Introduction

## Integral Equations and the Method of Green's Functions James V. Herod*

Page maintained by Evans M. Harrell, II, harrell@math.gatech.edu.

AN INTRODUCTION TO INTEGRAL OPERATORS

The following three problems illustrate, in a simple way, the primary concerns of this course. The first is a problem about matrices and vectors, and it will be our guide to solving integral equations and differential equations.

SAMPLE PROBLEM 1: Let

Suppose v is a vector in R2. If u is a vector then

if and only if

(b) v is a vector and u = Bv.

The equivalence of these two is easy to establish. Even more, given only statement (a), you should be able to construct B such that statement (b) is equivalent to statement (a).

SAMPLE PROBLEM 2: Let K(x,t) = 1 + x t. The function u is a solution for

if and only if

If one supposes u is as in (b), then the integral calculus should show that u satisfies (a). On the other hand, the task of deriving a formula for u from the relationship in (a) involves techniques which we will discuss in this course.

SAMPLE PROBLEM 3: Let

Suppose f is continuous on [0,1]. The function g is a solution for

(a) g''= -f and g(0) = g(1) = 0

if and only if

VERIFICATION OF SAMPLE PROBLEM 3.

(a)=>(b) Suppose that f is continuous on [0,1] and g'' = -f with g(0) = g(1) = 0. Suppose also that K is as given by sample problem (3). Then

Using integration by parts this last line can be rewritten as

= -(1-x)[x g'(x) -(g(x)-g(0)}

-x[-(1-x) g'(x) + (g(1) - g(x))]

= (1-x) g(x) + x g(x) = g(x).

To get the last line we used the assumption that g(1) = g(0) = 0.

(b)=>(a) Again, suppose that f is continuous and, now, suppose that

As you can see, it is not hard to show that these two statements are equivalent. Before the course is over then, given statement (a), you should be able to construct K such that statement (b) is equivalent to statement (a). Perhaps you can do this already.

SAMPLE PROBLEM 4: Let u(x,y) = e-y sin(x) for y >= 0 and all x. Then

Find B such that, if v is in R2, then these are equivalent:

(a) u is a vector and Au = v.

(b) v is a vector and u = Bv.

2. Let K be as in SAMPLE PROBLEM 2. Show that if u(x) =

3x2 - (25 + 12x )/6 then u solves the equation

3. Let

Suppose that f is continuous on [0,1]. Show these are equivalent:

4. Let u(r,\theta) = r sin(\theta). Show that

with u(1,[[theta]]) = sin([[theta]]).