Pde

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: with variable coefficients: ∂u ∂ =x ∂t ∂x x ∂u ∂x − 3u, u(1, t) = u(eπ , t) = 0, u(x, 0) = 3 sin(log x) + 7 sin(2 log x) − 2 sin(3 log x). 4.8 Numerical solutions to the eigenvalue problem* We can also apply Sturm-Liouville theory to study the motion of a string of variable mass density. We can imagine a violin string stretched out along the x-axis with endpoints at x = 0 and x = 1 covered with a layer of varnish which causes its mass density to vary from point to point. We could let ρ(x) = the mass density of the string at x for 0 ≤ x ≤ 1. If the string is under constant tension T , its motion might be governed by the partial differential equation ∂2u T ∂2u = , ∂t2 ρ(x) ∂x2 (4.34) which would be subject to the Dirichlet boundary conditions u(0, t) = 0 = u(1, t), for all t ≥ 0. (4.35) It is natural to try to find the general solution to (4.34) and (4.35) by separation of variables, letting u(x, t) = f (x)g (t) as usual. Substituting into (4.34) yields T g (t) T f...
View Full Document

This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online