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CHAPTER 1
INTRODUCTION
1.1 Initial-Value Problems
The theory of boundary-value problems for ordinary differential equations relies rather heavily on initial-value problems. Even more significant for the subject of this monograph is the fact that some of the most generally applicable numerical methods for solving boundary-value problems employ initial-value problems. Thus we must assume that the reader is somewhat familiar with the existence and uniqueness theory of, as well as numerical methods for solving, initial-value problems. We shall review here and in Section 1.3 some of the basic results required.
Since every nth-order ordinary differential equation can be replaced by an equivalent system of n first-order equations, we confine our attention to first-order systems of the form
u' = f(x; u).
Here u [equivalent to] (u1, u2, ..., un)T is an «-dimensional column vector with the dependent variables uk(x) as components; we say that u(x) is a vector-valued function; f(x; u) is vector-valued with components fk(x, u1, u2, ... un), which are functions of the n + 1 variables (x; u). An initial-value problem for the above system is obtained by prescribing at some point, say x = a, the value of u, say
u(a) = α.
The existence, uniqueness, and continuity properties of the solutions of such problems depend on the continuity or smoothness properties of the function f in a neighborhood of the initial point (a; α). We shall use as a measure of distance between two points in n-space the maximum norm
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However, all results are equally valid if we employ the Euclidean norm
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One of the basic results can now be stated as follows.
Theorem 1.1.1. Let the functionf (x; u) be continuous on the infinite strip
R: a ≤ x ≤ b, [absolute value of (u)] < ∞
and satisfy there a Lipschitz condition inuwith constant K, uniformly in x; that is,
|f(x; u) - f(x; v)| ≤ K|u - v| for all (x; u) and (x; v) [member of] R.
Then
(a) the initial-value problem
u' = f(x; u), u(a) = α (1.1.1)
has a unique solution u = u(x; α) defined on the interval
[a, b] [euqivalent to] {x | a ≤ x ≤ b};
(b) this solution is Lipschitz-continuous in α, uniformly in x; in fact we have
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Proof We merely sketch the proof; more details can be found in Ince (1944), pp. 62–72. If a solution exists, we obtain from (1.1.1) by integration
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Conversely, if u(x) is continuous and satisfies this integral equation, it is differentiable and hence, by differentiation, satisfies (1.1.1). We now construct a solution of this integral equation by means of the Picard iteration procedure
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From the Lipschitz continuity of f(x; u) we have
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Then, by induction, with |f(x; α)| ≤ M on [a, b], it follows that
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Now the sequence of continuous functions {u(v)(x)} can be shown to converge uniformly on [a, b] since
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The limiting function clearly satisfies the integral equation and hence (1.1.1); thus, existence is established for any α.
The uniqueness of this solution will follow from the continuity of u(x, β) by letting β -> α.
To demonstrate (b) we form
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Then, by the Lipschitz continuity of f, it follows that
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Calling
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we may write the above inequality as
E'(x) - KE(x) ≤ |α - β|.
This differential inequality may be "solved" by multiplying by the integrating factor e-K(x-a) and integrating over [a, x] to get
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The original inequality now yields (b).
In many problems of interest the function f(x; u) does not have the required continuity properties in the infinite strip R but rather in a finite domain RB: a ≤ x ≤ b, |u - α| ≤ B. In this case a similar theorem holds but the solution may exist only in some smaller interval a ≤ x ≤ b0 [equivalent to] min (b, B/M), ibid. (See Problems 1.1.4 and 1.1.5 for other, more powerful, generalizations.)
With little additional effort the result in Theorem 1.1.1 can be strengthened to show that in fact the solution u(x; a) is uniformly differentiable with respect to the initial values ak, k = 1, 2, ..., n. However, for some of our applications we require a still stronger result which shows that each derivative [partial derivative]u(x; α)/[partial derivative]αk is the solution of a specific initial-value problem for a linear system of ordinary differential equations. This is the content of the following theorem.
Theorem 1.1.2. In addition to the hypothesis of Theorem 1.1.1 let the Jacobian offwith respect touhave continuous elements on R; that is, the nth-order matrix
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is continuous on R. Then for any α the solutionu(x; α) of (1.1.1) is continuously differentiable with respect to αk, k = 1, 2, ..., n. In fact, the derivative [partial derivative]u(x; α)/[partial derivative]αk [equivalent to] [xi](k)(x) is the solution, on [a, b], of the linear system
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subject to the initial condition
[xi](a) = e(k). (1.1.2b)
(Here e(k) [equivalent to] (0, ..., 0, 1, 0, ..., 0)T is the kth unit vector in n-space.)
Proof. Again we sketch the proof; details can be found in Birkhoff and Rota (1962), pp. 123-124. For arbitrarily small |h| > 0 we define the difference quotient
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in terms of the indicated solutions of the initial-value problems (1.1.1) (with initial data α and α + he(k)). Then it follows that
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where we have used Taylor's theorem and |δ| -> 0 as h -> 0. But by (b) of Theorem 1.1.1, |η| must be bounded as h -> 0 and so the above equation formally tends to Equation (1.1.2a). We also note that η(a, h) = e(k) for all |h| ≥ 0. Since the system (1.1.2a) is linear, we can show that the solution η(x, h) of the above initial-value problem converges, as h -> 0, to the solution of (1.1.2). That the solution [xi](x) [equivalent to] [xi](x; α) of (1.1.2) is continuous in α follows easily as in the proof of Theorem 1.1.1.
The system (1.1.2a) is known as the variational equation for the system u' = f(x; u). We note that it can be obtained formally by assuming u to depend upon some parameter, say c, differentiating with respect to this parameter and setting [partial derivative]u/[partial derivative]c [equivalent to] [xi]. If this is done to both the system and the initial condition in (1.1.1), using c = ak as a parameter, we obtain the variational problem (1.1.2).
For some of our later studies of eigenvalue problems, we shall require additional results for initial-value problems which contain a parameter in the equations, that is, problems of the form
u' = f(λ, x; u), u(a) = α, (1.1.3)
where f is now a vector-valued function of the n + 2 variables (λ, x; u). If for all λ in λ* ≤ λ ≤ λ* the hypothesis of Theorem 1.1.1 is satisfied, and in addition f(λ, x; u) depends Lipschitz-continuously on λ, uniformly for (x; u) [member of] R, then the initial-value problem (1.1.3) has a unique solution, u(λ, x), which is Lipschitz-continuous in λ. Again, this can be strengthened to continuous differentiability of the solution with respect to λ. But if g(λ, x; u) = [partial derivative]f(λ, x; u)/[partial derivative]λ and F(λ, x; u) = [partial derivative]f(λ, x; u)/[partial derivative]u are continuous, we can show that [ci](x) [equivalent to] [partial derivative]u(λ, x)/[partial derivative]λ is a solution of the linear variational problem
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Note that this problem is obtained by formal differentiation of the initial-value problem (1.1.3). The initial condition in the variational problem (1.1.4) is homogeneous, while the variational equation is not; this is just the reverse of the situation in the variational problem (1.1.2).
(Continues…)
Excerpted from "Numerical Methods for Two-Point Boundary-Value Problems"
by .
Copyright © 1992 Herbert B. Keller.
Excerpted by permission of Dover Publications, Inc..
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