(untitled) (Journal of Nonlinear Modeling and Analysis, 2025)
The nitrogen uptake model for plant roots is an advection-diffusion equation 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 function of 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 and Dorroh, the nitrogen uptake model is homogenized and its eigenvalues are obtained from the Sturm-Liouville problem. Because the convergence of this eigenfunction expansion method is slow around the left boundary, i.e., root surface, we additionally consider the Laplace transform to solve the nitrogen uptake model. However, the solution after Laplace transform involves composite functions and numerical inverse Laplace transforms are introduced to obtain the final solutions. The analytical and numerical solutions show that the nitrogen concentration profiles along the distance from the root surface are convex upward and almost horizontal in the middle part with large gradients at both ends. The numerical simulation demonstrates that the eigenfunction expansion method can reach a satisfactory accuracy and the Laplace transform method with Stehfest inversion has higher calculation efficiency.