Math 19. Lecture 15Introduction to Advection (II)T. JudsonFall 20051 The Advection EquationRecall that an advection equation has the form∂u∂t= −c∂u∂t− ru.The solutions to this equations are of the formu(t, x) = e−rtf(x − ct),where f is any differentiable function in one variable and the choice of f isdetermined by initial a nd boundary conditions.2 Boundary and Initial ConditionsSuppose that we know u(0, x) = g(x) is an initial condition for∂u∂t= −3∂u∂t− ru.That is, the particle density is given by g(x) right before the explosion. Sinceevery solution to this PDE can be written in the formu(t, x) = e−rtf(x − 3t),we know thatg(x) = u(0, x) = f(x)oru(t, x) = e−rtg(x − 3t).13 Traveling Wave SolutionsFirst, observe that ut= −3ux− ru predicts the values for u(t, x) at all timest ≥ 0, and all of the points x. Then q(t, x) = 3u(t, x) is predictive when thevalue of u(t, 0) is specified for all t. If u(t, 0) = h(t), we say that this is aboundary condition for ut= −3ux− ru. Thus,h(t) = u(t, 0) = e−rtf(−3t),or if we make the substitution s = −3t,f(s) = e−rs/3h(−s/3).Therefore, our solution becomesu(t, x) = e−rte−r(x−3t)/3h((3t − x)/3).In the example of our meltdown, this resembles a traveling wave.Homework• Chapter 13. Exercises 1, 3, 5, 6, 7, 8; pp. 213–215.Reading and References• C. Taubes. Modeling Differential Equations in Biol ogy. Prentice Hall,Upper Saddle River, NJ, 2001. Chapter 13.• “Malaria: Focus on Mosquito Genes” pp.
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