Integral Equations and the Method of Green's Functions

James V. Herod*

*(c) Copyright 1993,1994,1995 by James V. Herod, herod@math.gatech.edu. All rights reserved.

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

Section 3.1 Language and Classification

In this beginning with second order equations, we will take a look at the language of partial differential equations. We should be sure that we all have the same vocabulary. Since there is some standard language for the identification of partial differential equations, let's agree with the language that every one else uses as a start.

Several parts are easy. Suppose that we have a partial differential equation in u. The order of the equation is the order of the highest partial derivative of u, the number of variables is simply the number of independent variables for u in the equation, and the equation has constant coefficients if u and the coefficients of all the partial derivatives of u are constant. If all the terms involving u are moved to the left side of the equation, then the equation is called homogeneous if the right side is zero, and non homogeneous if it is not zero. The system is linear if that left side involves u in only a "linear way."

Examples:

(1) 4u_xx - 24 u_xy + 11u_yy - 12u_x - 9u_y - 5u = 0

is a linear, homogeneous, second order partial differential equation in two variables with constant coefficients.

(2) u_t = x u_xx + y u_yy + u²

is a nonlinear, second order partial differential equation in two variables and does not have constant coefficients.

(3) u_tt - u_xx = sin( \pi t)

is a non-homogeneous equation.

In thinking of partial differential equations, it is a common practice to carry over the language that has been used for matrix or ordinary differential equations in as far as possible. Recall that one solved linear systems of algebraic equations such as the equation

3x + 4y = 0

5x + 6y = 0.

This equation could be re-written as a matrix system

One talked of the matrix equation or the linear homogeneous equation Au = 0, where u and 0 are understood to be vectors in the two dimensioned vector space on which the linear operator A is defined.

In a similar manner, in sophomore differential equations, one considered equations such as

x' = 3x + 4y

y' = 5x + 6y

and, perhaps, re-wrote these equations as a system z' = Az where A is the matrix

This is a first order, linear system; A is a linear operator on R² and we follow the vector value function u(t) as it runs through that space, changing in time as prescribed by the differential equation

du/dt = Az.

So, in partial differential equation, we consider linear equations

Lu = 0, or u' = Lu,

only now, L is a linear operator on a space of functions. For example, it may be that L(u) = u_xx + u_yy. And, a corresponding notion for the equation u' = Lu is u_t = u_xx + u_yy.

The notion that a linear operator can have domain a space of functions may seem alien at first, but the analogies from Rⁿ are more than superficial. This is an idea worthy of consideration.

We will be interested in a rather general second order, differential operator. In two variables, we consider the operator

                                                                              (3.1)

There is an analogous formula for three variables.

We suppose that u is smooth enough so that

                                                                              (3.2)

that is, we can interchange the order of differentiation. In this first consideration, the matrix A, the vector B, and the number C do not depend on u; we take the matrix A to be symmetric. Because of (3.2) we are free to arrange this, since the coefficient of the cross term is A₁₂ + A₂₁, and we may assign A₁₂ and A₂₁ any way we wish, so long as the sum has the right value.

Example

We write an example in the matrix representation for constant coefficient equations:

L[u] = 4u_xx - 24 u_xy + 11u_yy - 12u_x - 9_y - 5u

can be written as

We shall be interested in an equation which has the form

L(u) = f

or u' = L(u) + f.

In this section, u is a function on R² or R³ and the equations are to hold in an open, connected region D of the plane. We will also assume that the boundary of the region is piece-wise smooth, and denote this boundary by \boundary D. Just as in ordinary differential equations, in partial differential equations some boundary conditions will be needed to solve the equations. We will take the boundary conditions to be linear and have the general form

B(u) = a u + b u_\eta,

where u_\eta is the derivative taken in the direction of a normal to the boundary of the region.

The techniques of studying partial differential operators and the properties of these operators change depending on the "type" of operator. These operators have been classified into three principal types. The classifications are made according to the nature of the coefficients in the equation which defines the operator. The operator is called an elliptic operator if the eigenvalues of A are non-zero and have the same algebraic sign. The operator is hyperbolic if the eigenvalues have opposite signs and is parabolic if at least one of the eigenvalues is zero.

The classification makes a difference both in what we expect from the solutions of the equation and in how we go about solving them. Each type is modeled on an important equation of physics.

The basic example of a parabolic equation is the one dimensional heat equation. Here, u(t,x) represents the heat on a line at time t and position x. One should be given an initial distribution of temperature which is denoted u(0,x), and some boundary conditions which arise in the context of the problem. For example, it might be assumed that the ends are held at some fixed temperature for all time. In this case, boundary conditions for a line of length L would be u(t,0) = \alpha and u(t,L) = \beta . Or, one might assume that the ends are insulated. A mathematical statement of this is that the rate of flow of heat through the ends is zero:

The manner in which u changes in time is derived from the physical principle which states that the heat flux at any point is proportional to the temperature gradient at that point and leads to the equation

Geometrically, one may think of the problem as one of defining the graph of u whose domain is the infinite strip bounded in the first quadrant by the parallel lines x = 0 and x = L The function u is known along the x axis between x = 0 and x = L, To define u on the infinite strip, move in the t direction according to the equation

while maintaining the boundary conditions.

