Analytical solution of Nye-Tinker-Barber model by Laplace transform

Abstract

The Nye-Tinker-Barber model is a classical convection-diffusion model for nutrient uptake by plant roots in cylindrical coordinates and has one nonlinear left Robin boundary condition with Michaelis-Menten function of concentration. First the Michaelis-Menten function is fitted into a function of time by numerical concentration at root surface from difference scheme, and then the Laplace and numerical inverse Laplace transforms - Zakian inversion method are taken to obtain the approximate analytical solution. Compared with other solutions made by difference scheme, Stehfest inversion method and previous analytical methods, it is found that the analytical solution obtained by Laplace and Zakian inversion transforms has higher accuracy and computation efficiency. This analytical method can be extended to other nutrient uptake models with Michaelis-Menten function.

Keywords: Convection–diffusion equation; Laplace transform; Numerical inverse Laplace transform; Semi-infinite domain.