A course on ordinary differential equations.

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. G, I as well as the completing parts of C and E subjects are devoted to the covering of the specific objectives stated in the Introduction.

II.3. Course exposition, mathematical level and rigor 

The above topics are divided into two parts which have different objectives and exposition philosophy, namely:

Part 1. Fundamental methods and applications. This part deals with the above topics A to E. It pursues to transmit the basic knowledge of the course to the students. The exposition is based on the solution of first and second-order differential linear equations. We think that ODE are one of the mathematical fields having more intuitive projections into real problems so that a preliminary exposition in this context is highly recommended [1]. Special attention is paid to the linear ODE, the operational methods for differential systems of equations (substitution method, Cramer's rule etc.) to solve algebraic systems, the Laplace transform and the series method for solution about ordinary and regular singular points (Froebenius method [1-3]). Because of its closeness to Physics and Engineering, we also introduce a formal exposition about differential systems theory. The main purpose is to relate algebraic systems theory [10, 11, 13, 14, 21] (namely, the algebraic description of invariant ordinary differential equations or systems of equations: modes, eigenvectors, basis changes etc.) to physical systems. Special attention is also paid to the reduction method to deal with the general solution of second-order differential equations. Its generalization to n-th ODE is then given in Part II. The reduction method useful to find the general solution of a second-order differential equation only requires the knowledge of a particular solution rather than two as in the usual standard methods. Furthermore, it allows an easy-comparison of the general solution of a first-order linear differential homogeneous equation 2.1 with that of the second-order ODE (2.2) by applying the reduction method, [1] to (2.2). This idea can be used to transmit to physics/engineering students the effects of the unmodeled dynamics [19-22] in the forced case since, usually, high frequencies of the forcing term make the real model to vary with respect to the simplified one; i.e., eqn. 2.2 related to eqn. 2.l. Unmodeled dynamics appears frequently in practice when high frequency dynamics is present as, for instance, in the missile-antimissile maneuvers. Thus, when the forcing fields exhibit high frequency components the low-order nominal model is not useful for an appropriate problem description. Examples of this question in problems of linear control theory and in electrostatic and magnetostatic fields may be pointed out. Particular examples are the study of Maxwell's equations for sinusoidal electric and magnetic fields (see eqn. 3.1). Note that related differential equations are in partial derivatives but the problem can be equally emphasized at an empirical level, by assuming all the independent variables to be (quasi)- constant except for one of them, as for instance, time. In section III. 4 below, we fully discuss a related case, which is more feasible for the knowledge level of undergraduates, related to basic electronics of oscillators. That example is also very useful to discuss the existence and uniqueness of the solution as well as local and global stability. The proposed problems to be solved are specifically oriented to the following aspects:

* -The solution of linear ODE including the related use of the Laplace transform. Extensions to linear differential systems including methods to solve the fundamental matrix in the time-invariant case. Some specific topics merit important attention in physics and engineering as, for instance, the free and forced response of a system, the linearity, the superposition principle related to forcing terms and initial states etc.

* -Laguerre and Hermite polynomials, examples of Bessel's equation in Physics. Skin electronic effect; i.e. currents of high frequency which tend to circulate in the surfaces of electrical rods.

Part 2. fundamental theory and another methods. The main body of this part is devoted to the presentation of the existence and uniqueness theory of ODE. Because of its closeness to physics and engineering, we also introduce a formal exposition about differential systems theory. The formalism with nth-order linear differential equations is developed for obtaining equivalent differential systems of n first-order linear equations. The solution existence and uniqueness are addressed through the use of the iterative Picard's method for construction of the solution and Lipschitz's condition. In the classroom expositions, one takes the advantage of the student "s previous knowledge from Part 1:

* about first-order differential systems at an algebraic level, and

* about their equivalence with the n-th order differential equations as well as its importance for obtaining orthonormal sets of functions being of special interest in the Fourier series.

The relationships between Fourier expansions in nonlinear systems with the orthonormal sets are obtained as characteristics functions of Sturm-Liouville systems. The nonlinear stability is also studied. The difference about qualitative stability of nonlinear systems of equations related to its linearized forms about the equilibrium for the center singular point is emphasized.

III. Achievement of the course objectives

As it has been stated in the introduction, three main objectives are pursued, namely, to understand the basic theory including the solution of classroom problems, the projection of the stability theory on physical problems from a general point of view (independent of the particular problem) when possible and the teaching-learning of the existence theory related to the parametrical/order changes of the linear differential equations of systems of equations and their influences on the solutions.

