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che132a-rec-7

# che132a-rec-7 - 2 = ∇ V(a Make a contour plot of your...

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Chemical Engineering 132A Analytical Methods of Chemical Engineering Professor Mike Gordon, Fall 2011 Mathematica Assignment #7 : Visualizing PDE Solutions Commands for today Table[F[m,n],{m,1,20},{n,1,20}] make a 2D table for all combinations of m and n F[[m]][[n]] extract the (m,n) element from the 2D list called F Sum[expr,{m,1,20},{n,1,20}] take the double sum of expr over all values of m and n ContourPlot[f[x,y],{x,0,1},{y,0,1}] make a 2D contour plot of a 3D function f[x,y] Plot3D[f[x,y],{x,0,1},{y,0,1}] make a 3D plot of f[x,y] Today we are going to visualize the 2D solution to Laplace’s equation and investigate the 2D transient heat equation with homogeneous BCs. (1) Using what we derived in class, find the solution to the following electrostatics problem in 2D. Use 50 Fourier terms and generate a table of the coefficients before using them in your Sum[] statement.

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Unformatted text preview: 2 = ∇ V (a) Make a contour plot of your solution with the contours labeled (see the ContourPlot help file). Use the AspectRatio->1/2 option in the ContourPlot to scale the geometry properly. (b) Make a 3D plot of your solution. (2) Investigate the 2D transient heat equation on the unit square with homogeneous Dirichlet conditions on all sides and initial temperature distribution C y x T o 100 ) , , ( = . In other words, you’re solving: T t T 2 ∇ = ∂ ∂ α 1V 0V 0V 5V T(x,y) 2 1 T(x,y,t) 0°C 0°C 0°C 0°C 1 1 V(x,y) Let 1 = α . Use 10 Fourier terms for each axis and generate a table of coefficients using the nested “2-dimensional” Table command as mentioned above. (a) Plot your solution in 3D for t=0.001. Does your solution match the initial condition? (b) Animate your solution for t=0.001 to 0.1, step 0.001. Pretty cool eh?...
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che132a-rec-7 - 2 = ∇ V(a Make a contour plot of your...

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