Chow T.L. Mathematical Methods for Physicists: A Concise Introduction

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The solution blows up when   117:96 or   2:04. These are the eigenvalues of

the integral equation.

Fredholm found that if: (1) f x is continuous, (2) K x; t is piecewise contin-

uous, (3) the integrals

x; tdxdt;

tdt exist, and (4) the integrals

x; tdt and

t; xdt are bounded, then the following theorems apply:

(a) Either the inhomogeneous equation

uxf x

Kx; tu tdt

has a unique solution for any function f x is not an eigenvalue), or the

homogeneous equation

ux

Kx; tutdt

has at least one non-trivial solution corresponding to a particular value of .

In this case,  is an eigenvalue and the solution is an eigenfunction.

(b)If is an eigenvalue, then  is also an eigenvalue of the transposed equation

ux

Kt; xu tdt;

and, if  is not an eigenvalue, then  is also not an eigenvalue of the

transposed equation

uxf x

Kt; xu tdt:

(c)If is an eigenvalue, the inhomogeneous equation has a solution if, and only

if,

uxf xdx  0

for every function f x.

We refer the readers who are interested in the proof of these theorems to the

book by R. Courant and D. Hilbert (Methods of Mathematical Physics, Vol. 1,

Wiley, 1961).

Neumann series solutions

This method is due largely to Neumann, Liouville, and Volterra. In this method

we solve the Fredholm equation (11.2)

uxf x

Kx; tu tdt

416

SIMPLE LINEAR INTEGRAL EQUATIONS

by iteration or successive approximations, and begin with the approximation

uxu

xf x:

This approximation is equivalent to saying that the constant  or the integral is

small. We then put this crude choice into the integral equation (11.2) under the

integral sign to obtain a second approximation:

xf x

Kx; tf tdt

and the process is then repeat ed and we obtain

xf x

Kx; tf tdt  

Kx; tKt; t

f t

dt

dt:

We can continue iterating this process, and the resulting series is known as the

Neumann series, or Neumann solution:

uxf x

Kx; tf tdt  

Kx; tKt; t

f t

dt

dt :

This series can be written formally as

x

i1



x; 11:14

where

xu

xf x;

x

Kx; t

f t

dt

;

x

Kx; t

Kt

; t

f t

dt

;

x



Kx; t

Kt

; t

Kt

nÿ1

; t

f t

dt

dt

;

11:15

The series (11.14) will converge for suciently small , when the kernel Kx; t is

bounded. This can be checked with the Cauchy ratio test (Problem 11.4).

Example 11.2

Use the Neumann method to solve the integral equation

uxf x

ÿ1

Kx; tutdt; 11:16

417

SOME METHODS OF SOLUTION

where

f xx; Kx; tt ÿ x:

Solution: We begin with

xf xx:

Then

xx 

ÿ1

t ÿ xtdt  x 

Putting u

x into Eq. (11.16) under the integral sign, we obtain

xx 

ÿ1

t ÿ x t 



dt  x 

Repeating this process of substituting back into Eq. (11.16) once more, we obtain

xx 

We can improve the approximation by iterating the process, and the convergence

of the resulting series (solution) can be checked out with the ratio test.

The Neumann method is also applicable to the Volterra equation, as shown by

the following example.

Example 11.3

Use the Neumann method to solve the Volterra equation

ux1  

utdt:

Solution: We begin with the zeroth approximation u

x1. Then

x1  

tdt  1  

dt  1  x:

This gives

x1  

tdt  1  

1  tdt  1  x 



;

similarly,

x1  

1  t 





dt  1  t 







418

SIMPLE LINEAR INTEGRAL EQUATIONS

By induction

x

k1



When n !1, u

x approaches

uxe

x

Transformation of an integral equatio n into a diÿerential equation

Sometimes the Volterra integral equation can be transformed into an ordinary

diÿerential equation which may be easier to solve than the original integral equa-

tion, as shown by the following example.

