Math 19. Lecture 14Introduction to Advection (I)T. JudsonFall 20041 An Expl osion ExampleSuppose a meltdown at a nuclear reactor pumps radioactive pollution intothe air. A wind blows from west to east at 3 m/sec. Particles fall out of theair at a constant rate r. We wish to know the particle concentration eastand west of the explosion at location x and time t. Let the density function,u = u(t, x),be the number of particles per meter.The amount of particulate mater in the region between x a nd x + ∆x attime t is approximately u(t, x)∆x. The rate of change with respect to timeisddtu(t, x)∆x = q(t, x) − q(t, x + ∆x) + k(t, x)∆x,where• q(t, x) is the number of particles that pass x from left to right, so−q(t, x) is the number of particles that pass x from right to left.• q(t, x + ∆x) is the number of particles that pass x + ∆x from left toright, so −q(t, x+∆x) is the number of particles that pass x +∆x fromright to left.• k(t, x) is the net numb er of part icles created in [x, x + ∆x]. Thatis, k(t, x) is the number created minus the number destroyed. In ourexample,k(t, x) = −ru(t, x).1Thus,ddtu(t, x)∆x = q(t, x) − q(t, x + ∆x) + k(t, x)∆x,orddtu(t, x) = −q(t, x + ∆x) − q(t, x)∆x+ k(t, x).As ∆x → 0, this last expression becomes∂u∂t(t, x) = −∂q∂x(t, x) + k(t, x).In our example,k(t, x) = −ru(t, x)q(t, u) = 3u(t, x).Thus, we obtain the advection equation.∂u∂t= −3∂u∂x− ru.2 Solution s to the Advection EquationEvery solution to∂u∂t= −3∂u∂x− ru.can be written in the formu(t, x) = e−rtf(x − 3t),where f is any differentiable function in one variable. The choice of f isdetermined by initial and boundary conditions.Homework• Chapter 13. Exercises 1, 3, 5, 6, 7, 8; pp. 213–215.2Reading and References• C. Taubes. Modeling Differential Equations in Biology. Prentice Hall,Upper Saddle River, NJ, 2001. Chapter 13.• “Malaria: Focus o n Mosquito Genes” pp.
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