# It is therefore usually necessary to use numerical

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Once again, we separate variables, putting all the functions involving t on the left, all the functions involving x on the right: g (t) 1 d = g (t) ρ(x)σ (x) dx κ(x) df (x) dx 1 . f (x) As usual, the two sides must equal a constant, which we denote by λ, and our equation separates into two ordinary diﬀerential equations, g (t) = λg (t), 107 (4.29) and 1 d ρ(x)σ (x) dx κ(x) df (x) dx = λf (x). (4.30) Under the assumption that u is not identically zero, the boundary condition u(a, t) = u(b, t) = 0 yields f (a) = f (b) = 0. Thus to ﬁnd the nontrivial solutions to the homogeneous linear part of the problem, we need to ﬁnd the nontrivial solutions to the boundary value problem: 1 d ρ(x)σ (x) dx κ(x) df (x) dx = λf (x), f (a) = 0 = f (b). (4.31) We call this the eigenvalue problem or Sturm-Liouville problem for the diﬀerential operator 1 d d L= κ(x) , ρ(x)σ (x) dx dx which acts on the space V0 of well-behaved functions f : [a, b] → R which vanish at the endpoints a and b. The eigenvalues of L are the constants λ for which (4.31) has nontrivial solutions....
View Full Document

## This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online