Academic journal article Academic Exchange Quarterly

A Course on Ordinary Differential Equations

Academic journal article Academic Exchange Quarterly

A Course on Ordinary Differential Equations

Article excerpt

Abstract

This paper presents a planning course on ordinary differential equations (ODE) for graduating students of physics / engineering. It contains a classical body and some nonstandard topics such as the influence of unmodeled dynamics of a system on its behavior.

I. Introduction

A course on ordinary differential equations (ODE) is included in the teaching curricula of basic studies in physics and engineering. It is obvious that ODE are basic in the understanding of the properties of physical processes. The objectives are:

Objective 1 (classical objective): The understanding of the basic theory while solving specific differential equations.

Objective 2: Understanding of the physical phenomena related to local and global Lyapunov's stability from a general point of view, i.e. through the nature of the dynamical equations.

Objective 3: It seems to be convenient and useful for future applicants to a Master or Ph.D. Degree in Physics/Engineering to know the influence of the unmodeled dynamics in the behavior of a physical system. This is important since many dynamic equations are approximations after deleting small forcing terms or negligible modes.

II. Course distribution

II.1 Teaching curricula

In parallel with the course of ODE, other courses of quantum physics, geometry and optics as well as electromagnetic field theory are programmed. They have approximately the same number of hours for classroom explanations. The curricula for engineering studies are very similar to the various mathematics topics. That of physics is oriented to classical applied physics in curses of general physics, thermodynamics theory, thermotechnics, classical mechanics (statics/dynamics being stated directly from Newton's laws in mechanics with brief introductory guides to rational Lagrangian formulation and quantum mechanics), electromagnetic and electrical engineering. The specialization of the remaining years of related engineering curricula are very varied as metallurgic engineering, electrical engineering, mechanical engineering etc. while those of physics are two, respectively related to control theory and informatics, and theoretical physics and condensed matter. The course of ODE is fundamental towards the last years of these careers too.

II. 2 Course contents

The course lasts ninety clock hours of which twenty five are devoted to solving classroom problems. The distribution of the theoretical topics is as follows: A. introduction. (four hours)

B. linear and homogeneous differential equations. (six hours)

C. orden reduction: The basic new and specific idea is the construction of a general solution through a particular one and a transformation of variables. Importance and usefulness for modeling simplification (presence of unmodeled simplification). Examples concerned with deleting coefficients in a linear differential equation. (four hours)

D. Bessel's equation of order p: (six hours)

E. systems of n-th order linear differential equations: (eight hours)

F. Laplace transform: (six hours)

G. existence theory: (ten hours) G'. error transmission: It is seen how the errors propagate in the solution from errors in the initial conditions or caused by deleting, adding or modifying some of the coefficients of the differential equation / system of differential equations or originally generated by input disturbances (unmodeled dynamics and /or generalized forces) (seven hours)

H. Sturm- Liouville systems: (six hours)

I. nonlinear differential equations: (eight hours)

The basic and classical course which usually contains the above topics A,B, F, G, H and partly G (just the basic method of reduction in order to be able to solve Bessel's differential equation in D) and E (operative methodology to solve differential systems). The reminder of the course; i.e. …

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