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

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

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

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

$
u = f(x-ct)
$

i know a bout the charactertics solution $f(x-ct)$
but i wanna a solution that show the effect of bounday condition feeding on the solution.i mean a solution that is made by the IC/BC assumption
not just to supsitute into the $f(x-ct)$