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where an, f$n, and Yn(t) are the expansion coefficients for f (x), g(x), and F(x, t)/r(x) in terms of the eigenfunctions 01(x), (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 A.j,...,A.n and corresponding orthogonal eigenvectors ^(1),..., ^(n).
Consider the nonhomogeneous system of equations
Ax ó gx = b, (i)
where g 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?,(Ď), where b. = (b, ^(l)).
i=1 Ď Ď
(b) Assume that x = at^(i) and show that for Eq. (i) to be satisfied, it is necessary that
at = bt /(Xt ó g). Thus
x=? e. (ę)
provided that g is not one of the eigenvalues of A, g = *i for i = 1,..., n. Compare this
result with Eq. (13).
Greenís5 Functions. Consider the nonhomogeneous system of algebraic equations
Ax ó gx = b, (i)
5Greenís 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 Greenís functions as a means of solving boundary value problems, and developed the integral transformation theorems of which Greenís theorem in the plane is a particular case. However, these results did not become widely known until Greenís 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, p 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 ópI)ó1, which exists if p is not an eigenvalue of A. Then
x = (A ó pI)ó1b. (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 Greenís 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 p = 0 for simplicity.
28. (a) Show by the method of variation of parameters that the general solution of the differential equation
ó/ = f (x)
can be written in the form
y = $(x) = c1 + c2 x óI (x ó s) f (s) ds,
where c1 and c2 are arbitrary constants.
(b) Let y = 0 (x) also be required to satisfy the boundary conditions y(0) = 0, y(1) = 0. Show that in this case
c1 = 0, c2 = 1 (1 ó s) f(s) ds.
(c) Show that, under the conditions of parts (a) and (b), 0(x) can be written in the form
0(x) = ( s(1 ó x) f (s) ds + ( x(1 ó s) f (s) ds.
G(xs)- |s(1 - x^ 0 < 5 < x,
G(x, s) i(i - s), x < s < 1,
show that the solution takes the form
fi(x) = ( G(x, s) f (s) ds.
The function G(x, s) appearing under the integral sign is a Greenís function. The usefulness of a Greenís function solution rests on the fact that the Greenís function is independent of the nonhomogeneous term in the differential equation. Thus, once the Greenís function is determined, the solution of the boundary value problem for any nonhomogeneous term f (x) is obtained by a single integration. Note further that no determination of arbitrary constants is required, since 0(x) as given by the Greenís function integral formula automatically satisfies the boundary conditions.
29. By a procedure similar to that in Problem 28 show that the solution of the boundary value problem
ó (/' + y) = f (x), y(0) = 0, y(1) = 0
y = 0(x) = f G(x, s) f (s) ds, Jo
11.3 Nonhomogeneous Boundary Value Problems
sin s sin(1 ó x)
G(x, s) =
30. It is possible to show that the Sturm-Liouville problem
sin 1 sin x sin(1 ó s)
0 < s < x, x < s < 1.
L [y] = -[ p(x)/]' + q (x) y = f (x), (i)
a1 y(0) + a2 y (0) = 0, b1 y(1) + b2 y (1) = 0 (ii)
has a Greenís function solution
y = $(x) = ( G(x, s) f (s) ds, (iii)
provided that k = 0 is not an eigenvalue of L[y] = ky subject to the boundary conditions
(ii). Further, G(x, s) is given by
^ ^ \ó”J(s)”2(x)/p(x)W (jp y2)(x), 0 < s < ^ ,,
G(x, s) = (iv)
Ió y1(x)y2(s)/p(x) W(y1, y2)(x), x < s < 1,
where y is a solution of L [y] = 0 satisfying the boundary condition at x = 0, y2 is a solution of L[y] = 0 satisfying the boundary condition at x = 1, and W(y1, y2) is the Wronskian of y and y2.
(a) Verify that the Greenís function obtained in Problem 28 is given by formula (iv).
(b) Verify that the Greenís function obtained in Problem 29 is given by formula (iv).