We could also have a source term. Physically, this could be thought of as a heater (or refrigerator) adding or removing heat at some rate along the strip. Such an equation could be written as

Boundary and initial conditions would be as before. In order to rewrite this equation in the context of this course, we should conceive of the equation as L[u] = f , with appropriate boundary conditions. The operator L is

This is a parabolic operator according to the definition given above; in fact, the matrix A in (3.1) is given by

(A and Laplaceoperator, or Laplacian:

Laplace's equation states that

L[u] = 0

and Poisson's equation states that

L[u] = f(x,y)

for some given function f. A physical situation in which it arises is in the problem of finding the shape of a drum under force. Suppose that the bottom of the drum sits on the unit disc in the xy-plane and that the sides of the drum lie above the unit circle. We do not suppose that the sides are at a uniform height, but that the height is specified on the circle.

That is, we know u(x,y) for {x,y} on the boundary ot the drum. We also suppose that there is a force pulling down, or pushing up, on the drum at each point and that this force is not changing in time. An example of such a force might be the pull of gravity. The question is, what is the shape of the drum? As we shall see, the appropriate equations take the form: Find u if

with

u(x,y) specified for x² + y² = 1.

Laplace's and Poisson's equations also arise in electromagnetism and fluid mechanics as the equations for potential functions. Still another place where you may encounter Laplace's equation is as the equation for a temperature distribution in 2 or 3 dimensions, when thermal equilibrium has been reached. The Laplace operator is elliptic by our definition, for the matrix A in (3.1) is given by

Finally, the model for a hyperbolic equation is the one dimensional wave equation. One can think of this equation as describing the motion of a taunt string after an initial perturbation and subject to some outside force. Appropriate boundary conditions are given. To think of this as being a plucked string with the observer watching the up and down motion in time is not a bad perspective, and certainly gives intutive understanding. Here is another perspective, however, which will be more useful in the context of finding the Green function to solve this one dimensional wave equation:

As in example (a), the problem is to describe u in the infinite strip within the first quadrant of the xt-plane bounded by the x axis and the lines x = 0 and x = L. Both u and its first derivative in the t direction are known along the x axis. Along the other boundaries, u is zero. What must be the shape of the graph above the infinite strip?

To classify this as a hyperbolic problem, think of the operator L as

and re-write it in the appropriate form for classification. The matrix A in (3.1) is given by

(A physical derivation of the wave equation and further discussion of it)

There are standard procedures for changing more general partial differential equations to the familiar standard forms, which we shall investigate in the next section.

These very names and ideas suggest a connection with quadratic forms in analytic geometry. We will make this connection a little clearer. Rather than finding geometric understanding of the partial differential equation from this connection we will more likely develop algebraic understanding. Especially, we will see that there are some standard forms. Because of the nearly error free arithmetic that Mapleis able to do, we will offer syntax in Maplethat enables the reader to use this computer algebra system to change second order, linear systems into the standard forms.

If presented with a quadratic equation in x and y, one could likely decide if the equation represented a parabola, hyperbola, or ellipse in the plane. However, if asked to draw a graph of this conic section in the plane, one would start recalling that there are several forms that are easy to draw:

   a x² + b y² = c², and the special case x² + y² = c²,

   a x² - b y² = c², and the special case x² - y² = c²,

or
   y - a x² = 0 and x - b y² = 0.

These quadratic equations represent the familiar conic sections: ellipses, hyperbolas and parabolas, respectively. If a quadratic equation is given that is not in these special forms, then one may recall procedures to transform the equations algebraically into these standard forms. This will be the topic of the next section.

The purpose for doing the classification is that the techniques for solving equations are different in the three classes, if it is possible to solve the equation at all. Even more, there are important resemblance among the solutions of one class; and there are striking differences between the solutions of one class and those of another class. The remainder of these notes will be primarily concerned with finding solutions to hyperbolic, second order, partial differential equations. As we progress, we will see the importance of the equation being a hyperbolic partial differential equation to use the techniques of these notes.

Before comparing the similarity in procedures for changing the partial differential equation to standard form with the preceeding arithmetic, we pause to emphasize the differences in geometry for the types: elliptic, hyperbolic, and parabolic.

                                                             Figure 3.1

Here are three equations from analytic geometry:

x² + y² = 4 is an ellipse,

x² - y² = 4 is a hyperbola,

and x² + 2 x y + y² = 4 is a parabola.

Figure 3.1 contains the graphs of all three of these. Their shapes and their geometry, are strikingly different. Even more, naively, one might say that the graph of the third of those above is not the graph of a parabola. Indeed. It does, however, meet the criteria: b² - 4 a c = 0. One might think of the graph as that of a parabola with vertex at the "point at infinity."

The criterion for classifying second order, partial differential equations is the same: ask what is the character of b² - 4 a c in the equation

We now present solutions for three equations that have the same start -- the same initial conditions. However the equations are of the three types. There is no reason you should know how to solve the three equations yet. There is no reason you should even understand that solving the hyperbolic equation by the method of characteristics is appropriate. But, you should be able to check the solutions -- to see that they solve the specified equations. Each equation has initial conditions

u(0,y) = sin(y) and ux(0,y) = 0.

The equations are

Laplace's equation - elliptic;

a form of the wave equation - hyperbolic; and

is a parabolic partial differential equation,

These have solutions cosh(x) sin(y), cos(x) sin(y), and cos(x-y) x/2 + sin(y-x) respectively. Figure 3.2 has the graphs of these three solutions in order.

                                                             Figure 3.2 a

                                                             Figure 3.2 b

                                                             Figure 3.2 c

EXERCISES 3.1

Classify each of the following as hyperbolic, parabolic, or ellipt