• Feb 17th 2010, 05:58 AM
amazing
Hi every body
i wanna to find the solution of the advection equation $\displaystyle u(x,t)$with the following intial and boundary condtion.
can it be solved by variable seperation
$\displaystyle \dfrac {\partial u}{\partial t}+c\dfrac {\partial u}{\partial t}=0$
$\displaystyle u(x,0)=sin(kx)$ intial condition
$\displaystyle u(0,t)=sin(wt)$ boundary condition
• Feb 17th 2010, 06:03 AM
Jester
Quote:

Originally Posted by amazing
Hi every body
i wanna to find the solution of the advection equation $\displaystyle u(x,t)$with the following intial and boundary condtion.
can it be solved by variable seperation
$\displaystyle \dfrac {\partial u}{\partial t}+c\dfrac {\partial u}{\partial t}=0$
$\displaystyle u(x,0)=sin(kx)$ intial condition
$\displaystyle u(0,t)=sin(wt)$ boundary condition

I think probably what you meant is

$\displaystyle \dfrac {\partial u}{\partial t}+c\dfrac {\partial u}{\partial x}=0$

The solution does admit separation of variables but won't work with your IC and BC. The solution however is

$\displaystyle u = f(x-ct)$

i know a bout the charactertics solution $\displaystyle f(x-ct)$
not just to supsitute into the $\displaystyle f(x-ct)$