The approach we employ is the method of successive approximations, which was developed by the French mathematician Emile Picard (1856–1941). This method is based on the fact that the function y(x) satisfies the initial value problem in (1) on the open interval I containing if and only if it satisfies the integral equation
for all x in I. In particular, if y(x) satisfies Eq. (4), then clearly and differentiation of both sides in (4)—using the fundamental theorem of calculus—yields the differential equation .
To attempt to solve Eq. (4), we begin with the initial function
and then define iteratively a sequence of functions that we hope will converge to the solution. Specifically, we let
In general, is obtained by substitution of for y in the right-hand side in Eq. (4):
Suppose we know that each of these functions is defined on some open interval (the same for each n) containing and that the limit
exists at each point of this interval. Then it will follow that
and, hence, that
provided that we can validate the interchange of limit operations involved in passing from (9) to (10). It is therefore reasonable to expect that, under favorable conditions, the sequence defined iteratively in Eqs. (5) and (7) will converge to a solution y(x) of the integral equation in (4), and hence to a solution of the original initial value problem in (1).
To apply the method of successive approximations to the initial value problem
we write Eqs. (5) and (7), thereby obtaining
The iteration formula in (12) yields
and
It is clear that we are generating the sequence of partial sums of a power series solution; indeed, we immediately recognize the series as that of There is no difficulty in demonstrating that the exponential function is indeed the solution of the initial value problem in (11); moreover, a diligent student can verify (by using a proof by induction on n) that obtained in the aforementioned manner, is indeed the nth partial sum for the Taylor series with center zero for .
To apply the method of successive approximations to the initial value problem
we write Eqs. (5) and (7) as in Example 1. Now we obtain
The iteration formula in (14) yields
and
It is again clear that we are generating partial sums of a power series solution. It is not quite so obvious what function has such a power series representation, but the initial value problem in (13) is readily solved by separation of variables:
In some cases, it may be necessary to compute a much larger number of terms, either in order to identify the solution or to use a partial sum of its series with large subscript to approximate the solution accurately for x near its initial value. Fortunately, computer algebra systems such as Maple and Mathematica can perform the symbolic integrations (as opposed to numerical integrations) of the sort in Examples 1 and 2. If necessary, you could generate the first hundred terms in Example 2 in a matter of minutes.
In general, of course, we apply Picard’s method because we cannot find a solution by elementary methods. Suppose that we have produced a large number of terms of what we believe to be the correct power series expansion of the solution. We must have conditions under which the sequence provided by the method of successive approximations is guaranteed in advance to converge to a solution. It is just as convenient to discuss the initial value problem
for a system of m first-order equations, where
It turns out that with the aid of this vector notation (which we introduced in Section 7.1), most results concerning a single [scalar] equation can be generalized readily to analogous results for a system of m first-order equations, as abbreviated in (15). Consequently, the effort of using vector notation is amply justified by the generality it provides.
The method of successive approximations for the system in (15) calls for us to compute the sequence of vector-valued functions of t,
defined iteratively by
Recall that vector-valued functions are integrated componentwise.
Consider the m-dimensional initial value problem
for a homogeneous linear system with constant coefficient matrix A. The equations in (16) take the form
Thus
and
We have therefore obtained the first several partial sums of the exponential series solution
of (17), which was derived earlier in Section 8.1.
The key to establishing convergence in the method of successive approximations is an appropriate condition on the rate at which f(x, t) changes when x varies but t is held fixed. If R is a region in -dimensional (x, t)-space, then the function f(x, t) is said to be Lipschitz continuous on R if there exists a constant such that
if and are points of R. Recall that the norm of an m-dimensional point or vector x is defined to be
Then is simply the Euclidean distance between the points and .
Let and let R be the strip in the xy-plane. If and are both points of R, then
because for all t and if and are both in the interval [0, 2]. Thus f satisfies the Lipschitz condition in (20) with and is therefore Lipschitz continuous in the strip R.
Let on the rectangle R consisting of the points (x, t) in the xt-plane for which and Then, taking and we find that
Because as we see that the Lipschitz condition in (20) cannot be satisfied by any (finite) constant Thus the function f, though obviously continuous on R, is not Lipschitz continuous on R.
Suppose, however, that the function f(x, t) has a continuous partial derivative on the closed rectangle R in the xt-plane, and denote by k the maximum value of on R. Then the mean-value theorem of differential calculus yields
for some in so it follows that
because Thus a continuously differentiable function f(x, t) defined on a closed rectangle is Lipschitz continuous there. More generally, the multivariable mean value theorem of advanced calculus can be used similarly to prove that a vector-valued function f(x, t) with continuously differentiable component functions on a closed rectangular region R in (x, t)-space is Lipschitz continuous on R.
The function is Lipschitz continuous on any closed [bounded] region in the xt-plane. But consider this function on the infinite strip R consisting of the points (x, t) for which and x is arbitrary. Then
Because can be made arbitrarily large, it follows that f is not Lipschitz continuous on the infinite strip R.
If I is an interval on the t-axis, then the set of all points (x, t) with t in I is an infinite strip or slab in -space (as indicated in Fig. A.1). Example 6 shows that Lipschitz continuity of f(x, t) on such an infinite slab is a very strong condition. Nevertheless, the existence of a solution of the initial value problem
under the hypothesis of Lipschitz continuity of f in such a slab is of considerable importance.
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