Analytical solution of Nye–Tinker–Barber model by Laplace transform
Introduction
Crops cannot grow without nutrients, most of which come from fertilizers. Excess of fertilizer will result into environmental pollution and ecosystem degradation, and will lead to quality and quantity decreasing of crops (Mohammadipour and Souri, 2019, Souri, 2016, Mikula et al., 2020). For example, the surplus nitrogen element in fertilizer will lead to soil nitrate leaching, pollution of surface water and groundwater, and degradation of soil structure (Dimkpa et al., 2020, Poudel et al., 2002, Larbi et al., 2020). Therefore, it is an important issue for agroforestry to understand the absorption rate of nutrient concentrations by plants and increase the nutrient use efficiency of fertilizers and water resource.
The nutrient uptake model for plant roots belongs to convection–diffusion problem (CDP) in porous media. Assuming that a single cylindrical root is surrounded by an infinite range of soil and the solute concentration away from the root is given, Nye and Tinker (1977) and Barber (1995) proposed a classical convection–diffusion model (CDM) with Dirichlet right and Robin left boundary conditions of Michaelis–Menten (MM) flux in a semi-infinite domain, which we call the Nye–Tinker–Barber model (NTB model) (Ou, 2019).
There are already a lot of work on CDM in semi-infinite or finite domain under rectangular coordinates or cylindrical coordinates with traditional integral transformation, separation of variable, homotopy analysis, the Duhamel’s theorem, etc. (Guerrero et al., 2009, Kumar et al., 2010, Das and Mritunjay, 2019, Dilip Kumar et al., 2011, Yadav et al., 2011, Massabó et al., 2006, Chen et al., 2011, Ateş and Zegeling, 2017, Ahmed et al., 2020). However most of them focus on the first or second type of boundary conditions and it is difficult to apply them to NTB model. There is one approximate analytical method for solving NTB model: first obtain the outer and inner solutions, match them in uniform scale, and finally determine the global solution of concentration and uptake flux (Ou, 2019, Roose et al., 2001, Nowack et al., 2006, Roose and Kirk, 2009). But this method has complicated formulae and calculation, and is quite difficult to implement (Ou, 2019). In this paper, we try to solve NTB model with conventional technique – Laplace transform and original technique – turning MM function of concentration into a function of time.
We give the solution of concentration obtained by Laplace transform in Section 2. In Section 3, we compare the solution by Laplace transform with numerical solution and analytical solutions derived from other methods. The discussion about the MM function of time and conclusion are in Sections 4 Discussion, 5 Conclusion.
Section snippets
Approximate analytical solution
Nye and Tinker (1977) and Barber (1995) developed a CDM in semi infinite soil cylindrical coordinates, in which the left boundary is Robin boundary condition with MM kinetics function and the right boundary is Dirichlet boundary condition at infinity. The dimensionless NTB model is (Roose et al., 2001) where is the dimensionless concentration, is the polar radius, is the time, is the Péclet number, is the uptake
Numerical simulation
In order to verify solution (14) of model (1), (3), (4), (5), we need to compare it with the solution of original model (1)–(4). Model (1)–(4) is a classical CDM, its difference scheme and solution are highly accurate and can be taken as precision benchmark.
We adopt the first-order forward difference in time, the first-order forward difference for the first derivative of space, and the second-order central difference for the second derivative. is the value of at the grid points
Discussion
In order to apply the Laplace transform to NTB model (1), (3), (4), (5), we have to turn the MM function of concentration into the one of time and thus linearize the boundary condition and the model. We will briefly discuss the reasonability of this treatment in experimental field.
The nutrient flux of plant roots is usually expressed by the nutrient consumption rate in solution. The nutrient uptake of plants is usually described by MM kinetics (Caperon and Meyer, 1972) where is the
Conclusion
In this paper, we have proposed a method based on Laplace and numerical inverse Laplace transforms to solve the classical NTB model with MM flux. This method consists of turning the MM function of concentration into the one of time, Laplace transform and numerical inverse Laplace method. We have compared the deviation degree and computation efficiency among the numerical solution and the analytical solutions derived by Laplace transform, the Zakian inversion method, the Stehfest inversion
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by on Constrained Critical Points [grant number 11671085]; Researches on Non-linear Wave Problems for Multi-Scale Evolution Systems [grant number 11771082]; Researches on several bifurcation problems in singular perturbation systems [grant number 12271096]; Fujian Provincial Key Laboratory of Mathematical Analysis and its Applications [grant number ZCD1707233]; and Center for Applied Mathematics of Fujian Province(FJNU) [grant number ZGD200872301]; Blow-up dynamics of
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