Thus the region apb is the domain of dependence of p

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Unformatted text preview: 111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000λ dx/dt = λ dx/dt = 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 A B m 1 x Figure 10.1.2: Domain of dependence of a point P (x0 t0). The solution at P depends on the initial data on the line A B ] and the values of b within the region APB bounded by the characteristic curves dx=dt = 1 m. Example 10.1.3. Consider an initial value problem for the forced wave equation (10.1.5a) with the initial data u(x 0) = u0(x) ut(x 0) = u0(x) _ ;1 < x < 1: Transforming (10.1.5a) using (10.1.5b) yields the rst-order system (10.1.2) with A and b given by (10.1.5). Using (10.1.5b), The initial conditions become u1(x 0) = u0(x) _ u2(x 0) = au0 (x) x ;1 < x < 1: 6 Hyperbolic Problems With A given by (10.1.5), we nd its eigenvalues as 1 2 = a. Thus, the characteristics are x= a _ and the eigenvectors are 1 1 P = p 1 ;1 : 1 2 Since P;1 = P, we may use (10.1.7) to determine the canonical variables as ; + w2 = u1p u2 : w1 = u1p u2 2 2 From (10.1.8), the canonical form of the problem is q q (w1)t ; a(w1)x = p (w2)t + a(w2 )x = p : 2 2 The characteristics integrate to x = x0 ; at and along the characteristics, we have dwk = p q dt 2 Integrating, we nd x = x0 + at k = 1 2: 1 Z t q(x ; a )d w1(x t) = w (x0) + p 20 0 or 1 Z x0;at q( )d : 0 w1(x t) = w1 (x0 ) ; p a 2 x0 It's usual to eliminate x0 by using the characteristic equation to obtain 1 Z x q( )d : 0 w1(x t) = w1 (x + at) ; p a 2 x+at Likewise 1 Z x q( )d : 0 w2(x t) = w2 (x ; at) + p a 2 x;at The domain of dependence of a point P (x0 t0) is shown in Figure 10.1.3. Using the bounding characteristics, it is the triangle connecting the poin...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

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