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The main part of Theorem 11.3.1 is sometimes stated in the following way:
For a given value of i, either the nonhomogeneous problem (1), (2) has a unique solution for each continuous f (if i is not equal to any eigenvalue km of the corresponding homogeneous problem), or else the homogeneous problem (3), (2) has a nontrivial solution (the eigenfunction corresponding to km).
This latter form of the theorem is known as the Fredholm4 alternative theorem. This is one of the basic theorems of mathematical analysis and occurs in many different contexts. You may be familiar with it in connection with sets of linear algebraic equations where the vanishing or nonvanishing of the determinant of coefficients replaces the statements about i and km. See the discussion in Section 7.3.
Solve the boundary value problem
/ + 2 y =-x, (15)
y(0) = 0, y(1) + /(1) = 0. (16)
This particular problem can be solved directly in an elementary way and has the solution
_ Sin V2 x x (17)
y sin \[2 + \[2 cos \[2 2
The method of solution described below illustrates the use of eigenfunction expan-
sions, a method that can be employed in many problems not accessible by elementary procedures. To identify Eq. (15) with Eq. (1) it is helpful to write the former as
- / = 2 y + x. (18)
We seek the solution of the given problem as a series of normalized eigenfunctions of the corresponding homogeneous problem
f + Xy = 0, y(0) = 0, y(1) + y (1) = 0. (19)
4The Swedish mathematician Erik Ivar Fredholm (1866-1927), professor at the University of Stockholm, established the modern theory of integral equations in a fundamental paper in 1903. Fredholms work emphasized the similarities between integral equations and systems of linear algebraic equations. There are also many interrelations between differential and integral equations; for example, see Section 2.8 and Problem 21 of Section 6.6.
11.3 Nonhomogeneous Boundary Value Problems
These eigenfunctions were found in Example 2 of Section 11.2, and are
(x) = kn sinfnx, (20)
K = I - ^ (21)
n 1 + COs2 t/K )
and kn satisfies
sin + y[K cos^n = 0 (22)
Recall that in Example 1 of Section 11.1 we found that
k1 = 4.116, k2 = 24.14,
X3 = 63.66, kn = (2n 1)2^2/4 for n = 4, 5,....
We assume that y is given by Eq. (4),
y =J2 bn^n (x),
and it follows that the coefficients bn are found from Eq. (12),
n K 2
where the cn are the expansion coefficients of the nonhomogeneous term f (x) = x in Eq. (18) in terms of the eigenfunctions . These coefficients were found in Example 3 of Section 11.2, and are
2 V2 sin. fk~
- = V n (23)
n k(i + cos^vi;)1'2^
Putting everything together, we finally obtain the solution
sin A fln------------------I-----------------
y = 4^-------------------------27=" sinjInx. (24)
n=1 In(In - 2)(1 + cos 7in)
While Eqs. (17) and (24) are quite different in appearance, they are actually two different expressions for the same function. This follows from the uniqueness part of Theorem 11.3.1 or 11.3.2 since I = 2 is not an eigenvalue of the homogeneous problem (19). Alternatively, one can show the equivalence of Eqs. (17) and (24) by expanding the right side of Eq. (17) in terms of the eigenfunctions (x). For this problem it is
fairly obvious that the formula (17) is more convenient than the series formula (24). However, we emphasize again that in other problems we may not be able to obtain the solution except by series (or numerical) methods.
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
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(l, t) = 0 (26)
and the initial condition
u(x, 0) = f (x). (27)
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
[ p(x) X']' + q (x) X = kr (x) X, (28)
X! (0) h1 X(0) = 0, X'(1) + h2 X(1) = 0. (29)
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 k < k2 < < kn < and corresponding normalized eigenfunctions $1(x), 02(x),... ,0n(x),....
We will solve the given nonhomogeneous boundary value problem (25) to (27) by assuming that u(x, t) can be expressed as a series of eigenfunctions,
u(x, t) = ^2 bn(t)$n(x), (30)
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 (x) satisfies the boundary conditions (29).