# When x c n1 a c nn and a is hermitian the expression

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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 http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 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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