1. By trial and error, find a solution of the diffusion equation ut = uxx with the initial condition u(x, 0) = x2.

2. (a) Show that the temperature of a metal rod, insulated at the end x = 0, satisfies the boundary condition ∂u/∂x = 0. (Use Fourier’s law.)

(b) Do the same for the diffusion of gas along a tube that is closed off at the end x = 0. (Use Fick’s law.)

(c) Show that the three-dimensional version of (a) (insulated solid) or (b)

(impermeable container) leads to the boundary condition ∂u/∂n = 0.

3.Verify the divergence theorem (i.e. compute both sides) for the disc of radius 2 and the vector fields:

a) V(x,y)= (1,2)

b) V(x,y)= (y,-x)

c) V(x,y)= (0,3x)

d) V(x,y)= (6,2y)

Use the fact that the unit outward normal is (cost, sint) if the point of the boundary is (2cost, 2sint).

Subject | Mathematics |

Due By (Pacific Time) | 09/13/2013 09:15 am |

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