Applying Gauss' Law to other situations

[res3210]

Hey everyone,

I'm not sure if this belongs in the math or physics section of this forum, but I figure since my question is more related to the mathematical manipulation of what I am dealing with, I figured I would ask it here and then if it has to be moved, it can be.

My question has to do with applying gauss' law to a differential equation I am dealing with. The differential equation is the partial derivative of some function of x,y,z,t with respect to t is equal to a constant times dell^2 of that function plus another constant times the same function. The case I am considering is when the time derivative is equal to zero. So I have:

D*dell^2(n) + C*n = 0

So I am thinking I am basically dealing with a divergence of the vector dell(n) (n is a scalar function). However, I'm not sure how to apply that logic to the C*n part. Can I just subtract it to the other side and take the triple volume integral? Or does the divergence theorem not apply? I think it does because I am basically dealing with a diffusion of particles out from a spherical surface. The only difference is there are particles being generated inside of the sphere as well. That's the C*n term.

 

[UltrafastPED]

Can you type in the equation using LaTex or the special symbols?

 

[pasmith]

Hey everyone,

I'm not sure if this belongs in the math or physics section of this forum, but I figure since my question is more related to the mathematical manipulation of what I am dealing with, I figured I would ask it here and then if it has to be moved, it can be.

My question has to do with applying gauss' law to a differential equation I am dealing with. The differential equation is the partial derivative of some function of x,y,z,t with respect to t is equal to a constant times dell^2 of that function plus another constant times the same function. The case I am considering is when the time derivative is equal to zero. So I have:

D*dell^2(n) + C*n = 0

So I am thinking I am basically dealing with a divergence of the vector dell(n) (n is a scalar function). However, I'm not sure how to apply that logic to the C*n part. Can I just subtract it to the other side and take the triple volume integral? Or does the divergence theorem not apply? I think it does because I am basically dealing with a diffusion of particles out from a spherical surface. The only difference is there are particles being generated inside of the sphere as well. That's the C*n term.

If you integrate your PDE over a volume, then you can indeed replace the $\nabla^2 n$ term by an integral of $\nabla n$ over the boundary surface, but the integrals of the other terms will remain as volume integrals.

Usually such conservation PDEs are obtained by looking at $\frac{d}{dt} \int_V n\,dV$ and then using the divergence theorem to replace the surface term expressing the flux of $n$ out of $V$ with a volume integral.

 

[res3210]

Can you type in the equation using LaTex or the special symbols?

Sorry about that, I am typing from my smart phone, and I don't know how to use LaTex from this interface. Mainly because I'm not familiar with LaTex. If you know any good sources where I can learn it, I'd be willing to use it.