1D advection with damping/forcing

[Tunneller]

Hi,

I'm looking for a solution to

u_t + A u_x + B (u - f) = 0

where f is a given function linear in x and constant in t.

u(x,0) = f(x)

u(0,t) = f(0)

A and B constant. Does this equation have a name? It's almost the inviscid Burger, but has the damping term (B u) and force term (B f).

Also, any ideas on a solution? I've tried method of characteristics but I trip over the third term and/or the boundary conditions

Regards, John

 

[kosovtsov]

Your PDE is first order, so you need only one boundary condition.

The general solution to your PDE is as follows

$$u(t,x) = -e^{-Bt}[\int_0^tBf(A\xi-At+x)e^{B\xi}d\xi -F(-At+x)],$$

where F is an arbitrary function.

If the boundary condition is u(x,0) = f(x) then

$$u(t,x) = -e^{-Bt}[\int_0^tBf(A\xi-At+x)e^{B\xi}d\xi -f(-At+x)]$$

 

[Tunneller]

Wow! That's a lot more elegant than what I eventually struggled through... :-)

Using a Laplace' transform on the time, then I solved the ODE in x and transformed back. That did use both the boundaries though...

I think the reason I have two boundary statements is that I forgot to mention it is u(x,0) for x>0 and u(0,t) for t>0 so effectively forms a single boundary condition on the "quarter-plane". The solution takes one form for x-vt >0 and another for x-vt<0.

Appreciate the help, regards, John