The Solutions Of Mathematical Models Of Several Kinds Of Differential Equations

Differential equation is a long-standing theoretical subject.It is not only a hot topic in the field of mathematics,but also in other natural science fields(physics,chemistry,biology,etc).differential equations can also constitute mathematical models of some complex systems.Even today,some problems in society can also be transformed into differential equations,such as population models and infectious disease models.Therefore,the study of differential equations is closely related to human society and has very important practical significance.At present,there are several different emphases in the study of differential equations,but they are mainly concerned about the solutions of differential equations.The nonlinear partial differential equation,as a branch of differential equation,has a broad prospect of development.It has been widely used in mathematics,engineering technology and physics,and has become an indispensable tool for large-scale engineering research,design and verification.In this paper,we study the solutions of three mathematical models of differential equations:the tree-grass kinetic model is solved by Picard successive approximation method and the power series method,and the coupled homogeneous-catalytic reaction model and the chemical tubular reactor model are solved by the Adomian decomposition method respectively.This article mainly consists of the following six chapters:Chapter 1 briefly reviews the research background of differential equations and several methods to solve these equations,and introduces the research background of Picard successive approximation method,power series method and Adomian decom?position method.In addition,the main work and specific structure of this paper are described in detail.Chapter 2 is the preparation of knowledge.It mainly describes the basic ideas of Picard successive approximation method,power series method and Adomian decom-position method,as well as the definitions related to the initial and boundary value conditions.Chapter 3 analyzes a class of tree-grass dynamical model composed of two first-order ordinary differential equations by using the Picard successive approximation method and the power series method.Picard successive approximation method is based on the existence and uniqueness theorem of solutions of ordinary differential equations.Finally,the comparison of two methods for this model is given.In chapter 4,the analysis is performed for a mathematical model of the coupled homogeneous-catalytic reaction,which consists of two strongly nonlinear second-order partial differential equations.Based on the Adomian decomposition method,the ap-proximate analytical solution of the model is obtained by using the boundary conditions and choosing the correct solution direction.The distributions of temperature and fluid concentration are obtained by assigning dimensionless parameters in the model.Chapter 5 uses the Adomian decomposition method to acquire the approximate analytical solutions for a chemical tubular reactor model composed of a system of strongly nonlinear second-order partial differential equations with standard boundary conditions.By assigning the dimensionless parameters in the model,the distributions of temperature and process component concentration can be obtained.Chapter 6 makes a brief summary of this paper,and puts forward some problems that need to be solved.In addition,the research direction of the future is envisaged.