coefficient from a single flux data given at a fixed time t2 > t1 > 0. Linearized model of the nonlinear ion transport problem with variable diffusion and convection coefficients is analyzed. It is shown that the measured output (flux) data can not be given arbitrarily.