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**Unformatted text preview: **⎬
T⎝
A=
λ2 = (T /mL)(2 − √2) ,
−1
2 −1 ⎠ with
⎩
⎭
mL
0 −1
2
λ3 = (T /mL)(2 + 2) IG
R Y
P and a complete orthonormal set of eigenvectors is
⎛
⎛
⎞
⎞
⎛
⎞
1
1
1
1⎝
1 ⎝√ ⎠
1⎝ √ ⎠
x1 = √
−2 .
2 , x3 =
0 ⎠ , x2 =
2
2
2 −1
1
1 O
C The three corresponding normal modes are shown in Figure 7.6.3. Mode for (λ1 , x1 ) Mode for (λ2 , x2 ) Mode for (λ3 , x3 ) Figure 7.6.3 Copyright c 2000 SIAM Buy online from SIAM
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7.6 Positive Deﬁnite Matrices
http://www.amazon.com/exec/obidos/ASIN/0898714540 563 Example 7.6.2
Discrete Laplacian. According to the laws of physics, the temperature at time
t at a point (x, y, z ) in a solid body is a function u(x, y, z, t) satisfying the
diﬀusion equation
∂u
= K ∇2 u,
∂t where ∇2 u = ∂2u ∂2u ∂2u
+ 2+ 2
∂x2
∂y
∂z It is illegal to print, duplicate, or distribute this material
Please report violations to meyer@ncsu.edu is the Laplacian of u and K is a constant of thermal diﬀusivity. At steady
state the temperature at each point does not vary with time, so ∂u/∂t = 0 and
u = u(x, y, z ) satisfy Laplace’s equation ∇2 u = 0. Solutions of this equation
are often called harmonic functions. The nonhomogeneous equation ∇2 u = f
(Poisson’s equation) is addressed in Exercise 7.6.9. To keep things simple, let’s
conﬁne our attention to the following two-dimensional problem. D
E T
H Problem: For a square plate as shown in Figure 7.6.4(a), explain how to numerically determine the steady-state temperature at interior grid points when
the temperature around the boundary is prescribed to be u(x, y ) = g (x, y ) for
a given function g. In other words, explain how to extract a numerical solution
to ∇2 u = 0 in the interior of the square when u(x, y ) = g (x, y ) on the square’s
76
boundary. This is called a Dirichlet problem. IG
R Solution: Discretize the problem by overlaying the plate with a square mesh
containing n2 interior points at equally spaced intervals of length...

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