Solution to pde with directional derivative first order

[balugaa]

Homework Statement

trying to solve $$v.\nabla_x u + \sigma(x) u = 0$$

$$(x,v) \in \Gamma_-$$

$$\Gamma_- = \left\{(x,v) \in$$ X x V, st. $$-v.\nu(x) > 0\right\}$$

$$\nu(x) =$$outgoing normal vector to X

$$v =$$velocity

$$u =$$density

$$g(x) =$$Incoming boundary conditions

The Attempt at a Solution

I think that covers it .. Basically trying to work this out, not sure I am on the right track though, basically went through the usual Integration factor approach

i.e. saying
$$u = exp(-\int \sigma(x)) g(x)$$

Questions
Not sure how the directional derivative is dealt with ?
What does $$\nabla_x$$ mean?
with equations like these what's the best approach?

Please advise

 

[mighty2000]

.

As a fellow scientist, I can offer some advice on how to approach this problem. First, let's break down the equation and define some terms:

- The equation is a partial differential equation (PDE) with the variable u, which represents density, and the variables x and v, which represent position and velocity, respectively.
- The term v.\nabla_x u represents the directional derivative of u in the direction of v.
- The term \sigma(x) represents a coefficient that depends on the position x.
- The term \Gamma_- represents a boundary, specifically the part of the boundary where the outgoing normal vector is pointing in the opposite direction of v.
- The function g(x) represents the incoming boundary conditions, which specify the value of u at the boundary.

Now, to solve this PDE, we can use the method of characteristics. This involves finding curves in the (x,v) space along which the PDE reduces to an ordinary differential equation (ODE). We can then solve the ODE and use the solution to construct the solution to the PDE.

To find the characteristic curves, we can use the following equations:

dx/dt = v
dv/dt = -\sigma(x)

where t is a parameter along the characteristic curve. These equations can be solved to find the curves, and we can then use the solutions to construct the solution to the PDE.

As for the directional derivative, we can rewrite the equation as:

v.\nabla_x u = -\sigma(x) u

which is an ODE along the characteristic curves. We can then use the solution to this ODE to construct the solution to the PDE.

In general, for equations like these, the best approach is to first understand the physical meaning of the terms and then use appropriate mathematical methods, such as the method of characteristics, to solve the equation. I hope this helps!