Relationship between current response and time in ion transport problem including diffusion and convection. 1. An analytical approach

The mathematical models of the ion transport problem in a potential field are anayzed. Ion transport is regarded as the superposition of diffusion and convection. In the case of pure diffusion model the classical Gottrell’s result is studied for a constant as well as for the time dependent Dirichlet data at the electrode. Comparative analysis of the current response \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal I}_{\rm D}={\mathcal I}_{\rm D}(t)$$\end{document} and the classical Gottrellian \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal I}_{\rm G}={\mathcal I}_{\rm G}(t)$$\end{document} is given on the obtained explicit formulas. The approach is extended to find out the current response \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal I}_c={\mathcal I}_c(t)$$\end{document} corresponding to the diffusion-convection model. The relationship between the current response \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal I}_c={\mathcal I}_c(t)$$\end{document} and Gottrellian \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal I}_{\rm G}={\mathcal I}_{\rm G}(t)$$\end{document} is obtained in explicit form. This relationship permits one to compare pure diffusion and diffusion-convection models, including asymptotic behaviour of current response and an influence of the convection coefficient. The theoretical result are illustrated by numerical examples.