Sec. 2.3] A System of Ordinary Differential Equations' 57

2.3 A System of Ordinary Differential EquationsThe purpose of this section is to demonstrate the fourth part of applied

mathematics that we listed in Chapter 1: how the desire to solve scientificproblems motivates the development of a mathematical theory, and themanner in which such theories aid us in the solution of scientific problems.

We shall begin the discussion with a statement of certain theorems relatedto the initial value problem for mechanics. The proofs will follow. (As faras it is feasible, the proofs presented here will be those actually useful forconstructing the solutions.) At the end of the section we shall briefly discussthe general question of the value of such proofs to the applied mathematician. .

THE INITIAL VALUE PROBLEM: STATEMENT OF THEOREMS

For a given (initial) point Po(t, '1' 'z, ..., 'n) in the (n + I)-dimensionalspace of the points P(t, Zl, Zz, ..., zn)' the initial value problem for a systemof ordinary differential equations

dzk-;Ii = h(t; ZI, Zz, ..., zn); k = 1,2, . .., n (1)

is to find functions

Zm = gm(t), m = 1,2, .. ., n (2)

such that (1) is satisfied, and

gm(t) = 'm at t = t, m = 1,2, ..., n. (3)

We now state and subsequently prove certain theorems that are validfor the initial value problem. In the statement of these theorems and insubsequent discussions we shall frequently use Z and' to denote collectivelythe set of variables (ZI, Zz, . . ., zn) and ('1' 'Z, .. ., 'n).

Theorem 1 (Existence). Suppose that the functions* fk(t, z) are continuousin a rectangular parallelepiped defined by

R:lt-tl~a, IZk-'kl~b; k=I,2,...,n. (4)

This implies that there exists an upper bound M such that

Ihl ~M, k=I,2,...,n, (5)

for P(t, ZI, Z2, . . ., zn) in R. Suppose further that eachfk satisfies the followingLipschitz condition in R:

If(t,zl,...,zn)-f(t,zl,Z2,.",zn)1 ~K[lzI-Zll +...+ Izn-znl]. (6)

. Some mathematicians are always careful to distinguish the function f from f(x), the value

of this function at x, We do not find it profitable to emphasize this distinction.

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--

Sec. 2.3] A System of Ordinary Differential Equations 59

We shall take the initial conditions to be fixed and consider the dependenceof the solutions Zm = gm(t, A) on A. One might expect the nature of thisdependence to be the same as that of the functionsfk(t, z, A) in (8). Thus, ifthe h are analytic in A in the neighborhood of A = AO (i.e., if they have aconvergent Taylor series when 0 ~ I A - Ao I < r for some r), then the solution

functions gm should have the same property. For linear differential equations,this is generally true. That is, the solution of the equation.

dzdt = A(t, A)Z + b(t, A) (9)

is analytic in A, provided that A(t, A) and b(t, A) are analytic in A, even thoughthey may merely be continuous in t. (Note that the Lipschitz condition isautomatically satisfied for linear equations when the coefficients are con-tinuous.) In the nonlinear case, the situation is a little more complicated,as we can see from our experience with the development of a perturbationsolution for (2.1),

('l.."2.-1) dy~ f6 ~ = f(x, y, e).

In present terms, the formalt calculations there depended on the fact thatthe functions h(t, z, A) were analytic in z as well as in A. To keep the dis-cussion relatively simple, we shall therefore restrict our attention to thefirst derivative with respect to A.

If formal differentiation is justified, we would expect the functions

0um(t, A) = a;: zm(t, A) (10)

to satisfy the differential equation obtainable formally from the differentiationwith respect to A of the original equation (8), remembering that fk and Zmboth depend on A.§ Carrying out this differentiation, we obtain

~=L~Um+~' (11)dt m ozm OA

We note that (11) is a linear system in {Uk} whose coefficient functions areknown functions of t, since the set {zm(t, A)} is known. If the initial values 'mare independent of the parameter A, then the initial conditions on the {Uk} are

Uk = O. (12)

. This may be regarded as a single equation or a system of linear equations. In the latter case,A is a matrix, and z and b are vectors.

t A formal calculation is one that is presumably valid under suitable but unspecified conditions.§ Successive parametric differentiation of an equation forms the basis of an excellent way to

perform perturbation calculations. See Section 7.2.

