Thompson - Computing for Scientists and Engineers (523188), страница 42
Текст из файла (страница 42)
Nineteenth-century research using the differential calculus focused on mechanics and electromagnetism, while in the late twentieth century there have been themajor developments of microelectronics and photonics. Both the principles andpractical devices derived from these principles require for their design and understanding extensive use of differential equations.We now recall some of the main ideas of differential equations.
Formally speaking, a differential equation is an equation (or set of equations) involving one or morederivatives of a function, say y (x), and an independent variable x. We say that wehave solved the differential equation when we have produced a relation between yand x that is free of derivatives and that gives y explicitly in terms of x. This solution may be either a formula (analytical solution) or a table of numbers that relate yvalues to x values (numerical solution).Exercise 7.1(a) Given the differential equation(7.1)verify that a solution of this differential equation is(7.2)(b) Given a numerical table of x and corresponding y values that claim to represent solutions of a differential equation, what is necessarily incomplete about sucha differential equation solution? nGenerally, there will be more than one solution for a given differential equation.For example, there is an infinite number of straight lines having the same slope, a,but different intercepts, b.
They all satisfy the differential equation7.1DIFFERENTIAL EQUATIONS AND PHYSICAL. SYSTEMS223(7.3)The choice of appropriate solution depends strongly on constraints on the solution,such as the value of the intercept in the example of a straight line. These constraintsare called boundary values, especially when the independent variable refers to aspatial variable, such as the position coordinate x. The constraints may also be called initial conditions, which is appropriate if the independent variable is time.
(Forthose readers who are into relativistic physics, the distinction is ambiguous.)Knowing the constraints for a particular problem is often a guide to solving the differential equation. This is illustrated frequently in the examples that follow.Exercise 7.2(a) Show that the differential equation (7.1) has an infinite number of solutions,differing from each other by different choices of overall scale factor for y.(b) Given that a solution of the differential equation (7.3) is required to passthrough the y origin when x = 0, show that the solution is then uniquely determined once a is specified.(c) Make a logarithmic transformation from y in (7.1) to z = 1n (y) in order torelate the solutions to parts (a) and (b). nTime is most often the independent variable appearing in the differential equationexamples in this chapter and the next.
This probably arises from the predictive capability obtained by solving differential equations in the time variable. Thus, tomorrow’s weather and stock-market prices can probably be better predicted if differentialequations for their dependence on time can be devised.Notation and classificationWe now briefly review notations and classifications for differential equations. Weuse two notations for derivatives. The first is the compact notation that makes differentiation look like an operation, which it is, by writing(7.4)This notation we use most often within the text; for example Dxy denotes the firstderivative of y with respect to x, in which (by convention) we have dropped thesuperscript 1.
The second notation for derivatives is dy/dx, suggestive of divisionof a change in y by a change in x. We use this notation if we want to emphasize thederivative as the limit of such a dividend, as in numerical solution of differentialequations.The classification of differential equations goes as follows; it is mercifullybriefer than those used in biology or organic chemistry. Equations that involve total224INTRODUCTION TO DIFFERENTIAL EQUATIONSderivatives (such as Dxy) are called ordinary differential equations, whereas thoseinvolving partial derivatives (such as) are termed partial differential equations.
The aim of most methods of solution of partial differential equations is tochange them into ordinary differential equations, especially in numerical methods ofsolution, so we use only the latter in this book. This greatly simplifies the analyticaland numerical work. We note also that systems for doing mathematics by computer,such as Mathematica, are restricted to ordinary differential equations, both for analytical and numerical solutions. This limitation is discussed in Section 3.9 of Wolfram’s book on Mathematica.The order of a differential equation is described as follows. If the maximumnumber of times that the derivative is to be taken in a differential equation, that is,is n, then the differential equation is said to be ofthe maximum superscript innth order. For example, in mechanics if p is a momentum component, t is time, andF is the corresponding force component (assumed to depend on no time derivativeshigher than first), then Newton’s equation, Dtp = F, is first order in t.
