Partial Differential Equations for Cereal Seeds Distribution

Abstract

During the recent years all crop species achieved the best possible field distribution so a high yield is to be expected. In this paper the solutions of two different diffusion equations are determined, which describe the optimal distribution of cereal grains over a field. Therefore, there are two different partial differential equations of cereal seed distribution-distinction is made between the longitudinal spacing (seeds in a row), and transverse distance (between two rows), as well as the sowing depth. In particular, closed forms of solutions are derived in each case. Although the result of the diffusion equation with respect to the distribution of the lateral seed distance of two adjacent rows is already known, a new solving method is presented in this paper. By this method, the partial differential equation is reduced to an ordinary one, which is easier to solve. In this paper it is shown that the distribution of lateral resp. longitudinal and in-depth wheat seed distances is achieved by a normal Gauss function resp. a log-normal function. Furthermore, it is demonstrated that the fitting functions of the best experimental results of wheat seeding distributions are particular solutions of the individual differential equations. Normal Gauss function describes lateral distribution with R2 = 0.9325; RSME = 1.2450, and log-normal function describes longitudinal distribution with R2 = 0.9380; RSME = 1.4696 as well as depth distribution with R2 = 0.9225; RSME = 2.0187.

Keywords

• analytical solutions
• cereal seeds
• diffusion equations
• partial differential equation
• probability density function