T 454 find the function ux t dened for 0 x and t

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Unformatted text preview: e equation , ∂2u T ∂2u = , ∂t2 ρ ∂x2 or 96 ∂2u ∂2u = c2 2 , ∂t2 ∂x where c2 = T /ρ. Just as in the preceding section, we could approximate this partial differential equation by a system of ordinary differential equations. Assume that the string has length L = 1 and set xi = i/n and ui (t) = u(xi , t) = the displacement of the string at xi at time t. Then the function u(x, t) can be approximated by the vector-valued function u1 (t) u (t) u(t) = 2 · un−1 (t) of one variable, just as before. The wave equation is then approximated by the system of ordinary differential equations d2 u = c2 n2 P u, dt2 where P is the (n − 1) × (n − 1) matrix described in the preceding section. Thus the differential operator L= d2 dx2 is approximated by the symmetric matrix n2 P, and we expect solutions to the wave equation to behave like solutions to a mechanical system of weights and springs with a large number of degrees of freedom. Exercises: 4.4.1.a. Show that if f : R → R is any well-behaved function of one variable, u(...
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