Non linear wave propagation

(Difference between revisions)
 Revision as of 02:00, 25 December 2005 (view source)← Older edit Revision as of 23:28, 25 December 2005 (view source)Jola (Talk | contribs) (fixed up some formulas)Newer edit → Line 2: Line 2: :$\frac{\partial u}{\partial t}+ u \frac{\partial u}{\partial x}=0 :[itex] \frac{\partial u}{\partial t}+ u \frac{\partial u}{\partial x}=0$ [/itex] + == Domain == == Domain == - x=[-5,10] + :$x \in \left[-5,10\right]$ + == Initial Condition == == Initial Condition == - :$u(x,0)=0 ,x <=0$ + :$u(x,0) = - :[itex] u(x,0)=1 ,x >0$ + \begin{cases} + 0 & x \le 0 \\ + 1 & x > 0 + \end{cases} + [/itex] + == Boundary condition == == Boundary condition == - u[0]=0 + :$u(0,t)=0$ == Exact solution == == Exact solution == - :$u(x,t)=0 ,x<=0$ + :[itex]u(x,t) = - :[itex] u(x,t)=x/t, 0

Problem definition

$\frac{\partial u}{\partial t}+ u \frac{\partial u}{\partial x}=0$

Domain

$x \in \left[-5,10\right]$

Initial Condition

$u(x,0) = \begin{cases} 0 & x \le 0 \\ 1 & x > 0 \end{cases}$

Boundary condition

$u(0,t)=0$

Exact solution

$u(x,t) = \begin{cases} 0 & x \le 0 \\ x/t & 0 < x < t \\ 1 & \mbox{otherwise} \end{cases}$