Types of Integral Equations

It is evident that different types of boundary value problems give rise to different types of integral equations.

The inhomogeneous boundary value problem gave rise to Eq.(4.53),
whose form is

In this case, and are known functions, and is the unknown function.

The integration limits
and
are fixed. An integral equation
for
of the form Eq. (4.55) is called
*inhomogeneous Fredholm equation* of the *second kind*.
The expression
is called the ``kernel'' of the integral
equation.

A *homogeneous Fredholm equation of the second kind* is obtained by
dropping the function
,

Equation (4.54) and the subsequent eigenvalue equations are examples of such equations.

A *Fredholm equation of the first kind* has the form

whenever is a known function and is the unknown function.

Fredholm equations are based on definite integrals. If the integration limits
are variable, then the corresponding integral equations are *Volterra
equations*. An *inhomogeneous Volterra equation of the second kind*,
corresponding to Eq. (4.55), has the form

If , then one has a

where is known and is the unknown function. A Volterra integral equation may be viewed as a Fredholm equation whose kernel vanishes for . Thus letting

one finds that the Volterra Eq. (4.56) becomes

whose form is that of a Fredholm equation.

One of the prominent examples giving rise to Volterra's integral equations
are initial value problems. To illustrate this point, consider the motion
of a simple harmonic oscillator governed by the equation

and the initial conditions

The Green's function for this problem is depicted in Figure 4.5 on page . It is the response to the impulse , and it satisfies

or
| ||

(458) |

in spite of the fact that (Why? Hint: what is the second derivative of a function that depends only on ?). To obtain the integral equation multiply Eq. (4.57) by and Eq. (4.58) by . One finds

0 | |||

Subtraction yields a l.h.s. whose second derivative terms consolidate into a total derivative (Lagrange's identity!):

Next perform the integration , where signifies taking the limit of the integral as from the side for which . One obtains

or with the help of the property whenever ,

Here and are the initial amplitude and velocity of the simple harmonic oscillator, and they are now intrinsically incorporated in an inhomogeneous Volterra equation of the second kind. In this integral equation is the unknown function to be determined. However, the utility of this integral equation, which is based on the Green's function , is eclipsed by an integral equation which is based a another Green's function, say , satisfying

similar to Eq.(4.57). Following the same derivation steps, one finds that the -term gets cancelled.

The integral has diappeared. One is left with the solution to the problem one is actually trying to solve. The overall conclusion is this:

Consider the inhomogeneous Fredholm equation of the second kind,

Here is a parameter and is a known and given function. So is the integration kernel , which in this problem is given to be translation invariant, i.e. you should assume that , where is a given function whose Fourier transform

exists. SOLVE the integral equation by finding the function in terms of what is given.

Look up an integral equation of the *2nd kind,* either of the Volterra
or of the Fredholm type. Submit it and its solution.