It is therefore usually necessary to use numerical

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

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