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Unformatted text preview: ChBE 521 Homework 7 due Wednesday, November 11 1. How would you use ﬁnite diﬀerences to solve the boundary value problem in spherical coordinates d dr r2 dc(r) dr = kc(r) subject to the boundary conditions c(1) = 1, Solve the problem using N = 100. 2. Using ﬁnite diﬀerences (N = 50), solve the partial diﬀerential equation c(t, x) ∂c(t, x) ∂ 2 c(t, x) + = 0.1 2 ∂t ∂x (1 + c(x, t)) subject to the boundary conditions c(t, 0) = 1, and the initial condition c(0, x) = 0. 3. Using collocation (N = 10), solve the problem ∂ 2 c(t, x) ∂c(t, x) = 0.1 + c(x, t) ∂t ∂x2 subject to the boundary conditions ∂c(t, 0) = 0, ∂x and the initial condition c(0, x) = 1. 4. Using ﬁnite diﬀerences/CrankNicolson (N = 100) and Strang splitting, solve the the partial diﬀerential equation ∂c(t, x, y ) ∂ 2 c(t, x, y ) ∂ 2 c(t, x, y ) = +2 2 ∂t ∂x ∂y 2 subject to the boundary conditions ∂c(t, 0, y ) ∂c(t, 1, y ) ∂c(t, x, 1) ∂c(t, x, 1) = = = =0 ∂x ∂x ∂y ∂y and the initial condition c(0, x, y ) = = 1, 0, 1 x ≤ 0.5 and y ≤ 0.5 otherwise. ∂c(t, 1) = 0, ∂x c(t, 1) = 2, dc(0) = 0. dr ChBE 521 Homework 7 due Wednesday, November 11 5. Using the ﬁnite volume (upwind method) with N = 100, solve the advection equation ∂c(t, x) ∂c(t, x) + =0 ∂t ∂x subject to the boundary condition c(t, 1) = 0 and the initial condition c(0, x) = = 1, 0, x ≤ 0.1 otherwise (x > 0.1). 6. Suggest a strategy for simultaneously solving ∂c(t, x) ∂c(t, x) ∂ 2 c(t, x) + v (x) =D ∂t ∂x ∂x2 subject to the boundary conditions c(t, 0) = c(t, 1) = 0, and initial condition c(0, x) = c0 (x). Choose N = 4. 2 ...
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This note was uploaded on 11/30/2010 for the course CHBE CHBE521 taught by Professor Chrisrao during the Spring '10 term at University of Illinois, Urbana Champaign.
 Spring '10
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