? Transport equation: Suppose u(x, t) = f(x - 2t) is a solution to the transport

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Transport equation: Suppose u(x, t) = f(x - 2t) is a solution to the transport equation ut + 2ux = 0 where u is a density (mass/meter) and f is a function of one variable. Let M(t) = u(x, t)dx be a mass of in the interval [a(t), b(t)]. Find two functions a(t) and b(t) such that dM/dt = 0. Hint: there are many such functions, you must choose one example. Wave equation analysis: For the initial position f(x) = H(1 - x ) and initial velocity g(x) = H(1 - x - 10 ), the wave equation utt = uxx, with c = 1 on the real x-line has the solution u(x, t) = 1/2 [f(x + ct) + f(x - ct)] + 1/2c g( )d . Determine if the solution takes on the value u(x = 0, t = 5) = 0 or u(x = 0, t = 5) > 0. Hint: this can be answered by direct computation of the solution. Flux: Given a flux function J = -a u/ x - bu find a combination of values a > 0 and b > 0 such that (1) dM/dt > 0, (2) dM/dt = 0, and (3) dM/dt

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can you please specify , which one you want to get solved ?

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