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Sec. 2.3] A System of Ordinary Differential Equations 61

In the interval [xo, x], I G(x) - g(x) I has a maximum value that we shalldenote by IIG(x) - g(x) II. It then follows from (19) that

IG(x)-g(x)1 :sIIG(x)-g(x)1I 'Klx-xol, (20)

and hence

IIG(x) - g(x) II :S IIG(x) - g(x) II . Klx - Xo I, (21)

or[1 - K(x - xo)] . IIG(x) - g(x) II :S O.

If we now take the length I x - Xo I sufficiently small, equal to (2K) -1 for

example, we can make the first factor positive. Hence (21) requires that

IIG(x) - g(x) II = 0,

from which the desired result follows at once. 0PROOF OF THE EXISTENCE THEOREM aSc;.oL 7" ~ ..,. 3We have seen that the solution of the initial value problem for the differential

equation (13) satisfies the integral equation (15). Conversely, if we have asolution of the integral equation (15), we have a solution for the differentialequation, with the initial conditions implied. This can be verified by directcalculation (Exercise I). Thus it is sufficient to prove the existence theoremfor the integral equation.

To prove that (15) has a solution, we adopt the method of successiveapproximations. We start with a crude approximation y = Yo, which at leastsatisfies the initial condition.* We then calculate the sequence of functions

%

Yl(X) = Yo + f f(t, Yo) dt,%0

%

Y2(X) = Yo + f f[t, Yl(t)] dt,%0

... (22)

%

Yn(x) = Yo + f f[t, Yn-l(t)] dt,%0

We shall prove that this sequence converges uniformly to a continuousfunction in a suitably restricted interval I x - Xo I :S cx, which may bespecified as in Theorem I; i.e., cx = min (a, hjM). [Here M is the upper boundon If I that was introduced in (5).] The restriction to the range hjM is needed

. It is certainly not necessary to make this particular initial approximation. For example,

the same final result will obviously be obtained if Yt(x), defined in (22), is used as the initial

approximation.

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---

Sec. 2.3] A System of Ordinary Differential Equations 63

the functions smoother, and questions about the existence of a derivativeare avoided.

The proof required many technical steps. Since similar steps often occurin other problems, it is convenient to summarize them in some form so thatthey need not be continually repeated. This is one reason for developingthe theory of function spaces, in which an important concept is the normsuitably defined (in various ways) to replace the concept of distance in ordinaryEuclidean space. Another reason is that ordinary geometric ideas can beused to give intuitions about abstract distance properties.

In the present case, one possible norm is the quantity Ilg(x) - G(x)1I

first used in (20). Another possible measure of "distance" is the quantityD defined by

1 b

D2 = - f [G(x) - g(x)f dx. (29)b-a a

In words, D is the root-mean-squared difference between G and g. Thisnorm does not occur naturally here, but it will be used in Chapters 4 and 5.

Appendix 12.1 of II provides certain details concerning function spacesand related concepts. [In particular, Equation (59) of that Appendix givesan abstract characterization of distance that is exemplified by both of thenorms just defined.] Material in Chapter 12 of II on variational methodsfurnishes examples of the unity and clarity that can be achieved with such

concepts.

CONTINUOUS DEPENDENCE ON A PARAMETER OR INITIALCONDITIONS

Before we proceed to prove Theorems 3 and 3', let us note that if the solutionof the differential equation

dy- = f(x, y, ).), (30)dx

under the initial condition y = Yo for x = Xo, depends continuously on ).,then it depends continuously on (xo, Yo). For we can introduce the newvariables (~, ,,) = (x - Xo, Y - Yo) and solve the equation

d";If: = f(~ + Xo," + Yo,).) (31)

under the initial condition" = 0 for ~ = O. The function on the right-handside of (31) is continuous in (xo, Yo).

Return now to (30). Let us treat the equation in the integral formulation%

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Sec. 2.3] A System of Ordinary Differential Equations 65

these solutions is assured by the continuity of the functions involved.* Wewish to prove that

u(x, A) = ~. (39)

The difficult part of the proof is to establish the existence of the partialderivative ayjaA. This will be done by direct evaluation of the limit ofAyjAA as AA -10 o.

LetAy = y(x, A) - y(x, Ao), Af = f[x, y(x, Ao), A] -- f[x, y(x, Ao), Ao].

