Solve the PDE: [tex]u_t + xu_x = -tu, x \in R, t > 0; \: u(x,0) = f(x), x \in R[/tex]
Hi all. I'm feeling a bit stuck at the moment on a problem from my PDE course. The problem text is in the title, but I'll type it again here too, just in case:
My attempt to solve this was to use characteristic coordinates. To find out what \(\displaystyle \xi\) is, I first solved the ODE \(\displaystyle \dfrac{dx}{dt} = x\). Doing so revealed the solution \(\displaystyle x = Ce^t\). Hence, my characteristic coordinates are \(\displaystyle \xi = x - Ce^t\) and \(\displaystyle \tau = t\).
I then found the partial derivatives of u which are:
\(\displaystyle u_t = U_{\xi} \xi_t + U_{\tau} \tau_{t} = C \cdot e^t \cdot U_{\xi} + U_{\tau}\)
\(\displaystyle u_x = U_{\xi} \xi_x + U_{\tau} \tau_{x} = U_{\xi}\)
I plugged this all in and got:
\(\displaystyle C \cdot e^t \cdot U_{\xi} + U_{\tau} + xU_{\xi} = -\tau U\)
Simplifying and also plugging in \(\displaystyle x = Ce^t\) (I can do that, right?), I get:
\(\displaystyle $U_{\xi} \left(C \cdot e^t \right) + U_{\tau} = -\tau U$\)
But, that's as far as I can seem to go. This is the way we've solved example problems like this in class, so I think I'm headed in the right direction, but I'm just not seeing where to go next. Using the characteristic coordinates doesn't appear to have helped any. In fact, it actually might have even made things worse. Any help would be much appreciated. Thanks in advance!
Hi all. I'm feeling a bit stuck at the moment on a problem from my PDE course. The problem text is in the title, but I'll type it again here too, just in case:
My attempt to solve this was to use characteristic coordinates. To find out what \(\displaystyle \xi\) is, I first solved the ODE \(\displaystyle \dfrac{dx}{dt} = x\). Doing so revealed the solution \(\displaystyle x = Ce^t\). Hence, my characteristic coordinates are \(\displaystyle \xi = x - Ce^t\) and \(\displaystyle \tau = t\).
I then found the partial derivatives of u which are:
\(\displaystyle u_t = U_{\xi} \xi_t + U_{\tau} \tau_{t} = C \cdot e^t \cdot U_{\xi} + U_{\tau}\)
\(\displaystyle u_x = U_{\xi} \xi_x + U_{\tau} \tau_{x} = U_{\xi}\)
I plugged this all in and got:
\(\displaystyle C \cdot e^t \cdot U_{\xi} + U_{\tau} + xU_{\xi} = -\tau U\)
Simplifying and also plugging in \(\displaystyle x = Ce^t\) (I can do that, right?), I get:
\(\displaystyle $U_{\xi} \left(C \cdot e^t \right) + U_{\tau} = -\tau U$\)
But, that's as far as I can seem to go. This is the way we've solved example problems like this in class, so I think I'm headed in the right direction, but I'm just not seeing where to go next. Using the characteristic coordinates doesn't appear to have helped any. In fact, it actually might have even made things worse. Any help would be much appreciated. Thanks in advance!