Analytical Method of Nitrogen Uptake Model for Plant Roots (Journal of Nonlinear Modeling and Analysis, 2025)
The nitrogen uptake model for plant roots is an advection-diffusionequation subject to double Robin boundary conditions in Cartesian coordinates and its analytical method is expected to accurately estimate the quantity of nutrient uptake and fertilization. Firstly, the Michaelis-Menten (MM)kinetics function in the left boundary condition is changed into a functionof time by numerical fitting and the nonlinear left Robin boundary condition then becomes a linear one in order to use traditional analytical methods.Based on the eigenfunction expansion method originally built by Golz andDorroh, the nitrogen uptake model is homogenized and its eigenvalues areobtained from the Sturm-Liouville problem. Because the convergence of thiseigenfunction expansion method is slow around the left boundary, i.e., rootsurface, we additionally consider the Laplace transform to solve the nitrogenuptake model. However, the solution after Laplace transform involves composite functions and numerical inverse Laplace transforms are introduced toobtain the final solutions. The analytical and numerical solutions show thatthe nitrogen concentration profiles along the distance from the root surface areconvex upward and almost horizontal in the middle part with large gradientsat both ends. The numerical simulation demonstrates that the eigenfunctionexpansion method can reach a satisfactory accuracy and the Laplace transformmethod with Stehfest inversion has higher calculation efficiency.
Keywords: Nitrogen uptake; advection-diffusion equation; Robin boundaries; Golz and Dorroh’s method; numerical inverse Laplace transform.