Example 11.4

Consider the Volterra integral equation ux2x  4

t ÿ xutdt. Before we

transform it into a diÿerential equation, let us recall the following very useful

formula: if

I

b

a

f x;dx;

where a and b are continuous and at least once diÿerentiable functions of , then

dI 

d

 f b;

d

ÿ f a;

d



@f x;

@

dx:

With the help of this formula, we obtain

ux2  4 t ÿ xut

tx

utdt



 2 ÿ 4

utdt:

Diÿerentiating again we obtain

ux

ÿ4ux:

This is a diÿerentiation equation equivalent to the original integral equation, but

its solution is mu ch easier to ®nd:

uxA cos 2x  B sin 2x;

where A and B are integration constants. To determine their values, we put the

solution back into the original integral equation under the integral sign, and then

419

SOME METHODS OF SOLUTION

integration gives A  0 and B  1. Thus the solution of the original integral

equation is

uxsin 2x:

Laplace transform solution

The Volterra integral eq uation can sometime be solved with the help of the

Laplace transformation and the convolution theorem. Before we consider the

Laplace transform solution, let us review the convolution theorem. If f

x and

x are two arbitrary functions, we de®ne their convolution (faltung in German)

to be

gx

ÿ1

yf

x ÿ ydy:

Its Laplace transform is

Lgx  Lf

xLf

x:

We now consider the Volterra equation

uxf x

Kx; tutdt

 f x

gx ÿ tutdt; 11:17

where Kx ÿ tgx ÿ t, a so-called displacement kernel. Taking the Laplace

transformation and using the convolution theorem, we obtain

gx ÿ t utdt



 Lgx ÿ tLutGpUp;

where UpLut 

ÿpt

utdt, and similarly for Gp. Thus, taking the

Laplace transformation of Eq. (11.17), we obtain

UpFpGpUp

Up

Fp

1 ÿ Gp

Inverting this we obtain ut:

utL

ÿ1

Fp

1 ÿ Gp



420

SIMPLE LINEAR INTEGRAL EQUATIONS

Fourier transform solution

If the kernel is a displacement kernel and if the limits are ÿ1 and 1, we can use

Fourier transforms. Consider a Fredholm equation of the second kind

uxf x

ÿ1

Kx ÿ tutdt: 11:18

Taking Fourier transforms (indicated by overbars)



2

ÿ1

dxf xe

ÿipx





f p; etc:;

and using the convolution theorem

ÿ1

f tgx ÿ tdt 

ÿ1



f y



gye

ÿiyx

dy;

we obtain the transform of our integral equation (11.18):



up



f p



Kp



up:

Solving for



up we obtain



up



f p

1 ÿ 



Kp

If we can invert this equation, we can solve the original integral equation:

ux



2

ÿ1

f te

ÿixt

1 ÿ



2

Kt

: 11:19

The Schmidt±Hilbert method of solution

In many physical problems, the kernel may be symmetric. In such cases, the

integral equation may be solved by a method quite diÿerent from any of those

in the preceding section. This method, devised by Schmidt and Hilbert, is based

on considering the eigenfunctions and eigenvalues of the homogeneous integral

equation.

A kernel Kx; t is said to be symmetric if Kx; tKt; x and Hermitian if

Kx; yK*t; x. We shall limit our discussion to such kernels.

(a) The homogeneous Fredholm equati on

ux

Kx; tu tdt:

421

THE SCHMIDT±HILBERT METHOD OF SOLUTION

A Hermitian kernel has at least one eigenvalue and it may have an in®nite num-

ber. The proof will be omitted and we refer interested readers to the book by

Courant and Hibert mentioned earlier (Chapter 3).

The eigenvalues of a Hermitian kernel are real, and eigenfunctions belonging to

diÿerent eigenvalues are orthogonal; two functions f x and gx are said to be

orthogonal if

f *xgxdx  0:

To prove the reality of the eigenvalue, we multiply the homogeneous Fredholm

equation by u*x, then integrating with respect to x, we obtain

u*xuxdx  

Kx; tu*xutdtdx: 11:20

Now, multiplying the complex conjugate of the Fredholm equation by ux and

then integrating with respect to x,weget

u*xuxdx  *

K*x; t u* t u xdtdx:

Interchanging x and t on the right hand side of the last equation and remembering

that the kernel is Hermitian K*t; xKx; t, we obtain

u*xuxdx  *

Kx; tu t u* xdtdx:

Comparing this equation with Eq. (11.2), we see that   *, that is,  is real.

