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## Types of Integral Equations

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

A.  Fredholm Equations

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

 (455)

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.

B.  Volterra Equations

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

 (456)

If , then one has a homogeneous Volterra equation of the second kind. By contrast, a Volterra equation of the first kind has the form

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

 0 (457)

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: Picking the right Green's function for the problem speeds up the process of reaching one's goal.

Exercise 49.1 (TRANSLATION INVARIANT INTEGRATION KERNEL)

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.

Exercise 49.2

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

Lecture 35

Next: Singular Boundary Value Problem: Up: Boundary Value Problem via Previous: Eigenfunctions via Integral Equations   Contents   Index
Ulrich Gerlach 2010-12-09