III.1 Bibliographical support

The achievement of these objectives implies the use of mixed bibliography including the classical one about differential equations (see section I of the bibliography) and its corresponding mathematical support from another areas. Also, some bibliography related to stability and the importance of the unmodeled dynamics has been included in section HI of the bibliography. Concerning objectives 1-3, we have:

Objective 1 (basic classical objective): It is covered in [1] in a very pedagogical way including and abundant case-study material, solved examples and proposed problems.

Objective 2 (when the linearized system is / is not useful for local qualitative description of the nonlinear one about the equilibrium?): It is covered in [1], [15], the first reference presenting a lot of examples of electronics and control theory. In particular, [1] describes topics about limit cycles and some theorems of usefulness in typical specific nonlinear problems such as Vanderpol's equation. The different topological, configurations combining equilibrium points and limit cycles are intuitively focused on in [15]. Lyapunov's Theory is treated in detail in [11-13].

The importance of this objective arises from the following aspects. The stability of a nonlinear differential equation can be of different nature compared to that of its linearized version for some equilibrium points having associate linear systems with imaginary complex modes. A typical example is the case of the orbits around a center which are closed trajectories dependent on initial conditions. However, the trajectories of a nonlinear system, whose linearized version has a center equilibrium, can diverge with time (i.e., they can exhibit instability) or asymptotically vanish (i.e., they can exhibit local asymptotic stability) depending on each particular nonlinear system dealt with. The above discussion emphasizes that linearization is not useful on occasions to describe nonlinear systems around the equilibrium. The reason is that with the eigenvalues of the dynamics of a linearized system have zero real parts, second-order approximating terms corresponding to higher-order derivatives about the equilibrium have to be tested to elucidate local stability. Therefore, the same problem as for centers sometimes occurs for defective nodes whose linearized system possesses at least a zero eigenvalue. Global stability implies and it is implied by the local stability around the equilibrium in the linear case. Otherwise, a stable limit cycle can make the system to be (non asymptotically) globally Lyapunov's stable with local instability around the equilibrium exhibiting ultimate boundedness. Those topics might be emphasized with electronic examples.

The first above aspect is not clearly addressed is some western classical textbooks on control theory (see, for instance, [13]). In the listed bibliographical references related to stability, and in particular in [1], that topic is correctly treated. The stability theory is fundamental to understand physical and engineering phenomena, theoretical mechanics, control theory, thermodynamics etc.

Objective 3 (how the presence of neglected terms in the differential system (i.e. unmodeled dynamics) may lead to the usefulness for qualitative description of the simplified (i.e. nominal) description): It is of a crucial importance for non mathematicians from our point of view. However, it is not regularly included in a course on ODE. Its importance relies on the fact that many of the equations in physics are usually simplified in practice for an easy treatment. The solution of the real equation varies with respect to the nominal one and their respective stability degrees can be quite different. The simplified (nominal) system sometimes is not useful to qualitatively describe the asymptotic trajectory behavior when high-frequency modes become excited by the forcing term in steady-state, [2022]. Other situations of the effects of the unmodeled dynamics appear in electromagnetic fields, as Maxwell's equations 3.1, or others from mechanics like the effects of damping in vibrating systems. Take, for instance, the time variation of the vibrating string which leads to the well-known differential equation of harmonic motion (3.2) which can lead to underdamped, critically damped or overdamped motion related to the harmonic oscillation if no damping is present. Those examples can be included in the context of the general theory involving the reduction method as well as in that of the existence theory according to the following items.

III.2 Technical development of objective 3 via reduction method

It is usually applied to construct a general parametric from a particular one [1].

(a.1) Take the homogeneous linear equations of continuous coefficient real functions (3.3), which is of (n+m)-th order, with not all the extra coefficients in (3.3) related to the nominal one (3.4) being zero as well as its nominal n-th order version (3.4). Eqn. 3.3 can be usually interpreted as a high-frequency homogeneous model of the nominal homogeneous version eqn. 3.4 which describes appropriate the behavior for low-frequencies (i.e. in the absence of unmodeled dynamics). To figure out common practical situations, it may be supposed that the independent argument x is time. To compare both solutions, assume for simplicity n=m=1 and that a known nontrivial particular solution of (3.3) is used. The subsequent procedure is followed:

(a.2) Make the proposed change of variable and substitute in (3.3) to find the parametrical solution eqn. 3.5 via a linear combination of the known particular one with a second particular one being [1].

