Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

version of 28 January 2000

(Some remarks for the instructor).

You may wish to review the notion of a linear operator by referring to the Mathematica notebook for Chapter I.

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

**Model Problem XIII.1**. Find the *inverse*
of a matrix. Example:
Let **u** and **v** be vectors
in R^{2}, and

The function g is a solution for

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

VERIFICATION OF MODEL PROBLEM 3.

Using integration by parts this last line can be rewritten as

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

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

(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. In the next few chapters you will learn how, given statement (a), you can construct K such that statement (b) is equivalent to statement (a). Perhaps you can do this already.

**Model Problem XIII.4**.
Find the solution to a
boundary-value problem. Example:

with boundary data that u(x,0) = sin(x), and the "condition at infinity" that u(x,y) remains finite as y -> .

One of the unifying ideas of this course is that each of these model problems can be written in the form Lu=v for a linear operator L. It is a worthwhile exercise to reformulate each of these model problems in this form.

Most often, we shall take the interval on which our functions are defined to be [0,1]. Of course, we do not work in the class of

Then we have the inner product space we
called L^{2}( [0,1] ) in
Chapter II.
We recall that the
inner product
of two functions is
given by

and the norm of f is defined as the square-root of the inner product of a function with itself:

(Compare with the
norm
in R^{n}.)

It does not seem appropriate to study in detail the nature of
L^{2}[0,1] at this time. Rather, suffice it to say that the space is
large enough to contain all continuous functions - even functions which are
continuous except at a finite number of places. The interested student can
learn more about L^{2}[0,1] by looking
in standard books on real
analysis or Hilbert space.

Suppose { f_{p} } is a sequence of functions in L^{2}( [0,1]). It is
valuable to consider the possible meanings for
the statement that lim_{p} f_{p}(x) = g(x).
There are three useful interpretations for our purposes:

Sometimes we modify this to convergence

lim_{p} sup_{x} |f_{p}(x) - g(x)| = 0.

lim_{p} || f_{p} - g || = 0.

In this section we study one of the most common
types of integral equation. As an example,
given a function called the *kernel*

K: [0,1]x[0,1] -> R

and a function f: [0,1] -> R, we seek a function y such that for each x in [0,1],

Such equations are called Fredholm equations of the second kind. An equation of the form

is a Fredholm equation of the first kind.

The requirements in this section on K and f will be that

These requirements are met if K and f are continuous.

For simplicity, we denote by **K** the linear function given by

Note that **K** has a domain large enough to contain all functions y which
are continuous on [0,1]. Also, if y is continuous then **K**(y) is a
function and its value at x is denoted **K**(y)(x). In spoken conversation,
it is not so easy to distinguish the number valued function K and the function
valued **K.** The bold character will be used in these notes to denoted the
latter.

It is well to note the resemblance of this function **K** to the
multiplication of a vector u by a matrix A:

This formula has the same form as that for **K** given above.

It is a historical accident that differential equations were understood before integral equations. Often an integral equation can be converted into a differential equation or vice versa, so many of the laws of nature which we think of as differential equations might just as well have been developed as integral equations initially. In some instances it is easier to differentiate than to integrate, but at other times integral operators are more tractable.

In this course integral operators will be called upon to solve differential equations, and this is one of their main uses. They have many other uses as well, most notably in the theory of filtering and signal processing. In most of these applications the integral and differential operators are linear transformations. The analogy between linear transformations and matrices is deep and useful.

Just as a matrix has an adjoint, the integral operator **K** has an adjoint,
denoted **K***. The adjoint plays an important role in the theory and use of
integral equations.
(Review the
adjoint
for matrices.)

In order to understand **K***, one must consider < **K**(f), g >
and seek **K*** such that < **K**f, g > = < f, **K***g
>.

An examination of these last equations leads one to guess that **K*** is
given by

or, keeping t as the variable of integration,

Those last equations verified that

< **K**(f), g > = < f, **K***(g) >.

Care has to be taken to watch whether the "variable of integration" is t or x in the integrals involved.

