Download (direct link):
u(0, t) = 0, u(L, t) = 0, t > 0 (2)
and the initial condition
u(x, 0) = f (x), 0 < x < L, (3)
is typical of the problems considered there. A crucial part of the process of solving such problems is to find the eigenvalues and eigenfunctions of the differential equation
X" + \X = 0, 0 < x < L (4)
with the boundary conditions
X(0) = 0, X(L) = 0 (5)
X'(0) = 0, X'(L) = 0. (6)
The sine or cosine functions that result from solving Eq. (4) subject to the boundary conditions (5) or (6) are used to expand the initial temperature distribution f (x) in a Fourier series.
In this chapter we extend and generalize the results of Chapter 10. Our main goal is to show how the method of separation of variables can be used to solve problems somewhat more general than that of Eqs. (1), (2), and (3). We are interested in three types of generalizations.
First, we wish to consider more general partial differential equations, for example, the equation
r(x)ut = [p(x)ux]x - q(x)u + F(x, t). (7)
This equation can arise, as indicated in Appendix A of Chapter 10, in the study of heat conduction in a bar of variable material properties in the presence of heat sources. If p and r are constants, and if the source terms qu and F are zero, then Eq. (7) reduces to Eq. (1). The partial differential equation (7) also occurs in the investigation of other phenomena of a diffusive character.
A second generalization is to allow more general boundary conditions. In particular, we wish to consider boundary conditions of the form
ux(0, t) - h1u(0, t) = 0, ux(L, t) + h2u(L, t) = 0. (8)
Such conditions occur when the rate of heat flow through an end of the bar is proportional to the temperature there. Usually, h1 and h2 are nonnegative constants, but
in some cases they may be negative or depend on t. The boundary conditions (2) are obtained in the limit as h 1 and h2 ^ to. The other important limiting case,
h1 = h2 = 0, gives the boundary conditions for insulated ends.
The final generalization that we discuss in this chapter concerns the geometry of the region in which the problem is posed. The results of Chapter 10 are adequate only for a rather restricted class of problems, chiefly those in which the region of interest is rectangular or, in a few cases, circular. Later in this chapter we consider certain problems posed in a few other geometrical regions.
Let us consider the equation
r(x)ut = [p(x)ux]x - q(x)u
11.1 The Occurrence of Two-Point Boundary Value Problems
obtained by setting the term F(x, t) in Eq. (7) equal to zero. To separate the variables we assume that
u(x, t) = X(x)T(t), (10)
and substitute for u in Eq. (9). We obtain
r(x) XT = [p(x) X']' T - q(x) XT (11)
or, upon dividing by r(x) XT,
T [ p(x) XT q (x)
T r (x) X r (x)
= -X. (12)
We have denoted the separation constant by —X in anticipation of the fact that usually it will turn out to be real and negative. From Eq. (12) we obtain the following two ordinary differential equations for X and T:
[ p(x) X ]' — q (x) X + Xr (x) X = 0, (13)
T' + XT = 0. (14)
If we substitute from Eq. (10) for u in Eqs. (8) and assume that hi and ^ are constants, then we obtain the boundary conditions
X'(0) — h1 X(0) = 0, X'(L) + h2 X(L) = 0. (15)
To proceed further we need to solve Eq. (13) subject to the boundary conditions (15). While this is a more general linear homogeneous two-point boundary value problem than the problem consisting of the differential equation (4) and the boundary conditions
(5) or (6), the solutions behave very much the same way. For every value of X, the
problem (13), (15) has the trivial solution X(x) = 0. For certain values of X, called
eigenvalues, there are also other nontrivial solutions, called eigenfunctions. These eigenfunctions form the basis for series solutions of a variety of problems in partial differential equations, such as the generalized heat conduction equation (9) subject to the boundary conditions (8) and the initial condition (3).
In this chapter we discuss some of the properties of solutions of two-point boundary value problems for second order linear equations. Sometimes we consider the general linear homogeneous equation
P (x)/+ Q(x) y+ R(x) y = 0, (16)
investigated in Chapter 3. However, for most purposes it is better to discuss equations in which the first and second derivative terms are related as in Eq. (13). It is always possible to transform the general equation (16) so that the derivative terms appear as in Eq. (13) (see Problem 11).
Boundary value problems with higher order differential equations can also occur; in them the number of boundary conditions must equal the order of the differential equation. As a rule, the order of the differential equation is even, and half the boundary conditions are given at each end of the interval. It is also possible for a single boundary condition to involve values of the solution and/or its derivatives at both boundary points; for example,
y(0) - y(L) = 0.