problems

mathgeek · #1 $Old$ September 7th, 2009, 07:11 PM

1) a) find the general solution u(x,t) to
ut+xux=u

b) what is the soln that satisfies the auxilliary conidtion u(x,0)=x^2?

2) suppose fx is a conitinuously differentiable vector field in R3 that satisfies the bound

|fx|<= 1/1+|x|^3

show that the triple integral (radiant).(Fdx)=0.

3) a solid, homogeneous object occupies the region D is contained R^3, and is completely insulated. its initial temperature in given by u(x,0)=f(x), for some function f. so u satisfies the heat equation ut=K().(delta u) with boundary condition partial of u/partial n=0 on partial of D. after a long time, the object reaches a steady, uniform temperature. in terms of f, what is this temperature?

Coomast · #2 $Old$ September 13th, 2009, 04:54 AM

hello mathgeek,

You can separate this equation using the following assumption:

$u(x,y)=X(x)\cdot T(t)$

Meaning that we assume that the solution can be written as the product of two functions X and T, each respectively depending on only x and t. After substituting this in the equation you get:

$XT'+xX'T=XT$

After dividing by XT and rewriting this becomes:

$x\frac{X'}{X}=1-\frac{T'}{T}$

Because the left and right hand side are only depending on x and t respectively, these must be equal to a constant, say $\alpha$ This gives rise to two ordinary differential equations which can be solved fairly easy. We have for these:

$X(x)=K_2x^{\alpha}$
$T(t)=K_1e^{(1-\alpha)\cdot t}$

The solution is now:

$u(x,t)=K\cdot x^{\alpha}\cdot e^{(1-\alpha)\cdot t}$

Can you take it from here to solve the next part of the question?

coomast