Runge-kutta Numerical Methods And Applications To Differential Equation Models

To solve practical problems, we often need to build mathematical models and will meet the differential equation models, especially. For some simple and typical of differential equation models, we can find the analytical solutions. Unfortunately, for complex mathematical models, it is difficult or impossible to find its analytical solutions. Therfore, we often use numerical methods to solve the numerical solutions of differential equations. Runge-Kutta method is an important one of numerical methods.This dissertation is concerned with numerical solutions of ordinary differential equations, especially, Runge-Kutta method is introduced in detail, and some simple applications are given. The main focus is on Four-step Runge-Kutta method. Also, the iteration algorithm and convergnece of numerical solutions are given. At last, we use Matlab to simulate numerical computations and graphics for numerical solutions of differential equations is given to illustrate main results.This dissertation is composed of five parts. In the first part, the background and significance of the problem are described and then, the main contents are introduced.The second part is concerned with Runge-Kutta's background, basic formula, convergence, absolute stability of regional and local errors.Under the suitable assumptions, the coexistence model of two species is established in the chapter 3. By using a standard fourth-step Runge-Kutta method, iteration algorithm of numerical solutions is obtained. Matlab is used to simulate numerical solutions and graphics are given to illustrate its relations of interaction of two species.An economic dynamic model is constructed under the suitable assumptions. The iteration algorithm of numerical solutions is obtained by using a standard fourth-step Runge-Kutta method. Matlab is used to simulate numerical solutions and graphics are given to illustrate the main results.The dynamical model of infectious diseases is established under the suitable assumptions in Chapter 5. By using a standard fourth-step Runge-Kutta method, iteration algorithm and convergence of numerical solutions are given. Matlab is used to simulate numerical solutions. Finally, The analysis on key factors and some exploration on disease controllability are given.