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MSO 203b Assignment-05 November 12, 2012.

1. Let = {(r, ) R2 | 0 r < 1, < } denote the unit disk in R2. Solve theLaplace equation u = 0 with

(a) Dirichlet boundary condition u(1, ) = 1 + sin2 + 3 cos .

(b) Neumann boundary condition

u

(1, ) = cos 2 + 2 cos .

2. Assuming azimuthal symmetry of the function u, solve the Laplace equation u = 0 inthe unit sphere

= {(r,,) R3 | 0 r < 1, 0 < , 0 < 2}

with boundary conditions

(a) u(1, , ) = 1.

(b) u(1, , ) = 2 cos2 + cos .

3. Use separation of variable to solve for u(x, t) in the heat equation

ut uxx = 0 in (0, ) (0,)u(0, t) = u(, t) = 0 in (0,)

u(x, 0) = 4 sin x + 2 sin 2x + 7 sin 3x in (0, ).

4. Use separation of variable to solve for u(x, t) in the wave equation

utt uxx = 0 in (0, ) (0,)u(0, t) = u(, t) = 0 in (0,)

u(x, 0) = 5 sin x + 12 sin 2x + 6 sin 3x in (0, )ut(x, 0) = 0 in (0, ).

5. Use separation of variable to solve for u(x, t) in the wave equation

utt c2uxx = 0 in (0, ) (0,)

u(0, t) = u(, t) = 0 in (0,)u(x, 0) = 0 in (0, )

ut(x, 0) = sin 3x in (0, ).

6. Use Duhamels principle to solve the inhomogeneous wave equation

utt c2uxx = sin 3x in (0, ) (0,)

u(0, t) = u(, t) = 0 in (0,)u(x, 0) = ut(x, 0) = 0 in (0, ).

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MSO 203b Page 2 of 2 November 12, 2012.

7. Solve the wave equation

utt = 16uxx in R (0,)u(x, 0) = 6 sin2 x in R

ut(x, 0) = cos 6x in R.

8. Solve the inhomogeneous wave equation

utt uxx = x

2 t in R (0,)u(x, 0) = ut(x, 0) = 0, in R