Introduction to Asymptotic Approximations
We will be interested in this book in using what are known as asymptotic expansions to find approximate solutions of differential equations. Usually our efforts will be directed toward constructing the solution of a problem with only occasional regard for
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Introduction to Asymptotic Approximations
1.1 Introduction We will be interested in this book in using what are known as asymptotic expansions to find approximate solutions of differential equations. Usually our efforts will be directed toward constructing the solution of a problem with only occasional regard for the physical situation it represents. However, to start things off, it is worth considering a typical physical problem to illustrate where the mathematical problems originate. A simple example comes from the motion of an object projected radially upward from the surface of the Earth. Letting x(t) denote the height of the object, measured from the surface, from Newton’s second law we obtain the following equation of motion: gR2 d2 x =− , 2 dt (x + R)2
for 0 < t,
(1.1)
where R is the radius of the Earth and g is the gravitational constant. We will assume the object starts from the surface with a given upward velocity, so x(0) = 0 and x (0) = v0 , where v0 is positive. The nonlinear nature of the preceding ordinary differential equation makes finding a closed-form solution difficult, and it is natural to try to see if there is some way to simplify the equation. For example, if the object does not get far from the surface, then one might try to argue that x is small compared to R and the denominator in (1.1) can be simplified to R2 . This is the type of argument often made in introductory physics and engineering texts. In this case, x ≈ x0 , where x0 = −g for x0 (0) = 0 and x0 (0) = v0 . Solving this problem yields 1 x0 (t) = − gt2 + v0 t. 2
(1.2)
One finds in this case that the object reaches a maximum height of v02 /2g and comes back to Earth when t = 2v0 /g (Fig. 1.1). The difficulty with M.H. Holmes, Introduction to Perturbation Methods, Texts in Applied Mathematics 20, DOI 10.1007/978-1-4614-5477-9 1, © Springer Science+Business Media New York 2013
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1 Introduction to Asymptotic Approximations
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Figure 1.1 Schematic of solution x0 (t) given in (1.2). This solution comes from the linearization of (1.1) and corresponds to the motion in a uniform gravitational field
this reduction is that it is unclear how to determine a correction to the approximate solution in (1.2). This is worth knowing since we would then be able to get a measure of the error made in using (1.2) as an approximation and it would also be possible to see just how the nonlinear nature of the original problem affects the motion of the object. To make the reduction process more systematic, it is first necessary to scale the variables. To do this, let τ = t/tc and y(τ ) = x(t)/xc , where tc is a characteristic time for the problem and xc is a characteristic value for the solution. We have a lot of freedom in picking these constants, but they should be representative of the situation under consideration. Based on what is shown in Fig. 1.1, we take tc = v0 /g and xc = v02 /g. Doing this the problem transforms into the following: 1 d2 y =− , dτ 2 (1 + εy)2
for 0 < τ,
(1.3)
y(0) = 0
y (0) = 1.
(1.4)
where and
v02 /Rg
In (1.3), the
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