Note that the spreadsheet project consists of 20 % of the HW.
2 pages and 6 Problems.
1.
[Wave equation and numerical methods]
Consider the following initial-boundary value problem for the wave equation on the interval [0 , L ] : u tt
= u xx
, 0 < x < L, t > u (0 , t ) = 0 and u ( L, t ) = 0 , u ( x, 0) = f ( x ) , u t
( x, 0) = g ( x ) ,
0 t > 0
0 ≤ x ≤ L.
(a) For L = 1 , f ( x ) = sin( πx ) and g ( x ) = 0 , use the eigenfunction expansion method to find the solution u ( x, t ) explicitly.
(b) [Spreadsheet project]
Now use a spreadsheet program and a finite difference approximation to solve the problem numerically, taking ∆ x = 0 .
05 and ∆ t = 0 .
005 (as usual, you can use the examples on the website, or build your own from previous spreadsheets, but remember to properly account for the initial velocity condition). Plot the solution at t = 0 .
05 , 0 .
1 , 0 .
15 , 0 .
2 , 0 .
25 and turn that in.
Hint: For explanation for numerical methods, you may look at http://www.math.ubc.ca/%7Eyhkim/yhkim-home/teaching/Math257/fd.pdf
(see the link in the Lecture Notes, supplementary notes, in the course webpage) and also http://www.math.ubc.ca/˜costanza/WaveTutorial.html
You can use the example sheet in the course webpage.
(c) [Spreadsheet project]
Do the same thing as in (b) for L = 1 , g ( x ) = 0 , and f ( x ) =
(
1 0 .
45 ≤ x ≤ 0 .
55 ,
0 otherwise .
2.
[Laplace equation: Dirichlet boundary value problem]
Find the solution u ( x, y ) (explicitly) for the following boundary value problem for the Laplace equation.
u xx
+ u yy
= 0 , u ( x, 0) = x,
0 < x < 1 ,
0 ≤ x ≤ 1 , u ( x, 2) = 0 0 ≤ x ≤ 1 , u (0 , y ) = 0 = u (1 , y ) , 0 ≤ y ≤ 2 .
0 < y < 2
3.
[Steady State Temperature distribution] Find the steady state temperature on a 1 × 1 metal plate if the temperature at one end is kept fixed at T o and the other 3 ends are kept fixed at 0 o
.

4.
[Laplace equation: Dirichlet boundary value problem: in class, Friday March 25 and Monday,
March 28]
Find explicitly the solution of the Dirichlet problem u xx
+ u yy
= 0 for 0 < x < 1 , 0 < y < 1 u ( x, 0) = 1 , u ( x, 1) = − 1 , u (0 , y ) = 1 , u (1 , y ) = − 1 .
Hint: This type of problem u xx
+ u yy
= 0 in the rectangle 0 < x < a, 0 < y < b u (0 , y ) = g
1
( y ) , u ( a, y ) = g
2
( y ) , u ( x, 0) = f
1
( x ) , u ( x, b ) = f
2
( x ) was discussed in the later part of the lecture on Friday, March 25. The idea is to decompose the original problem to the following four simpler problems
( I ) · · · · · ·
( u xx
+ u yy
= 0 for 0 < x < 1 , 0 < y < 1 u ( x, 0) = 1 , and u = 0 along the other three sides
( II ) · · · · · ·
( u xx
+ u yy
= 0 for 0 < x < 1 , 0 < y < 1 u ( x, 1) = − 1 , and u = 0 along the other three sides
( III ) · · · · · ·
( u xx
+ u yy
= 0 for 0 < x < 1 , 0 < y < 1 u (0 , y ) = 1 , and u = 0 along the other three sides
( IV ) · · · · · ·
( u xx
+ u yy
= 0 for 0 < x < 1 , 0 < y < 1 u (1 , y ) = − 1 , and u = 0 along the other three sides
If u
I
, u
II
, u
III
, u
IV are the solutions to the corresponding problem, ( I ) , ( II ) , ( III ) , ( IV ) , respectively, then u = u
I
+ u
II
+ u
III
+ u
IV is the solution for the original problem.
5.
[Laplace equation: mixed boundary condition: in class, Monday, March 28] Find explicitly a solution u ( x, y ) to the following problem: u xx u x
+ u
(0 yy
, y )
= 0 for 0 < x < 1 , 0 < y < 2
= 0 , u x
(1 , y ) = 0 , 0 ≤ y ≤ 2 u ( x, 0) = 0 , u ( x, 2) = x, 0 ≤ x ≤ 1 .
Hint: You can consult [Notes, Example 23.1]
6.
[Laplace equation: Neumann boundary value problem: in class, Monday, March 28] Find an explicit solution u ( x, y ) of the following problem: u xx
+ u yy
= 0 for 0 < x < 1 , 0 < y < 1 u x
(0 , y ) = 0 , u x
(1 , y ) = cos πy + cos 2 πy, u y
( x, 0) = 0 , u y
( x, 1) = 0 , 0 ≤ x ≤ 1 .
0 ≤ y ≤ 1