We now prove the orthogonality. Let 

, 

be two diÿerent eigenvalues and

x; u

x, the corresponding eigenfunctions. Then we have

x

Kx; tu

tdt; u

x

Kx; tu

tdt:

Now multiplying the ®rst equation by u

x, the second by 

x, and then

integrating with respect to x, we obtain



xu

xdx  



Kx; tu

tu

xdtdx;



xu

xdx  



Kx; tu

tu

xdtdx:

11:21

Now we interchange x and t on the right hand side of the last integral and because

of the symmetry of the kernel, we have



xu

xdx  



Kx; tu

tu

xdtdx: 11:22

422

SIMPLE LINEAR INTEGRAL EQUATIONS

Subtracting Eq. (11.21) from Eq. (11.22), we obtain



ÿ 



xu

xdx  0: 11:23

Since 

6 

, it follows that

xu

xdx  0: 11:24

Such functions may always be nomalized. We will assume that this has been done

and so the solutions of the homogeneous Fredholm equation form a complete

orthonomal set:

xu

xdx  

: 11:25 

Arbitrary functions of x, including the kernel for ®xed t, may be expanded in

terms of the eigenfunctions

Kx; t

x: 11:26

Now substituting Eq. (11.26) into the original Fredholm equation, we have

t

Kt; xu

xdx  

Kx; tu

xdx

 

xu

xdx  



 

 u

t=

and for our homogeneous Fredholm equation of the second kind the kernel may

be expressed in terms of the eigenfunctions and eigenvalues as

Kx; t

n1

xu

t



: 11:27

The Schmidt±Hilbert theory does not solve the homogeneous integral equation;

its main function is to establish the propert ies of the eigenvalues (reality) and

eigenfunctions (orthogonality and completeness). The solutions of the homoge-

neous integral equation come from the preceding section on methods of solution.

(b) Solution of the inhomogeneous equation

uxf x

Kx; tutdt: 11:28

We assume that we have found the eigenfunctions of the homogeneous equation

by the methods of the preceding section, and we denote them by u

x. We may

423

THE SCHMIDT±HILBERT METHOD OF SOLUTION

now expand both ux and f x in terms of u

x, which forms an orthonormal

complete set.

ux

n1



x; f x

n1

x: 11:29

Substituting Eq. (11.29) into Eq. (11.28), we obtain

n1



x

n1

x

Kx; t

n1



tdt



n1

x

n1



x



tu

tdt;



n1

x

n1



x





;

from which it follows that

n1



x

n1

x

n1



x



: 11:30

Multiplying by u

x and then integrating with respect to x from a to b, we obtain



 ÿ

 

=

; 11:31

which can be solved for 

in terms of ÿ







ÿ 

; 11:32

where ÿ

is given by



f tu

tdt: 11:33

Finally, our solution is given by

uxf x

n1



x



 f x

n1



ÿ 

x; 11:34

where ÿ

is given by Eq. (11.33), and 

6 .

When  for the inhomogeneous equation is equal to one of the eigenvalues, 

of the kernel, our solution (11.31) blows up. Let us return to Eq. (11.31) and see

what happens to 



 ÿ

 



=

 ÿ

 

424

SIMPLE LINEAR INTEGRAL EQUATIONS

Clearly, ÿ

 0, and 

is no longer determined by ÿ

. But we have, according to

Eq. (11.33),

f tu

tdt  ÿ

 0; 11:35

that is, f x is orthogonal to the eigenfunction u

x. Thus if   

, the inho-

mogeneous equation has a solution only if f x is orthogonal to the correspond-

ing eigenfunction u

x. The general solution of the equation is then

uxf x

x

n1

f tu

tdt



ÿ 

x; 11:36

where the prime on the summation sign means that the term n  k is to be omitted

from the sum. In Eq. (11.36) the 

remains as an undetermined constant.

Relation between diÿerential and integral equations

We have shown how an integral equation can be transformed into a diÿerential

equation that may be easier to solve than the original integral equation. We now

show how to transform a diÿerential equation into an integral equation. After we

become familiar with the relation between diÿerential and integral equations, we

may state the physical problem in either form at will. Let us consider a linear

second-order diÿerential equ ation

 Atx

 Btx  gt; 11:37

with the initial condition

xax

; x

ax

Integrating Eq. (11.37), we obtain

ÿ

dt ÿ

Bxdt 

gdt  C

The initial conditions require that C

 x

. We next integ rate the ®rst integral on

the right hand side by parts and obtain

ÿAx ÿ

B ÿ A

xdt 

gdt  Aax

 x

Integrating again, we get

x ÿ

Axdt ÿ

ByÿA

y



xydydt



gydydt  Aax

 x



t ÿ ax

425

DIFFERENTIAL AND INTEGRAL EQUATIONS