1D advection with damping/forcing

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In summary, we have a first order PDE with a damping term and a force term, similar to the inviscid Burger equation. The general solution can be expressed as 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 the solution becomes u(t,x) = -e^{-Bt}[\int_0^tBf(A\xi-At+x)e^{B\xi}d\xi -f(-At+x)]. The solution takes one form for x-vt
  • #1
Tunneller
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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
 
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  • #2
Your PDE is first order, so you need only one boundary condition.

The general solution to your PDE is as follows

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

where F is an arbitrary function.

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

[tex]u(t,x) = -e^{-Bt}[\int_0^tBf(A\xi-At+x)e^{B\xi}d\xi -f(-At+x)][/tex]
 
  • #3
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
 

1. What is 1D advection with damping/forcing?

1D advection with damping/forcing is a mathematical model used to describe the movement of a substance or property in one direction, while taking into account the effects of both damping and forcing. This model is commonly used in atmospheric and oceanic sciences to study the transport of heat, moisture, and pollutants.

2. How is damping incorporated into the 1D advection model?

Damping is incorporated into the 1D advection model through the addition of a term that represents the loss of the substance or property being transported. This can be due to factors such as friction, diffusion, or other dissipative processes.

3. What is the role of forcing in 1D advection?

Forcing in 1D advection refers to the external factors that drive the movement of the substance or property being transported. This can include wind, temperature gradients, or other external forces that influence the direction and speed of advection.

4. What are some real-world applications of 1D advection with damping/forcing?

1D advection with damping/forcing is used in a variety of fields, including atmospheric and oceanic sciences, meteorology, and environmental engineering. It can be used to model air and water pollution, weather patterns, and the transport of nutrients and sediments in aquatic ecosystems.

5. How is 1D advection with damping/forcing different from other advection models?

Unlike other advection models that only consider advection in one direction, 1D advection with damping/forcing takes into account the effects of both damping and forcing. This allows for a more accurate representation of the transport processes and their impact on the substance or property being studied.

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