(a.3) Compute the solution for (3.4) on a small interval and extend it by prolongation for all time while obtaining the error between the solutions of (3.3) and (3.4).

(a.4) Then, discuss the asymptotic stability of both solutions to (3.3) and (3.4) and then the conditions guaranteeing asymptotic convergence of the mutual error to zero.

III.3 Case study

An electronic oscillator consisting of a bipolar-transistor basic amplifier and a third-order associated RC network is approximated with a time-invariant ODE (3.4.) The stability of this linear equation around the equilibrium operation point is discussed. If amplifier saturates, improved stability results are obtained linked to the existence of several equilibrium points. By changing the basic amplifier parameters (i.e., the slope and saturation point) and the RC ones, the two standard operation modes (i.e. amplifier/oscillator) may be discussed. Key issues of objectives1-2 are covered with this case study. Now, an (unsuitable in practice) parasitic emitter-collector capacitor is assumed to operate significantly at high forcing frequencies. Such a capacitor ranges in practice from 10 exp (-6) to 10 exp (-8) of those used in the RC- network. A fourth-order ODE (3.3) is obtained by incorporating such a mode as unmodeled dynamics. Its solution may be calculated from either four particular solutions or from an auxiliary dynamic system or via the reduction method. One can interpret how the dynamics generated by the extra capacitor is negligible under a sinusoidal forcing term of low frequency in (3.3). However, it cannot be neglected if the capacitor value or the forcing frequency are sufficiently large (objective 3).

References

I. Basic bibliography

1. S.L. Ross, Differential equations, John Wiley and Sons, New York, 1979.

2. S.L. Boyce and R.C. DiPrima, Elementary differential equations and boundary value problems, John Wiley and Sons, New York, 1965.

3. F. Brauer and J. Nohel, Ordinary differential equations, Benjamin Co., New York, 1973.

4. M. Braun, Differential equations and their applications, Springer- Verlag, Berlin, 1993.

5. E. Coddington, An Introduction to ordinary differential equations, Prentice-Hall, Englewood Cliffs, 1961.

6. E.A. Coddington and R. Carlson, Linear differential equations, SIAM, Philadelphia, 1997.

7. G.F. Carrier and C.A. Pearson, Ordinary differential equations, SIAM, Philadelphia, 1991.

8. F. Ayres Jr., Differential equations. Theory and problems, Me.Graw-Hill, 1969.

9. F. Simmons, Differential equations with applications, Mc. Graw-Hill, 1972.

II. Advanced bibliography

10. R. Bellmann, Methods of nonlinear analysis. Vol. I, Academic Press, New York, 1970.

11. L. Elsgoltz, Ecuaciones diferenciales y cqlculo variacional (in Spanish). Original version in Russian, MIR, Moscow, 1981.

12. F. Brauer and J. Nohel, Qualitative theory of ordinary differential equations, W.A. Benjamin, New York, 1969.

13. W. Cunningham, Introduction to nonlinear analysis, McGraw-Hill, New York, 1958.

14. J. Stoker, Nonlinear vibrations, Interscience, New York, 1950.

15. G. Sansone and R. Conti, Nonlinear differential equations, Pergamon Press, New York, 1964.

16. R. Cole, Theory of ordinary differential equations, Wiley, New York, 1958.

17. W. Hurewicz, Lectures on differential equations, Wiley and Sons, New York, 1958.

18. U.M. Ascher, R.M.M. Matteiji and R. Russell, Numerical solution of boundary problems for ordinary differential equations, SIAM, Philadelphia, 1997.

19. E. Coddington and N. Levinson, Theory of ordinary differential equations, Mc-Graw-Hill, New York, 1955.

III. On the problem of unmodeled dynamics

20. I. Stakgold, Boundary value problems of mathematical physics SIAM, Philadelphia, 2000.

21. M. Marcos, I. Iparaguirre and M. Delasen," A note on stability and a class of singular point", Int. J. of Electrical Engineering Education, Vol. 21, pp. 276-277, 1984.

22. M. Delasen, " Project-based course on nonlinear control", 31st FIE, pp. S1C-16/S1C-22, Reno, NE, 2001.

M. De La Sen, University of Basque Country, Spain

The author is Professor of Electrical Engineering at the University of Basque Country.