Assignment # 6
Math 456 - Spring 2005
Due: Tuesday, April 11
1. (20 pts)
(a) Show that v(x, y) = F(yā3x), where Fis an arbitrary differentiable
function, is a general solution of the conservation model vx+ 3vy= 0.
(b) Find the particular solution which satisfies the condition v(0, y) = 4 sin(y).
2. (20 pts) Use the method of separation of variables to solve the boundary value
problem,
īux= 4uy
u(0, y) = 8eā3y
3. (15 pts) Use the method of separation of variables to solve the heat equation
in the domain 0 ā¤xā¤3 for t > 0,
ļ£±
ļ£“
ļ£“
ļ£²
ļ£“
ļ£“
ļ£³
ut= 2uxx
u(x, 0) = 5 sin(4Ļx)ā3xsin(8Ļx) + 2 sin(10Ļx)
u(0, t) = 0
u(3, t) = 0
4. (15 pts) Use an explicit forward numerical method to approximate the solution
to the following conservation law
ļ£±
ļ£“
ļ£“
ļ£²
ļ£“
ļ£“
ļ£³
ut+ux= 0 1 >x>0, t > 0
u(x, 0) = 2 + x1ā„xā„0
u(0, t) = 2 āt t > 0
u(1, t) = 0 t > 0
Obtain all the solutions from t= 0 until t= 1 with a time step-size k=.2 for
0ā¤xā¤1 with a space step-size h=.1.
5. (15 pts) Use an explicit backward numerical method to approximate the solu-
tion to the previous problem with exactly the same parameters. Produce the
corresponding table of solutions in the entire domain.
6. (15 pts) Use an explicit centered numerical method to approximate the solution
to the following conservation law
ļ£±
ļ£“
ļ£“
ļ£²
ļ£“
ļ£“
ļ£³
ut=uxx 0<x<1, t > 0
u(x, 0) = 100 sin(Ļx) 0 <x<1
u(0, t) = 0 t > 0
u(1, t) = 0 t > 0
Obtain all the solutions from t= 0 until t= 1 with a time step-size k=.25
for 0 ā¤xā¤1 with a space step-size h=.2.
