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Unformatted text preview: MATH 300 Fall 2007 Advanced Boundary Value Problems I Solutions to Problem Set 1 To Be Completed by: Friday September 28, 2007 Department of Mathematical and Statistical Sciences University of Alberta Question 1. [p 11, #1.2.4] Derive the diffusion equation for a chemical pollutant. (a) Consider the amount of the chemical in a thin region between x and x + Δ x. (b) Consider the amount of the chemical between x = a and x = b. Solution: (a) A thin rod of constant cross-sectional area A lies along the x-axis with one end at x = 0 and the other end at x = L. We assume that no chemical pollutant flows through the lateral surface of the rod. For 0 ≤ x ≤ L and t ≥ , we let u ( x, t ) be the density or concentration of chemical pollutant per unit volume at position x at time t. Now consider the amount of chemical pollutant contained in a small slice of the rod from position x to x + Δ x, as in the figure below. x x = x = Δ x A x + L Let φ ( x, t ) be the flux of the chemical at position x at time t, that is, the amount of the chemical per unit surface area flowing to the right per unit time at position x at time t (assume φ is positive if the chemical is flowing to the right). Now, if there are no chemical sources inside the slice, the rate of change of the amount of chemical in the slice is equal to the amount of chemical per unit time flowing into the slice minus the amount of chemical per unit time flowing out of the slice, that is, ∂u ∂t ( x, t ) · A Δ x ≈ [ φ ( x, t )- φ ( x + Δ x, t )] A, so that ∂u ∂t ( x, t ) ≈ φ ( x, t )- φ ( x + Δ x, t ) Δ x , and letting Δ x → , we have ∂u ∂t =- ∂φ ∂x . ( I ) In solids, chemicals spread from regions of high concentration to regions of low concentration, and according to Fick’s law of diffusion , the flux is proportional to ∂u ∂x , that is, φ =- k ∂u ∂x where the positive constant k is the chemical diffusivity of the rod. If the chemical concentration is increasing to the right, so ∂u ∂x > , then the atoms of chemicals migrate to the left, and vice-versa, and using Fick’s law in the conservation law ( I ) , we get ∂u ∂t = k ∂ 2 u ∂x 2 . ( II ) This is the one-dimensional diffusion equation , the partial differential equation that governs dif- fusion of chemicals in a one-dimensional rod. (b) Now we give an alternate derivation where we consider the total amount of chemical U ( t ) at time t in a finite segment from x = a to x = b by adding the amounts in infinitesimal slices to get U ( t ) = Z b a u ( x, t ) A dx. The rate of change of the amount of chemical with respect to t is d dt Z b a u ( x, t ) A dx = [ φ ( a, t )- φ ( b, t )] A that is, it is the difference between the rate at which the chemical flows into the end x = a and the rate at which the chemical flows out of the end x = b. Again, there is no flow of chemical through the lateral sides of the rod and there are no sources of the chemical in the rod. Therefore Z b a ∂u ∂t ( x, t...
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- Spring '00
- Laplace, Boundary value problem, Partial differential equation, boundary condition