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Unformatted text preview: MTH 201 Problem Set #9
Due: Thursday, June 10, 5:00pm 1. Solve the following diffusion equation.
u t u
2 x 2 u (0, t ) 0, u ( L , t ) 0 1 u ( x , 0) 0 if if 0 x L/2 L/2 x L 2. Solve the following Laplace equation.
2 x u x 2 u
2 y 2 0 u x
x0 0 u ( x , 0) x , u ( x , b ) 0 3. Solve the following wave equation.
2 t 2 c 2 u
2 x 2 u (0, t ) 0, u ( L , t ) 0 u ( x , 0) 1 4 x ( L x ), u t
t0 0 4. Find the steady-state temperature distribution u ( r , ) in a semicircular plate of radius a by solving the following Laplace equation in the polar coordinate system.
2 r 2 1 u r r 1 u
2 r 2 2 0, 0 r a , 0 u (a , ) u0 u ( r , 0) 0, u ( r , ) 0 5. Find the temperature distribution u ( r , t ) in a circular plate of radius a by solving the following heat equation in the cylindrical coordinate system.
u 2u 1 u 2 t r r r u (a, t ) 0 u ( r , 0) f ( r ) 6. Find the steady-state temperature distribution u ( r , ) in a hemisphere of radius a by solving the following Laplace equation in the spherical coordinate system. (Here, is azimuth angle.) 1 2 u 1 u r 2 sin 0, 0 r a , 0 2 r r r r sin 2 u (r , / 2) 0 u ( a , ) f ( ) (Hint: Pn ( 0) 0 only if n is odd.) Note: ① Please put your homework in the paper box located in front of my office (501-5 EB1). The box will be labeled with the course name. ② I will take away the box at 5:00pm on the due date. No late homework will be accepted. ③ There will be a penalty of 1 point if your homework is not stapled. ...
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This note was uploaded on 11/29/2010 for the course MATH.SCI MAS201 taught by Professor Limmikyoung during the Spring '10 term at 카이스트, 한국과학기술원.
- Spring '10