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Unformatted text preview: Math 124A October 05, 2010 Viktor Grigoryan 3 Method of characteristics revisited 3.1 Transport equation A particular example of a first order constant coefficient linear equation is the transport, or advection equation u t + cu x = 0, which describes motions with constant speed. One way to derive the transport equation is to consider the dynamics of the concentration of a pollutant in a stream of water flowing through a thin tube at a constant speed c . Let u ( t,x ) denote the concentration of the pollutant in gr/cm (unit mass per unit length) at time t . The amount of pollutant in the interval [ a,b ] at time t is then Z b a u ( x,t ) dx. Due to conservation of mass, the above quantity must be equal to the amount of the pollutant after some time h . After the time h , the pollutant would have flown to the interval [ a + ch,b + ch ], thus the conservation of mass gives Z b a u ( x,t ) dx = Z b + ch a + ch u ( x,t + h ) dx. To derive the dynamics of the concentration u ( x,t ), differentiate the above identity with respect to b to get u ( b,t ) = u ( b + ch,t + h ) . Notice that this equation asserts that the concentration at the point b at time t is equal to the con- centration at the point b + ch at time t + h , which is to be expected, due to the fact that the water containing the pollutant particles flows with a constant speed. Since b is arbitrary in the last equation, we replace it with x . Now differentiate both sides of the equation with respect to h , and set h equal to zero to obtain the following differential equation for u ( x,t ). 0 = cu x ( x,t ) + u t ( x,t ) , or u t + cu x = 0 . (1) Since equation (1) is a first order linear PDE with constant coefficients, we can solve it by the method of characteristics. First, we rewrite the equation as (1 ,c ) u = 0 , which implies that the slope of the characteristic lines is given by dx dt = c 1 . Integrating this equation, one arrives at the equation for the characteristic lines x = ct + x (0) , (2) where x (0) is the coordinate of the point at which the characteristic line intersects the x-axis. The solution to the PDE (1) can then be written as...
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