MHB How Can You Solve the Scaled Transport Equation in the First Quadrant?

[onie mti]

am given a scaled transport equation
u_t(x,t) + u_x(x,t)=0 x>0; t>0
u(x,0)=0 x>0
u(0,t)= sint t>0

how can I begin to find a solution in the quadrant {x.0,t>0} to this problem, am really struglling:(

 

[evinda]

am given a scaled transport equation
u_t(x,t) + u_x(x,t)=0 x>0; t>0
u(x,0)=0 x>0
u(0,t)= sint t>0

how can I begin to find a solution in the quadrant {x.0,t>0} to this problem, am really struglling:(

You have to use the method: separation of variables.

 

[chisigma]

am given a scaled transport equation
u_t(x,t) + u_x(x,t)=0 x>0; t>0
u(x,0)=0 x>0
u(0,t)= sint t>0

how can I begin to find a solution in the quadrant {x.0,t>0} to this problem, am really struglling:(

A PDE of the form...

$\displaystyle u_{t} + c\ u_{x} = 0\ (1)$

... can be solved with the auxiliary variables $\xi= x + c\ t$ and $\eta = x - c\ t$. Applying the chain rule You arrive to the equivalent PDE...

$\displaystyle 2\ c\ \frac{\partial{u}}{\partial{\xi}} = 0\ (2)$

... the solution of which is...

$\displaystyle u = f(\eta) = f(x - c\ t)\ (3)$

... where $f(*,*) \in C^{1}$ is arbitrary. In Your case is...

$\displaystyle u(x,t)= - \sin (x - t)\ (4)$

Kind regards

$\chi$ $\sigma$