Models and Mathematical Problems

Abstract

As already mentioned in Chapter 1, we refer to mathematical models that can be expressed as systems of ordinary differential equations and deal with the solution of problems in mechanics and, in general, in applied sciences. Therefore, it is important to deal with mathematical aspects related to the analysis of differential equations and with the solution of initial and boundary-value problems. This chapter refers to the above topics and is organized in seven sections: Section 2.2 deals with a formal classification of ordinary differential equations. Linear and nonlinear properties are identified together with the analysis of some additional properties of the system. Physical interpretations are often related to this type of classification. Section 2.3 deals with the mathematical statement of problems, which is obtained by linking the evolution equation to the conditions that are necessary to its solution. If these conditions are given at the initial value of the independent variable, say t = 0, one has an initial-value problem, also called a Cauchy problem. Otherwise, if these quantities are given at both ends of the range of the independent variable, say t = 0 and t = T, one has a boundary-value problem. The solution of problems has to satisfy both the differential equation and the initial and/or the boundary conditions. This section is mainly devoted to the analytic treatment of the Cauchy problem and will report the basic theorems which provide the conditions necessary to ensure existence, uniqueness, and regularity of the solution. Section 2.4 deals with solution methods of the initial-value problem. Analytic methods can be developed in order to solve linear problems, while computational schemes are needed for nonlinear problems. Actually, this matter is proposed as a concise guide to mathematical methods and to the related literature. The aim of this guide is to provide the necessary elements for the interaction with the scientific computation based on Mathematica ® Section 2.5 deals with the representation of solutions and applications both in the linear and in the nonlinear case. Here, a systematic use of scientific computation is developed. As we shall see, the computation is useful not only in the nonlinear case, but also for the fast treatment of linear problems. Section 2.6 deals with boundary-value problems for ordinary differential equations and the problems related to their solution, both analytical and numerical. The last section suggests some problems which are proposed to the reader’s attention for practicing with mathematical methods.