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where f is a given continuous function on 0 < x < 1, and ä is not an eigenvalue of the corresponding homogeneous problem.
4. Consider Legendre’s equation (see Problems 22 through 24 of Section 5.3)
The eigenfunctions of this problem are the odd Legendre polynomials ô1 (x) = P1 (x) = x, ô2(x) = P3(x) = (5x3 — 3x)/2,..., ôï(x) = P2n_ 1(x),... corresponding to the eigenvalues Y1 =2, Y2 = 4 • 3,...,kn = 2n(2n — 1), ....
(a) Show that
xôm (x^n (x) dx = 0, m = n.
— (xy)+ (k2/x) y = Yxy, y, y bounded as x ^ 0, y(1) = 0,
(d) Determine the coefficients in the formal series expansion
xôm(x^n(x) dx = 0, m = n.
f (x) = J2 an ôï (x).
(e) Find a formal solution of the nonhomogeneous problem
— (xy,)' + (k2/x)y = äxy + f (x), y, y bounded as x ^ 0, y(1) = 0,
[(1 — x2)/]' = Yy
subject to the boundary conditions
y(0) = 0, y, y bounded as x ^ 1.
m = n.
11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 663
(b) Find a formal solution of the nonhomogeneous problem
- [(1 - x2)/]' = ëÓ + f (x),
y(0) = 0, y, y bounded as x ^ 1,
where f is a given continuous function on 0 < x < 1, and /ë is not an eigenvalue of the corresponding homogeneous problem.
5. The equation
(1 - x2 )Ó - xy + Xy = 0 (i)
is Chebyshev’s equation; see Problem 10 of Section 5.3.
(a) Show that Eq. (i) can be written in the form
-[(1 - x2)l/2/]' = X(1 - x2)-1/2y, -1 < x < 1. (ii)
(b) Consider the boundary conditions
y, y bounded as x ^-1, y, y bounded as x ^ 1. (iii)
Show that the boundary value problem (ii), (iii) is self-adjoint.
(c) It can be shown that the boundary value problem (ii), (iii) has the eigenvalues X0 = 0,
= 1, X2 = 4,...,Xn = n2The corresponding eigenfunctions are the Chebyshev
polynomials Tn(x): TQ(x) = 1, T1 (x) = x, T2(x) = 1 - 2x2, .... Showthat
^ Tm (x) T (x)
-1 (1 - x2)1/2 Note that this is a convergent improper integral
dx = 0, m = n. (iv)
11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion
In this chapter we are interested in extending the method of separation of variables developed in Chapter 10 to a larger class of problems—to problems involving more general differential equations, more general boundary conditions, or different geometrical regions. We indicated in Section 11.3 how to deal with a class of more general differential equations or boundary conditions. Here we concentrate on problems posed in various geometrical regions, with emphasis on those leading to singular Sturm-Liouville problems when the variables are separated.
Because of its relative simplicity, as well as the considerable physical significance of many problems to which it is applicable, the method of separation of variables merits its important place in the theory and application of partial differential equations. However, this method does have certain limitations that should not be forgotten. In the first place, the problem must be linear so that the principle of superposition can be invoked to construct additional solutions by forming linear combinations of the fundamental solutions of an appropriate homogeneous problem.
As a practical matter, we must also be able to solve the ordinary differential equations, obtained after separating the variables, in a reasonably convenient manner. In some problems to which the method of separation of variables can be applied in principle, it
Chapter 11. Boundary Value Problems and Sturm-Liouville Theory
is of very limited practical value due to lack of information about the solutions of the
ordinary differential equations that appear.
Furthermore, the geometry of the region involved in the problem is subject to rather severe restrictions. On one hand, a coordinate system must be employed in which the variables can be separated, and the partial differential equation replaced by a set of ordinary differential equations. For Laplace’s equation there are about a dozen such coordinate systems; only rectangular, circular cylindrical, and spherical coordinates are likely to be familiar to most readers of this book. On the other hand, the boundary of the region of interest must consist of coordinate curves or surfaces, that is, curves or surfaces on which one variable remains constant. Thus, at an elementary level, one is limited to regions bounded by straight lines or circular arcs in two dimensions, or by planes, circular cylinders, circular cones, or spheres in three dimensions.