- #1
fahraynk
- 186
- 6
Homework Statement
$$
U_t+U_x+\frac{1}{x} = 0\\
U(x,0)=\phi(x)
$$
Homework Equations
The Attempt at a Solution
I learned somewhat of an algorithm for method of characteristics. It works for a different problem :
$$U_t + U_x - KU = 0 \\
U(x,0)= \phi(x) \\$$
Method :
$$ x_s = 1\\
t_s = 1\\
x=s+x_0\\
t=s+t_0=s+0\\
x-t=x_0\\
U_s -KU = 0\\
U=Ae^{Ks}\\
U=\phi(x-t)e^{kt}$$
For this problem... its not working -_-
The first part is the same, mainly :
$$
x_s=1\\
t_s = 1\\
x=s+x_0\\
t=s\\
x-t=x_0\\
$$
But the next part...
$$U_s + \frac{1}{x} = 0$$
I think x should be x0, so:
$$U_s + \frac{1}{x_0} = U_s + \frac{1}{x-s} $$or$$ U_t +\frac{1}{x-t} = 0$$
(since x=s+x0 or x=t+x0)
$$U= ln(x-t)$$
I'm not sure what to do here... can someone point me in the right direction please...
The books answer is $$-ln(t+1) + \phi(x-t)$$
I think the main transformation from $$U_x+U_t +KU = U_s + KU$$ comes from the fact that if U(x,t) and x(s), t(s), than $$U_s = U_x*x_s + U_t*t_s$$
with $$x_s = 1 \\t_s=1$$
So shouldn't this be the same for the next problem, with x = x0... ?