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ESAM 411-2 Differential Equations of Mathematical Physics Homework 3

ESAM 411-2 Differential Equations of Mathematical Physics Homework 3
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Due 2 pm, March 17, 2020

You must show details and when possible you should verify that the answer is correct. You must work alone. Your work must be neatly written.

1.(3 points) Find and discuss (include an xt diagram) the solution of uxx ? c?2utt = 0

for0<t<,0<x<. Supposethattheinitialconditionsareu(x,0)=0 forx>0andut(x,0)=0forxabutut(x,0)=1forx>a. Assumethatthe boundary condition at x = 0 is ux(0, t) = 0 for t > 0. Also find the behavior of the solution as t → ∞.

2. (3 points) Solve the problem
?2u ???2u ?2u??

?t2? ?x2+?y2 =F(x,y)cosωt for0<x<π,0<y<π,andt>0. Lettheboundaryconditionsbe

and the initial conditions

u(0,y,t) ?u(π,y,t)

?x

= =

u(x,π,t) = 0 (1) ?u(x,0,t) = 0, (2)

?y

u(x,y,0) = ?u(x,y,0) = 0. ?t

3. (3 points) Solve the two dimensional wave equation ?2u ? c2?2u = 0

?t2

for all (x, y) and t > 0. The initial conditions are u(x, y, 0) = xy and ut(x, y, 0) = 1. If something can be integrated, do it!

4. (6 points) Solve the two dimensional wave equation ?2u ? ?2u = 0

?t2
in the region 0 < x < π, y > 0, t > 0, with the initial conditions

u(x,y,0) = ?u(x,y,0) = 0, ?t

and the boundary conditions
u(0,y,t) = u(π,y,t) = 0,

and

The answer is:

and
u(x,y,t) = sin(x)?2 ?? 1 1?ω2 cosh(y 1?ω2)?2sinh(y 1?ω2) sin(ωt)?

?u(x, 0, t) + 2u(x, 0, t) = sin(x). ?y

u(x,y,t) = 0 for t < y,

? ??√ √ √ ?? ?

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for t > y.

?π 0
+61 sin(x) ??ey + 3e?y ? 4e?2y+t3?? ,

ω(ω2+3) ?