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Unformatted text preview: Exercise 2.5.2):
n := 100; rho[x ] := 1/(x + .1);
m := Table[Max[2Abs[ij],0], { i,n1 } ,{ j,n1 } ];
p := m  4 IdentityMatrix[n1];
q := DiagonalMatrix[Table[(1/rho[i/n]), { i,1,n1 } ]];
a := n∧2 q.p; eigenvec = Eigenvectors[N[a]];
ListPlot[eigenvec[[n1]]]
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.
 Winter '14
 Equations

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