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BOUNDARY VALUE PROBLEMS AND FOURIER EXPANSIONS
By Charles R. MacCluer Dover Publications, Inc.
Copyright © 2004 Charles R. MacCluer
All rights reserved.
ISBN: 978-0-486-15317-9
CHAPTER 1
Preliminaries
We begin with a review of partial derivatives and their two chain rules. Several examples of partial differential equations (PDEs) are solved. The relaxing temperatures within a block are obtained by a numerical model which reveals the PDE governing heat conduction. Finally, the important V operator is introduced and the 'big six' PDEs are displayed.
§1.1 Partial Derivatives
The most interesting problems involve multiple degrees of freedom: vibrating structures, changing temperatures within solids, voltage potentials within regions, orbiting electrons, growing economies or populations, etc. Astonishingly accurate models of these complicated phenomena have been given using partial differential equations — as relations among the various partial derivatives of measured quantities.
Recall that the partial derivative at (x0, y0) of a function f of the two real variables x, y with respect to (say) the first variable x is the limit
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(1.1)
when it exists. The symbol fx is a common alternate notation for the partial derivative [partial derivative]f/[partial derivative]x. So for instance, if f(x, y) = x3 + y2 + x7y5 + 1, then fx = 3x2 + 7x6y5, while fy = 2y + 5x7y4. The partial fx represents the sensitivity of f to changes in x while holding y fixed.
More generally, the directional derivative of f at (x0, y0) in the direction of the unit vector v = (a, b) is the limit
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(1.2)
when it exists. The existence of the directional derivative in all directions v from (x0, y0) is guaranteed when f is differentiable at (x0, y0, i.e., when f is well approximated by a plane nearby (x0, y0) — see [Apostol]. It is easy to derive the following formula for the directional derivative:
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(1.3)
(Exercise 1.1). This formula for the directional derivative generalizes to n variables x = (x1, x2, ..., xn):
Dvf(x0) = [nabla]f(x0) · v,
(1.4)
where [nabla]f is the gradient of f, i.e, the vector field
[nabla]f = ([partial derivative]f/[partial derivative]x1, [partial derivative]f/[partial derivative]x2, ..., [partial derivative]f/[partial derivative]xn),
and where · is dot product. See §1.5. Because
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(1.5)
the gradient points in the direction of the maximal increase of f.
The formula for the directional derivative is, in turn, a special case of the first of two chain rules for partial derivatives.
First Chain Rule. Suppose x = x(t) is a curve in n-space that is differentiable at t = t0, and that f(x) is a function of n variables differentiable at x0 = x(t0). Then
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(1.6)
In more familiar notation,
df/dt = [partial derivative]f/[partial derivative]x dx/dt + [partial derivative]f/[partial derivative]y dy/dt + [partial derivative]f/[partial derivative]z dz/dt.
Consequently,
the gradient is normal to all contour surfaces f = c.
The first chain rule is itself a special case.
Second Chain Rule. Suppose f(u) is a differentiable function of the n variables u = (u1, u2, ..., un), while each ui is itself a differentiable function of the r variables x = (x1, x2, ..., xr). Then
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(1.7)
Review these chain rules by working Exercises 1.2-1.6 and 1.19-1.22.
§1.2 Several Example PDEs
Let us work through several simple examples of partial differential equations, each of some physical importance.
Example 1. Consider the simple second order PDE
uxy = 0, -∞ < x, y < ∞
(1.8)
Since (ux)y = 0, ux cannot depend on y, and so ux = h(x). But then integrating with respect to x yields that u must be of the form
u(x, y) = f(x) + g(y).
(1.9)
Conversely note that any function of the form (1.9) with f and g differentiable is indeed a solution of (1.8).
If we happen to know the values of u(x, y0) and u(x0, y) on the two lines x = x0 and y = y0, then we can recover the functions f and g uniquely within constants that add to 0 (Exercise 1.7). We will see this simple PDE again when we solve the all-important wave equation in Chapter 4.
Example 2. The first order PDE
aux + buy = 0, -∞ < x, y < ∞,
(1.10)
where a and b are constants, is called the transport equation for reasons explained below. Observe that this equation is a geometric statement: The directional derivative
[nabla]u · (a, b) = 0
of u is zero in the direction (a, b), i.e., u = u(x, y) is constant along curves with tangent vector (a, b). This means u is constant along each line ay - bx = c. But then u = u(x, y) is determined solely by the value c, i.e.,
u(x, y) = f(ay - bx).
(1.11)
Conversely, note that any u of the form (1.11) clearly satisfies the PDE (1.10) as long as f is differentiable. These lines ay - bx = c are called the characteristic lines of the PDE (1.10).
Physical realization of the transport equation. Think of fluid flowing through a pipe with velocity v = v(x, t) at location x at time t. This fluid is carrying immiscible particles of density p = p(x, t) per unit length at location x at time t. The total particle mass m within the tube between x = a and x = b is therefore
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(1.12)
By Leibniz's principle (§3.2) we may differentiate past the integral to find the instantaneous rate of change of the mass within this length of pipe:
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(1.13)
On the other hand, this gain of mass is the net flow in from the left less the flow out from the right, i.e.,
mt = ρ(a, t)v(a, t) - ρ(b, t)v(b, t).
(1.14)
Equating and dividing by b - a gives that
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(1.15)
In the limit, as b -> a, we have conservation of mass:
(ρv)x = -ρt,
(1.16)
or in the case velocity v is constant, we have transport:
vρx + ρt = 0.
(1.17)
(Continues...)
Excerpted from BOUNDARY VALUE PROBLEMS AND FOURIER EXPANSIONS by Charles R. MacCluer. Copyright © 2004 Charles R. MacCluer. Excerpted by permission of Dover Publications, Inc..
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