# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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646

Chapter 11. Boundary Value Problems and Sturm Liouville Theory

Nonhomogeneous Heat Conduction Problems. To show how eigenfunction expansions can be used to solve nonhomogeneous problems for partial differential equations, let us consider the generalized heat conduction equation

This problem was previously discussed in Appendix A of Chapter 10 and in Section

11.1. In the latter section we let u(x, t) = X(x)T(t) in the homogeneous equation obtained by setting F(x, t) = 0, and showed that X(x) must be a solution of the boundary value problem

If we assume that p, q, and r satisfy the proper continuity requirements and that p(x) and r(x) are always positive, the problem (28), (29) is a Sturm-Liouville problem as discussed in Section 11.2. Thus we obtain a sequence of eigenvalues X1 < X2 < ¦ ¦¦ < Xn < ¦ ¦¦ and corresponding normalized eigenfunctions ô1(x), ô2^),... ,ôï (x),....

We will solve the given nonhomogeneous boundary value problem (25) to (27) by assuming that u(x, t) can be expressed as a series of eigenfunctions,

and then showing how to determine the coefficients bn (t). The procedure is basically the same as that used in the problem (1), (2) considered earlier, although it is more complicated in certain respects. For instance, the coefficients bn must now depend on t, because otherwise u would be a function of x only. Note that the boundary conditions (26) are automatically satisfied by an expression of the form (30) because each ôï (x) satisfies the boundary conditions (29).

Next we substitute from Eq. (30) for u in Eq. (25). From the first two terms on the right side of Eq. (25) we formally obtain

r(x)ut = [p(x)ux]x - q(x)u + F(x, t)

(25)

with the boundary conditions

ux(0, t) - h1u(0, t) = 0, ux(1, t) + h2u(1, t) = 0

(26)

and the initial condition

u(x, 0) = f (x).

(27)

- [p(x) X] + q(x) X = Xr(x) X,

X(0) - h1 X(0) = 0, X'(1) + h2X(1) = 0.

(28)

(29)

TO

(30)

d TO TO

[p(x)ux]x- q(x)u = p(x^bn(t)ôÏ(x) - q(x)^Zbn(t)ôï(x)

TOTO

TO

= Y2 bn(t){[p(xWn(x)] - q(x)ôï(x)}.

(31)

Since [p(x)ô'n(x)]; - q(x)ôn(x) = —Xnr(x)ôn(x), we obtain finally

TO

[p(x)ux]x - q(x)u = ~r(x)Y2 bn(´)Õïôï(x).

n=1

(32)

11.3 Nonhomogeneous Boundary Value Problems

647

Now consider the term on the left side of Eq. (25). We have

O

d v >

r(x)ut = r(x)—2_^ bn(²)Ô„(x)

n= 1

= r (x)Y2 b'n (³)Ôâ (x). (33)

n=1

We must also express the nonhomogeneous term in Eq. (25) as a series of eigenfunctions. Once again, it is convenient to look at the ratio F(x, t)/r(x) and to write

F(x, t) O

—— = Y1 Yn (t )ôï (x)> (34)

r( x) n n

n=1

where the coefficients are given by

f1 F (x, t)

Yn(t) = r(x)———Ôï(x) dx

Ë r(x)

= f F(x, t)ôï(x) dx, n = 1, 2,.... (35)

0n

Since F(x, t) is given, we can consider the functions y„ (t) to be known.

Gathering all these results together, we substitute from Eqs. (32), (33), and (34) in Eq. (25), and find that

O O O

r (x )E bn (t)Ôï (x) = -r (x )? bn (t )Xn Ôï (x) + r (x)Y2 Yn (¿)Ô„ (x). (36)

n=1 n= 1 n= 1

To simplify Eq. (36) we cancel the common nonzero factor r(x) from all terms, and write everything in one summation:

E [b'n(t) + Xnbn(t) - Yn(t)tyn(x) = 0. (37)

n=1

Once again, if Eq. (37) is to hold for all x in 0 < x < 1, it is necessary that the quantity in square brackets be zero for each n (again see Problem 14). Hence bn(t) is a solution of the first order linear ordinary differential equation

bn (t) + Xnbn (t) = Yn (t), n = 1, 2,..., (38)

where y„(t) is given by Eq. (35). To determine bn(t) completely we must have an initial condition

bn (0) = an, n = 1, 2,... (39)

for Eq. (38). This we obtain from the initial condition (27). Setting t = 0 in Eq. (30) and using Eq. (27), we have

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