In terms ofthe mass, m, and the displacement component, x say, Newton’s equation,is a second-order differential equation in variable t.Exercise 7.3(a) Write Newton’s force equation for a single component direction as a pair offirst-order ordinary differential equations.(b) Show how any nth-order differential equation can be expressed in terms of nfirst-order differential equations. nThe result in (b) is very important in numerical methods of solving differential equations, as developed in Chapter 8. There remains one important item of terminology.The degree of a differential equation is the highest power to which a derivativeappears raised in that equation. Thus, if one identifies the highest value of m appearing in (Dxy)m in a differential equation, one has an mth-degree differentialequation. If m = 1, one has a linear differential equation.
Generally, it is preferable to be able to set up differential equations as linear equations, because then oneisn’t battling algebraic and differential equations simultaneously. The distinctionbetween the order of a differential equation and its degree is important.We consider mainly ordinary, first- and second-order, linear differential equations. Since there are also interesting systems that cannot be so described, we sometimes go beyond these, as for the logistic-growth equation (Section 7.3) and forcatenaries (Section 8.2).Homogeneous and linear equationsTwo distinct definitions of “homogeneous” are used in the context of differentialequations.
The first definition is that a differential equation is homogeneous if it isinvariant under multiplication of x and y by the same, nonzero, scale factor. Forexample, Dxy = x sin (y/x) is homogeneous according to this definition. Such7.2 FIRST-ORDER LINEAR EQUATIONS: WORLD-RECORD SPRINTS225equations are not very common in the sciences, because in order to have the invariance, x and y must have the same dimensions, which is uncommon if one is discussing dynamical variables, as opposed to geometrical or other self-similar objects.The other definition of a homogeneous differential equation, usually applied tolinear differential equations, is that an equation is homogeneous if constant multiplesof solutions are also solutions.
For example,= - ky, the differential equationfor simple harmonic motion, is a homogeneous differential equation. The power ofsuch linear differential equations is that their solutions are additive, so we speak ofthe linear superposition of solutions of such equations. Almost all the fundamentalequations of physics, chemistry, and engineering, such as Maxwell’s, Schrödinger’s, and Dirac’s equations, are homogeneous linear differential equations.Electronics devices are often designed to have a linear behavior, using the termin the same sense as a linear differential equation.
Such a device is termed linear ifits output is proportional to its input. For example, in signal detection the primaryamplifiers are often designed to be linear so that weak signals are increased in thesame proportion as strong signals. It is therefore not surprising that much of thetheory of electronics circuits is in terms of linear differential equations. The differential equations for mechanical and electrical systems that we discuss in Section 8.1are linear equations.Nonlinear differential equationsDifferential equations that are nonlinear in the dependent variable, y in the above discussions, are termed nonlinear, no matter what their order or degree.
In general,such differential equations are difficult to solve, partly because there has not beenextensive mathematical investigation of them. In science, however, such nonlinearequations are recently of much interest. This is because linear differential equationstypically describe systems that respond only weakly to an external stimulus, whichitself is not very strong. On the contrary, as an example of systems that should bedescribed by nonlinear equations, many optical materials respond nonlinearly whenstrong laser beams are focused on them.The example of a nonlinear differential equation that we explore in this chapter isthe logistic-growth equation in Section 7.3. It illustrates how the methods of solving nonlinear differential equations depend quite strongly on the problem at hand.7.2 FIRST-ORDER LINEAR EQUATIONS:WORLD-RECORD SPRINTSWe motivate our study of first-order linear differential equations by studying a realistic problem from athletic competition, namely the kinematics involved in sprinting.Our analysis is phenomenological in that it is concerned only with the kinematics andnot with the dynamics, physiology, and psychology that determine the outcome ofathletic contests.
So we need only a model differential equation and a tabulation ofworld-record times for various distances. We do not need treadmills for the dynam-226INTRODUCTION TO DIFFERENTIAL EQUATIONSics, specimen bottles for the physiology, and post-race interviews for the psychology. In the language of circuit theory and mathematical modeling, we are making a“lumped-parameter” analysis.Kinematics of world-record sprintsIn 1973, J. B. Keller published an analysis of competitive running sprints in whichhe showed that for world-class runners in races up to nearly 300 m distance theirspeed, v, at time t into the race can be well described in terms of the accelerationformula for world-record sprints(7.5)He found that for men’s world records in 1972 the appropriate constants were an acceleration parameter A = 12.2 m s-2 and a “relaxation time” = 0.892 s.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
