Simple Applications Of Ordinary Differential Equations To Mathematical Modeling

It is well known that all of the substances in nature have, their own rules in the movement and evolution. Although the forms of material movement vary greatly, the different material always moves in time and space, one can always figure out their common properties, namely, the evolution rules of the common variables. In order to qualitatively and quantitatively research the specific movement and evolution, the relevant factors in the material movement and the process of evolution are needed to be represented using mathematical varibles and mathematical equations are employed to describe their relations. This kind of quantization process is the procedure of mathematical modeling, that is, identify the relations among different variables according to the movement.Due to various of practical problems, the functions often can not be represented directly according to some slightly complicated processes of movement, but it is easy to set up the relationship between variable and their derivatives (or differentials); This relationship is called Differential Equations in mathematics. The application of Ordinary Differential Equations to mathematical modeling has its deep background in that the theory of Ordinary Differential Equations is a powerful tool to solve practical problems in mathematical modeling, which is a subject with vital background application Ordinary Differential Equation theory has a long history and it is a systematic theory which will be perfected gradually in continuously further development. The main reason for this is that its sources are deeply rooted in various practical problems.The author has made detailed and integrated descriptions as to the applications of Ordinary Differential Equations to mathematical modeling by consulting reference literature, materials and books. The main results are given as follows:1. A short introduction on the development of mathematical modeling and Ordinary Differential Equation. We introduce different definitions of mathematical models according to different practical problems and the classification of mathematical modeling based on the definitions. Detailed steps in mathematical modeling are concluded and the key point in every step is pointed out, where the main mathematical method to solve is Ordinary Differential Equation theory. Finally, an practical example, how to walk in the rain to maximally avoid the rain, is given to show how to do the mathematical modeling combined with Ordinary Differential Equation theory.2. A detailed description on the principle,method and process of mathematical modeling using Ordinary Differential Equation theory is given. First, the detailed steps to mathematical modeling by Ordinary Differential Equation theory are first introduced. We also present some methods to use and the key points which should be mentioned. By two typical examples, counterfeit and advertising, the process of mathematical modeling are explained in detail which implies that the practical application of mathematical modeling. Second, mathematical modeling based on first-order nonlinear Ordinary Differential Equation is introduced to consider the mathematical modeling of two classical problems, Brachistochrone and biotic population dynamics. Main procedure of mathematical modeling is applied to investigate practical problems. Third, second-order nonlinear Ordinary Differential Equations are introduced to construct the mathematical modeling in chasing problem and vibration problem. We establish the second-order chasing dynamics model as well as vibration equation.3. The stability of Ordinary Differential Equations models are introduced. The fundamental theory on stability of Ordinary Differential Equations are discussed, equilibriums and stability of one-dimensional and two-dimensional of Ordinary Differential Equations are presented, respectively. The mathematical model to maintain continual harvest in the fishing and the competition among biotic population model are presented. The stability of both models are analyzed and two problems are solved based on the stability analysis...