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In each of Problems 24 and 25 use the method indicated in Problem 23 to solve the given
boundary value problem.
11.3 Nonhomogeneous Boundary Value Problems
24. u=u —2, 25. u=u —n2 cos n x,
t xx 5 t xx 5
u(0, t) = 1, u(1, t) = 0, ux(0, t) = 0, u(1, t) = 1,
u(x, 0) = x2 - 2x + 2 u(x, 0) = cos(3nx/2) - cos nx
26. The method of eigenfunction expansions is often useful for nonhomogeneous problems related to the wave equation or its generalizations. Consider the problem
r(x)utt = [p(.x)ux]x - q(x)u + F(x, t), (i)
ux(0, t) - h1u(0, t) = 0, ux(1, t) + h2u(1, t) = 0, (ii)
u(x, 0) = f (x), ut(x, 0) = g(x). (iii)
This problem can arise in connection with generalizations of the telegraph equation (Prob-
lem 16 in Section 11.1) or the longitudinal vibrations of an elastic bar (Problem 25 in Section 11.1).
(a) Let u(x, t) = X(x)T(t) in the homogeneous equation corresponding to Eq. (i) and show that X(x) satisfies Eqs. (28) and (29) of the text. Let Xn and ôï (x) denote the eigenvalues and normalized eigenfunctions of this problem.
(b) Assume that u(x, t) = Y b (´)ô (x), and show that b (t) must satisfy the initial value
n=1 n n n
b'O + xnbn (t) = Yn (t), bn (0) = an, bn (0) = en,
where an, âï, and Yn(t) are the expansion coefficients for f (x), g(x), and F(x, t)/r(x) in terms of the eigenfunctions ô1^),. ..,ôï (x),....
27. In this problem we explore a little further the analogy between Sturm-Liouville boundary value problems and Hermitian matrices. Let A be an n x n Hermitian matrix with eigenvalues X1,...,Xn and corresponding orthogonal eigenvectors ^(1),..., ^(n).
Consider the nonhomogeneous system of equations
Ax - /xx = b, (i)
where x is a given real number and b is a given vector. We will point out a way of solving Eq. (i) that is analogous to the method presented in the text for solving Eqs. (1) and (2).
(a) Show that b = b.?,(l), where b. = (b, ^(l)).
i=1 ‘ ‘
(b) Assume that x =Y à³^ and show that for Eq. (i) to be satisfied, it is necessary that
at = /(Xj - /ë). Thus
^ (b, )) c(j) (-.ë
x =L~,-------------? (% (ii)
i=1 Xi - ë
provided that ë is not one of the eigenvalues of A, ë = X. for i = 1,..., n. Compare this result with Eq. (13). i
Green's5 Functions. Consider the nonhomogeneous system of algebraic equations
Ax - xx = b, (i)
5Green’s functions are named after George Green (1793-1841) of England. He was almost entirely self-taught in mathematics, and made significant contributions to electricity and magnetism, fluid mechanics, and partial differential equations. His most important work was an essay on electricity and magnetism that was published privately in 1828. In this paper Green was the first to recognize the importance of potential functions. He introduced the functions now known as Green’s functions as a means of solving boundary value problems, and developed the integral transformation theorems of which Green’s theorem in the plane is a particular case. However, these results did not become widely known until Green’s essay was republished in the 1850s through the efforts of William Thomson (Lord Kelvin).
Chapter 11. Boundary Value Problems and Sturm-Liouville Theory
where A is an n x n Hermitian matrix, ä is a given real number, and b is a given vector. Instead of using an eigenvector expansion as in Problem 27, we can solve Eq. (i) by computing the inverse matrix (A —ä²)—1, which exists if ä is not an eigenvalue of A. Then
x = (A — ä²)—1b. (ii)
Problems 28 through 36 indicate a way of solving nonhomogeneous boundary value problems that is analogous to using the inverse matrix for a system of linear algebraic equations. The Green’s function plays a part similar to the inverse of the matrix of coefficients. This method leads to solutions expressed as definite integrals rather than as infinite series. Except in Problem 35 we will assume that ä = 0 for simplicity.