- #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... ?