In summary, if K is the kernel associated with the linear operator **K**,
then the kernel associated with **K*** is given by K*(x,y) [[equivalence]]
K(y,x). It is of value to compare how to get **K*** from **K** with the
process of how to get A* from A:

A*_{p,q} = A_{q,p}.

Consistent with the rather standard notation we have adopted above, it is clear that a briefer representation of the equation

is the concise equation y = **K**(y) + f, or (1 - **K** ) y = f.

**Example XIII.5:** Suppose that

To get K*, let's use other letters for the argument of K* and K to avoid
confusion. Suppose that 0 < u < v < 1. Then, K*(u,v) = K(v,u) = 0. In
a similar manner, K*(u,v) = (u-v)^{2} if 0 < v < u < 1. Note
that K* is not K.

The discussion of this example has been algebraic to this point. Consider this geometric notion that is suggested by the alternate name for "self-adjoint", namely, some call K "symmetric" if K(x,t) = K(t,x). The geometric name suggests a picture and the picture is the graph of K. The K of this example is not symmetric in x and t. Its graph is not symmetric about the line x = t. The function K is different from the function K*.

A first understanding of the problem of solving an integral equation

y = **K**y + f

can be gotten by referring to the Fredholm Alternative Theorems in this context.

(Review the alternative theorem for matrices.)

I. Exactly one of the following holds:

(a)(**First Alternative**) if f is in L^{2}{0,1}, then

has one and only one solution.

(b)(**Second Alternative**)

has a nontrivial solution.

II. (a) If the first alternative holds for the equation

then it also holds for the equation

(b) In either alternative, the equation

and its adjoint equation

have the same number of linearly independent solutions.

III. Suppose the second alternative holds. Then

has a solution if and only if

for each solution z of the adjoint equation

Comparing this context for the Fredholm Alternative Theorems with an understanding of matrix examples seems irresistible. Since these ideas will re-occur in each section, the student should pause to make these comparisons.

**Example XIII.6**: Suppose that E is the linear space of continuous functions on
the interval [-1,1]. with

and that

The equation y = **K**(y) has a non-trivial solution: the constant function
1. To see this, one computes

One implication of these computations is that the problem y = **K**y + f is
a second alternative problem. It may be verified that y(x) = 1 is also a
nontrivial solution for y = **K***y. It follows from the third of the
Fredholm alternative theorems that a necessary condition for y = **K**y + f
to have a solution is that

Note that one such f is f(x) = x + x^{3}.

**XIII.1**.
Reformulate each of Model Problems XIII.2-XIII.4
in the form Lu = v. I.e., carefully identify the
to which u and
v belong, as well as how the operator L acts.
For Model Problem XIII.4 make v a nonzero quantity related to the boundary condition sin(x) by choosing the
vector spaces for the
linear operator intelligently.

**XIII.2**.
Suppose K(x,t) =1 + 2 x t^{2} on [0,1]x[0,1] and y(x) = 3 -
x. Compute **K**(y) and **K***(y). Ans:
(5+3x)/2, (15+14x^{2})/6

**XIII.3**.
Let

Find B such that, if v is in R^{2}, then these are equivalent:

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

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

**XIII.4**. Let K be as in
Model Problem XIII.2. Show that if u(x) =
3x^{2 }- (25 + 12x )/6 then u solves the equation

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

**XIII.6**. Let u(r, =
r sin( ). Show that

with u(1,) = sin().

**XIII.7**.. Suppose K(x,t) = x t if 0 < x < t < 1, and = x t^{2} if 0
< t < x < 1)). For y(x) = 3 - x, compute **K**(y) and
**K***(y).

Ans: **K**[y](x) = - x^{5}/4 +
4x^{4}/3 - 3x^{3}/2 + 7x/6.

**XIII.8**.

(1) Suppose that E is the linear space of continuous functions on [0,1] with

and that

(2) Show that y = **K**y has non-trivial solution the constant function 1.

(3) Show that y = **K***y has non-trivial solution the function
+ 2
cos( x).

(4) What conditions must hold on f in order that

y = **K**y + f

should have a solution?

Find examples of sequences of functions which converge

Link to