G(x, y) = (9.2)

1

’ L), 0 ¤ y ¤ x ¤ L,

y(x

L

so that

d2

’ 2 G(x, y) = δ(x ’ y). (9.3)

dx

Then we can pretend that V (x)u(x) in the di¬erential equation is a known

source term, and substitute it for “f (x)” in the usual Green function solution.

We end up with

L

u(x) + » G(x, y)V (y)u(y) dx = 0. (9.4)

0

255

256 CHAPTER 9. INTEGRAL EQUATIONS

This integral equation for u has not not solved the problem, but is equivalent

to the original problem. Note, in particular, that the boundary conditions

are implicit in this formulation: if we set x = 0 or L in the second term, it

becomes zero because the Green function is zero at those points. The integral

equation then says that u(0) and u(L) are both zero.

An intitial value problem: Consider essentially the same di¬erential equation

as before, but now with intial data:

’u + V (x)u = 0, u(0) = 0, u (0) = 1. (9.5)

In this case, we claim that the inhomogeneous integral equation

x

u(x) ’ (x ’ t)V (t)u(t) dt = x, (9.6)

0

is equivalent to the given problem. Let us check the claim. First, the initial

conditions. Rewrite the integral equation as

x

(x ’ t)V (t)u(t) dt,

u(x) = x + (9.7)

0

so it is manifest that u(0) = 0. Now di¬erentiate to get

x

u (x) = 1 + V (t)u(t) dt. (9.8)

0

This shows that u (0) = 1, as required. Di¬erentiating once more con¬rms

that u = V (x)u.

These examples reveal that one advantage of the integral equation for-

mulation is that the boundary or intial value conditions are automatically

encoded in the integral equation itself, and do not have to be added as riders.

9.2 Classi¬cation of Integral Equations

The classi¬cation of linear integral equations is best described by a list:

A) i) Limits on integrals ¬xed ’ Fredholm equation.

ii) One integration limit is x ’ Volterra equation.

B) i) Unkown under integral only ’ Type I.

ii) Unknow also outside integral ’ Type II.

C) i) Homogeneous.

9.3. INTEGRAL TRANSFORMS 257

ii) Inhomogeneous.

For example,

L

u(x) = G(x, y)u(y) dx (9.9)

0

is a Type II homogeneous Fredholm equation, whilst

x

(x ’ t)V (t)u(t) dt

u(x) = x + (9.10)

0

is a Type II inhomogeneous Volterra equation.

The equation

b

f (x) = K(x, y)u(y) dy, (9.11)

a

an inhomogeneous Type I Fredholm equation, is analogous to the matrix

equation

Kx = b. (9.12)

On the other hand, the equation

1 b

u(x) = K(x, y)u(y) dy, (9.13)

» a

a homogeneous Type II Fredholm equation, is analogous to the matrix eigen-

value problem

Kx = »x. (9.14)

Finally,

x

f (x) = K(x, y)u(y) dy (9.15)

a

an inhomogeneous Type I Volterra equation, is the analogue of a system of

linear equations involving an upper triangular matrix.

9.3 Integral Transforms

When a Fredholm Kernel is of the form K(x ’ y), with x and y taking values

on the entire real line, then it is translation invariant, and we can solve the

integral equation by using the Fourier transformation

∞

u(x)eikx dx

u(k) = F (u) =

˜ (9.16)

’∞

dk

∞

’1

u(k)e’ikx

u(x) = F (˜) =

u ˜ (9.17)

2π

’∞

258 CHAPTER 9. INTEGRAL EQUATIONS

Integral equations involving translation invariant Volterra kernels usually

succumb to a Laplace transform

∞

u(x)e’px dx

u(p) = L(u) =

˜ (9.18)

0

1 γ+i∞

’1

u(p)epx dp.

u(x) = L (˜) =

u ˜ (9.19)

2πi γ’i∞

The Laplace inversion formula is the Bromwich contour integral, where γ is

chosen so that all the sigularities of u(p) lie to the left of the contour. In

˜

practice one ¬nds the inverse Laplace transform by using a table of Laplace

transforms, such as the Bateman tables of integral transforms mentioned in

the introduction to chapter 8.

For kernels of the form K(x/y) the Mellin transform,

∞

u(x)xσ’1 dx

u(σ) = M(u) =

˜ (9.20)

0

1 γ+i∞

’1

u(σ)x’σ dσ,

u(x) = M (˜) =

u ˜ (9.21)

2πi γ’i∞