Lecture10

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Unformatted text preview: ts (x0 t0), (x0 ; at0 0), and (x0 + at0 0). (Actually, with q being a function of x only, the domain of dependence only involves values of q(x) on the subinterval (x0 ; at0 0) to (x0 + at0 0).) 0 1 10.1. Conservation Laws t 7 P(x 0 ,t 0) 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 dx/dt = a dx/dt = -a 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 x 0 - at 0 x 0 + at x 0 Figure 10.1.3: The domain of dependence of a point P (x0 t0) for Example 10.1.3 is the triangle connecting the points P , (x0 ; at0 0), and (x0 + at0 0). Transforming back to the physical variables 1 (w + w ) = p w0(x + at) + w0(x ; at)] + 1 Z x+at q( )d 1 u1(x t) = p 1 2 2 2a x;at 2 21 Zx Zx 1 (w ; w ) = p w0(x + at) ; w0(x ; at)] ; 1 1 u2(x t) = p 1 2 q( )d + q( )d ]: 2 2a x+at 2 21 x;at Suppose, for simplicity, that u0(x) = 0, then _ 10 0 u1(x 0) = 0 = p w1 (x) + w2 (x)] 2 10 0 u2(x 0) = au0 (x) = p w1 (x) ; w2 (x)]: x 2 Thus, 0 0 0 p2 w1 (x) = ;w2 (x) = aux(x) and a u0 (x + at) ; u0 (x ; at)] + 1 Z x+at q( )d u1(x t) = 2 x x 2a x;at Zx Zx a u0 (x + at) + u0 (x ; at)] ; 1 u2(x t) = 2 x x 2a x+at q( )d + x;at q( )d ]: Since u2 = aux, we can integrate to nd the solution in the original variables. In order to simplify the manipulations, let's do this with q(x) = 0. In this case, we have u2(x t) = a u0 (x + at) + u0 (x ; at)] x 2x 8 Hyperbolic Problems hence, u(x t) = 1 u0 (x + at) + u0(x ; at)]: 2 The solution for an initial value problem when 8 < x + 1 if ; 1 x 0 0 u (x) = : 1 ; x if 0 x 1 0 otherwise is shown in Figure 10.1.4. The initial data splits into two waves having half the initial amplitude and travel...
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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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