Numerical Integration Approach for Nonlinear Differential Equation in Growth Modelling

The nonlinear ordinary differential equation (ODE) is a common mathematical model for real-world problems. However, its analytical solution is hard to find and may not exist due to the nonlinear and complex structures. Thus, an approximate method is usually employed in mathematical modelling to obtain its solution. This study applies numerical integration techniques, namely Gaussian quadrature and Simpson’s rule methods, to solve nonlinear ODE, which is a hyperbolic growth model. We first discuss the ODE model and then substitute its exact solution model into the ODE model to obtain the model’s numerical solution using numerical integration approaches. Next, we aim to predict the solution of the nonlinear growth model by proposing two linear models and integrating them iteratively. We introduce a least square optimization problem and derive a set of first-order necessary conditions for estimating the model parameter optimally. A gradient descent method is employed to iterate and update the solution of the linear model. The numerical integration techniques are efficient, while the proposed method has proved to be an alternative approach to handling nonlinear ODEs, especially for a nonlinear growth model, since the optimal linear model solution satisfactorily approximates the growth model solution with a small mean square error value.