3

= (‚x (q + »)u) v + 3(q + »)‚x (uv) + (‚x (q + »)v) u.

Here, in passing from the ¬rst to second line, we have used the di¬erential

equation obeyed by u and v. We can re-express the second line as

13

(q‚x + ‚x q ’ ‚x )R(x) = ’2»‚x R(x). (5.140)

2

5.6. LOCALITY AND THE GELFAND-DIKII EQUATION 143

This is known as the Gelfand-Dikii equation. Using it we can ¬nd an ex-

pansion for the diagonal element R(x) in terms of q and its derivatives. We

√

begin by observing that for q(x) ≡ 0 we know that R(x) = 1/(2 »). We

therefore conjecture that we can expand

1 b1 (x) b2 (x) bn (x)

+ · · · + (’1)n

R(x) = √ 1 ’ + ··· .

+

(2»)2 (2»)n

2»

2»

If we insert this expansion into (5.140) we see that we get the recurrence

relation

13

(q‚x + ‚x q ’ ‚x )bn = ‚x bn+1 . (5.141)

2

We can therefore ¬nd bn+1 from bn by means of a single integration. Re-

markably, ‚x bn+1 is always the exact derivative of a polynomal in q and its

derivatives. Further, the integration constants must be be zero so that we

recover the q ≡ 0 result. If we carry out this process, we ¬nd

b1 (x) = q(x),

3 q(x)2 q (x)

’

b2 (x) = ,

2 2

5 q(x)3 5 q (x)2 5 q(x) q (x) q (4) (x)

’ ’

b3 (x) = + ,

2 4 2 4

35 q(x)4 35 q(x) q (x)2 35 q(x)2 q (x) 21 q (x)2

’ ’

b4 (x) = +

8 4 4 8

7 q (x) q (3) (x) 7 q(x) q (4)(x) q (6) (x)

’

+ + , (5.142)

2 4 8

and so on. (Note how the terms in the expansion are graded: Each bn

is homogeneous in powers of q and its derivatives, provided we count two

x derivatives as being worth one q(x).) Keeping a few terms in this series

expansion can provide an e¬ective approximation for G(x, x), but, in general,

the series is not convergent, being only an asymptotic expansion for R(x).

A similar strategy produces expansions for the diagonal element of the

Green function of other one-dimensional di¬erential operators. Such gradient

expansions also exist in in higher dimensions but the higher-dimensional

Seeley-coe¬cient functions are not as easy to compute. Gradient expansions

for the o¬-diagonal elements also exist, but, again, they are harder to obtain.

144 CHAPTER 5. GREEN FUNCTIONS

Chapter 6

Partial Di¬erential Equations

Most di¬erential equations of physics involve quantities depending on both

space and time. Inevitably they involve partial derivatives, and so are partial

di¬erential equations (PDE™s).

6.1 Classi¬cation of PDE™s

We will focus on second order equations in two variable such as the wave

equation

‚2• 1 ‚2•

’ 2 2 = f (x, t), (Hyperbolic) (6.1)

‚x2 c ‚t

Laplace or Poisson™s equation

‚2• ‚2•

+ 2 = f (x, y), (Elliptic) (6.2)

‚x2 ‚y

or Fourier™s heat equation

‚2• ‚•

’κ = f (x, t). (Parabolic) (6.3)

‚x2 ‚t

What do the names hyperbolic, elliptic and parabolic mean? Recall from

high-school co-ordinate geometry that a quadratic curve

ax2 + 2bxy + cy 2 + f x + gy + h = 0 (6.4)

represents a hyperbola, an ellipse or a parabola depending on whether the

discriminant, ac ’ b2 , is less than zero, greater than zero, or equal to zero.

145

146 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS

Later in life we learn to say that this means that the matrix

ab

(6.5)

bc

has signature (+, ’), (+, +) or (+, 0).

Similarly, the equation

‚2• ‚2• ‚2•

a(x, y) 2 + 2b(x, y) + c(x, y) 2 + (lower orders) = 0, (6.6)

‚x ‚x‚y ‚y

is said to hyperbolic, elliptic, or parabolic at a point (x, y) if

a(x, y) b(x, y)

= (ac ’ b2 )|x,y , (6.7)

b(x, y) c(x, y)

is less than, greater than, or equal to zero, respectively. This classi¬cation

helps us understand what sort of initial or boundary data we need to specify

the problem.

There are three broad classes of boundary conditions:

a) Dirichlet boundary conditions: The value of the dependent vari-

able is speci¬ed on the boundary.

b) Neumann boundary conditions: The normal derivative of the de-

pendent variable is speci¬ed on the boundary.

c) Cauchy boundary conditions: Both the value and the normal deriva-

tive of the dependent variable are speci¬ed on the boundary.

Less commonly met with are:

d) Robin boundary conditions: The value of a linear combination of

the dependent variable and the normal derivative of the dependent

variable is speci¬ed on the boundary.

Cauchy boundary conditions are analogous to the initial conditions for a

second-order ordinary di¬erential equation. These are given at one end of

the interval only. The other three classes of boundary condition are higher-

dimensional analogues of the conditions we impose on an ODE at both ends

of the interval.

Each class of PDE™s requires a di¬erent class of boundary conditions in

order to have a unique, stable solution.

1) Elliptic equations require either Dirichlet or Neumann boundary con-

ditions on a closed boundary surrounding the region of interest. Other

boundary conditions are either insu¬cient to determine a unique solu-

tion, overly restrictive, or lead to instabilities.

6.1. CLASSIFICATION OF PDE™S 147

2) Hyperbolic equations require Cauchy boundary conditions on a open

surface. Other boundary conditions are either too restrictive for a

solution to exist, or insu¬cient to determine a unique solution.

3) Parabolic equations require Dirichlet or Neumann boundary condi-

tions on a open surface. Other boundary conditions are too restrictive.

6.1.1 Cauchy Data

Given a second-order ordinary di¬erential equation

p0 y + p 1 y + p 2 y = f (6.8)

with initial data y(a), y (a) we can construct the solution incrementally. We

take a step δx = and use the initial slope to ¬nd y(a + ) = y(a) + y (a).