# Thus the region apb is the domain of dependence of p

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online