a

lima’0 ya = y in L2 [0, 1] .

The derivative of these functions also converges in L2 .

y™

y™

a

a

ya ’ y in L2 [0, 1] .

If we want L to be closed, we should therefore extend the domain of de¬nition

of L to include functions with non-vanishing endpoint derivative. We can also

use this method to add to the domain of L functions that are only piecewise

di¬erentiable ” i.e. functions with a discontinuous derivative.

Now consider what happens if we try to extend the domain of

d

D(L) = {y, y ∈ L2 : y(0) = 0},

L= , (9.109)

dx

to include functions that do not vanish at the endpoint. Take a sequence of

functions ya that vanish at the origin, and converge in L2 to a function that

does not vanish at the origin:

9.6. SOME FUNCTIONAL ANALYSIS 275

y

ya

1 1

a

lima’0 ya = y in L2 [0, 1].

Now the derivatives converge towards the derivative of the limit function ”

together with a delta function near the origin. The area under the functions

|ya (x)|2 grows without bound and the sequence Lya becomes in¬nitely far

from the derivative of the limit function when distance is measured in the L2

norm.

y™

a

δ(x)

1/a

a

ya ’ δ(x), but the delta function is not an element of L2 [0, 1] .

We therefore cannot use closure to extend the domain to include these func-

tions.

This story repeats for di¬erential operators of any order: If we try to

impose boundary conditions of too high an order, they are washed out in the

process of closing the operator. Boundary conditions of lower order cannot

be eliminated, however, and so make sense as statements involving functions

in L2 .

276 CHAPTER 9. INTEGRAL EQUATIONS

9.7 Series Solutions

9.7.1 Neumann Series

The geometric series

S = 1 ’ x + x2 ’ x3 + · · · (9.110)

converges to 1/(1 + x) provided |x| < 1. Suppose we wish to solve

(I + »K)• = f (9.111)

where K is a an integral operator. It is then natural to write

• = (I + »K)’1 f = (1 ’ »K + »2 K 2 ’ »3 K 3 + · · ·)f (9.112)

where

K 2 (x, y) = K 3 (x, y) =

K(x, z)K(z, y) dz, K(x, z1 )K(z1 , z2 )K(z2 , y) dz1dz2 ,

(9.113)

and so on. This Neumann series will converge, and yield a solution to the

problem, provided that » K < 1.

9.7.2 Fredholm Series

A familiar result from high-school algebra is Cramer™s rule which gives the

solution of a set of linear equations in terms of ratios of determinants. For

example, the system of equations

a11 x1 + a12 x2 + a13 x3 = b1 ,

a21 x1 + a22 x2 + a23 x3 = b2 ,

a31 x1 + a32 x2 + a33 x3 = b3 ,

has solution

b a12 a13 a b1 a13 a a12 b1

11 1 11 1 11

x1 = b2 a22 a23 , x2 = a21 b2 a23 , x3 = a21 a22 b2 ,

D D D

b3 a32 a33 a31 b3 a33 a31 a32 b3

where

a11 a12 a13

D = a21 a22 a23 .

a31 a32 a33

9.7. SERIES SOLUTIONS 277

Although not as useful as standard Gaussian elimination, Cramer™s rule is

useful as it is a closed-form solution. It is equivalent to the statment that

the inverse of a matrix is given by the transposed matrix of the co-factors,

divided by the determinant.

A similar formula for integral quations was given by Fredholm. The

equations he considered were of the form

(I + »K)• = f. (9.114)

We motivate Fredholm™s formula by giving an expansion for the determinant

of a ¬nite matrix. Let

···

1 + »K11 »K12 »K1n

···

»K21 1 + »K22 »K2n

D(») = det (I + »K) ≡ , (9.115)

. . .

..

. . .

.

. . .

· · · 1 + »Knn

»Kn1 »Kn2

then n

»m

D(») = Am , (9.116)

m!

m=0

where A0 = 1, A1 = tr K ≡ Kii ,

i

Ki 1 i 1 Ki 1 i 2 Ki 1 i 3

n n

Ki 1 i 1 Ki 1 i 2

A2 = , A3 = Ki 2 i 1 Ki 2 i 2 Ki2 i3 . (9.117)

Ki 2 i 1 Ki 2 i 2

Ki 3 i 1 Ki 3 i 2 Ki 3 i 3