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If ä = Xm and cm = 0, then Eq. (11) is satisfied regardless of the value of bm; in other words, bm remains arbitrary. In this case the boundary value problem (1), (2) does have a solution, but it is not unique, since it contains an arbitrary multiple of the eigenfunction ôï³.
Since cm is given by Eq. (9), the condition cm = 0 means that
¥ f(x)Ôm(x) dx = 0. (14)
Thus, if ä = Xm, the nonhomogeneous boundary value problem (1), (2) can be solved only if f is orthogonal to the eigenfunction corresponding to the eigenvalue Xm.
The results we have formally obtained are summarized in the following theorem.
Chapter 11. Boundary Value Problems and Sturm Liouville Theory
The nonhomogeneous boundary value problem (1), (2) has a unique solution for each continuous f whenever i is different from all the eigenvalues of the corresponding homogeneous problem; the solution is given by Eq. (13), and the series converges for each x in 0 < x < 1. If i is equal to an eigenvalue km of the corresponding homogeneous problem, then the nonhomogeneous boundary value problem has no solution unless f is orthogonal to ôò; that is, unless the condition (14) holds. In that case, the solution is not unique and contains an arbitrary multiple of ôò (x).
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 Fredholm 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
Ó + 2 y =-x, y(0) = 0, y(1) + /(1) = 0.
This particular problem can be solved directly in an elementary way and has the solution
sin Ó¯ + Ó¯ cos Ó¯ 2
The method of solution described below illustrates the use of eigenfunction expansions, 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
- Ó = 2y + x.
We seek the solution of the given problem as a series of normalized eigenfunctions ôï of the corresponding homogeneous problem
Ó + ky = 0, y(0) = 0, y(1) + /(1) = 0. (19)
The 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. Fredholm’s 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
Ôï (x) = kn sin ^nx,
1 + cos2 s/K J
kn ' 1 ' cos2
and kn satisfies
si^/kn + ë/kn co4 Ê = 0
Recall that in Example 1 of Section 11.1 we found that
Ë3 = 63^66,
k1 = 4^116, k2 = 24Ë4,
We assume that y is given by Eq. (4),
Ó =J2 bn ôï(x), n= 1
and it follows that the coefficients bn are found from Eq. (12),
k = (2n — 1) n /4 for n = 4, 5, • • • •
n kn — 2’
where the cn are the expansion coefficients of the nonhomogeneous term f (x) = x in Eq. (18) in terms of the eigenfunctions ôï. These coefficients were found in Example 3 of Section 11.2, and are
2\fl sin óË
Cn = kn (1 + cos2 Jkn )1/2 •
Putting everything together, we finally obtain the solution
Ó = 4E
=1 Ëï (Ëï — 2)(1 + cos2 s/^n)
sin, k x.
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 k = 2 is not an eigenvalue ofthe 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 ôï (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.