3 ux y satises the nonhomogeneous boundary condition

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Unformatted text preview: Exercise 2.5.2): n := 100; rho[x ] := 1/(x + .1); m := Table[Max[2-Abs[i-j],0], { i,n-1 } ,{ j,n-1 } ]; p := m - 4 IdentityMatrix[n-1]; q := DiagonalMatrix[Table[(1/rho[i/n]), { i,1,n-1 } ]]; a := n∧2 q.p; eigenvec = Eigenvectors[N[a]]; ListPlot[eigenvec[[n-1]]] If we run this program we obtain a graph of the shape of the lowest mode, as shown in Figure 4.1. Note that instead of approximating a sine curve, our numerical approximation to the lowest mode tilts somewhat to the left. 114 Chapter 5 PDE’s in Higher Dimensions 5.1 The three most important linear partial differential equations In higher dimensions, the three most important linear partial differential equations are Laplace’s equation ∂2u ∂2u ∂2u + 2 + 2 = 0, ∂x2 ∂y ∂z the heat equation ∂u = c2 ∂t ∂2u ∂2u ∂2u + 2+ 2 ∂x2 ∂y ∂z , ∂2u = c2 ∂t2 ∂2u ∂2u ∂2u + 2+ 2 ∂x2 ∂y ∂z , and the wave equation, where c is a nonzero constant. Techniques developed for studying these equations can often be applied to closely related equations. Each of these...
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This document was uploaded on 01/12/2014.

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