From (33)Ay x Ay Ix Af~ = I f,,(x, y, A) ~ dx + :6::::i ax, (40)

Xo Xo

where y is a value intermediate between y(x, Ao) and y(x, A). We have usedthe continuity of the partial derivative f,,(x, y, A) so that the mean valuetheorem can be applied. We also have

x x

u(x, Ao) = I f,,[x, y(x, Ao), Ao]U(X, Ao) dx + I f).[x, y(x, Ao), Ao] dx. (41)Xo Xo

By combining (40) and (41), we obtain for the difference

= ( 1 ) - y(x, A) - y(x, Ao)(42)w - U X, 11.0 A - Ao

the integral equation

xw = f f,,(x, y, A)W dx + D, (43)

Xo

where

D = Ix [/,,(x, y(x, Ao), A)- /y(x, y, Ao)]U dx + Ix [ f).(X, y(x, Ao), Ao) - ~ dx.Xo Xo ~J

Because of the continuity of the functions f" and f)., the quantity D can bemade as small as we wish by reducing AA. Since I /y(x, y, A) I is bounded byK, one can prove, from (43), that

lim W = 0 (44)4).-tO

by the same sort of reasoning as that previously used. The reader shouldcomplete the proof (Exercise 4).

. The Lipschitz condition is satisfied for (38), because the right-hand side is linear in the

dependent variable u.

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--

Sec. 2.3] A System of Ordinary Differential Equations 67

Let us divide the interval (xo, x) into n subintervals (each of length h) andwrite

Y = Yo + IX'f[~, Y(~)] d~ +... + Ix f[~, Y(~)] d~. (49)Xo Xn-1

This is still exact, but we shall now evaluate each of the integrals in (49)approximately by writing

Xk+1I f[~, Y(~)] d~ = hf(Xk' yJ. (50)Xk

Here Yk is the value of Y obtained by using the approximation (50) in (49)up to the point Xk; i.e., we have

Ixl Y1 = Yo + f(xo, Yo) d~,

XoIX2 Y2 = Y1 + f(X1, yJ d~, (51)

XI

xy(x) = y" = Y"-l + I f(X"-l' y,,-J d~.

Xn-1

The solution is then the broken line C L in Figure 2.2. This is taken as anapproximation to the true integral curve ,C. Since there is an error in Y ateach approximation, the accumulation of errors would, in general, make theapproximating curve C L deviate farther and farther from the true curve C

Y

CCL

Yo

xXo

FIGURE 2.2. The simplest finite difference method provides the brokentine C L as an approximation to the actual solution C of a differentialequation.

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--

Sec. 2.3] A System of Ordinary Differential Equations 69

established. The magnitude of the contribution is an increasing function ofexisting doubt in the theory. Indeed, if the doubt is serious, the appliedmathematician himself will try to resolve the controversy by fashioningappropriate proofs.

At the educational level, the future applied mathematician should beexposed to a considerable body of mathematical theory and proof (althoughcare must be taken not to overemphasize this aspect). He should learn typicalconditions under which the operations he may have to perform are valid.Moreover, he should become aware of additional mathematical conceptsthat may someday form a suitable framework for his theories or calculations.For example, some applied mathematicians find use for the unifying geometriccharacter of the function space concepts, developed in recent decades. Togive a more classical example, the Gibbs phenomenon, which can arise inpractical calculations, cannot be appreciated without a knowledge of thedistinction between pointwise and uniform convergence. (See Section 4.3.)And, to cite another possibility, it could be that a student will first becomeaware of the calculational usefulness of an integral equation formulation, orof a successive approximations approach, by virtue of having studied certainconstructive existence theorems.

Examples of the relation between pure and applied mathematics areprovided by our discussion of theorems for differential equations. Thus itis a beautiful theoretical result that under suitable conditions solutionsdepend continuously on parameters. But this result misses a very importantscientific issue, for it merely shows that the solution is changed by an arbitrarilysmall amount over a fixed time interval by a sufficiently small change in theparameter. Left untouched is the question of the ultimate effect of a givensmall change in a parameter, a question that could well be missed entirelyby one who accepts an impressive theorem as the last word. Poincare beganthe study of long term and "ultimate" effects. Much more has since beenaccomplished, both formally and rigorously, but the issues are by no meansresolved. [See Moser (1973).]

Another example of the relation between pure and applied mathematicsstems from the fact that standard existence and analyticity theorems forsystems of ordinary differential equations are valid only for a sufficientlyshort interval of time. But if these equations describe the trajectories ofinteracting particles, it seems likely that the theorems should usually holdfor arbitrarily long time intervals. For a discussion of classical results onsuch matters see Chapter 16 of E. T. Whittaker's Analytical Dynamics(New York: Cambridge V.P., 1927).

To achieve a degree of balance, we presented a few formal proofs in thischapter. But such proofs can be found in many fine books. Hence proofs aremainly omitted in the remainder of the present work, since we concentrateon the relatively unexplored interaction between science and mathematicsthat is the core of an applied mathematician's profession.

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