- #1
NeoDevin
- 334
- 2
I need a little help solving an equation here, I don't really know where to start. If anyone has any advice on solving (or even simplifying) such a beast, it would be much appreciated.
[tex]\frac{\partial}{\partial t}N + \frac{1}{\tau}N = -a(t)\frac{\partial}{\partial u}N - u\frac{\partial}{\partial x}N -v\frac{\partial}{\partial y}N + \frac{1}{\tau}D(u,v)\int_{-\infty}^\infty\int_{-\infty}^\infty Ndudv[/tex]
Where
[tex]N = N(u,v,x,y,t)[/tex]
[itex]D(u,v)[/itex] Is a given normalized distribution function. I get to choose this, if you need a particular example, use:
[tex]D(u,v) = \frac{1}{2\pi v_{th}^2}e^{-(u^2+v^2)/2v_{th}^2}[/tex]
[tex]a(t) = A_0\cos(\omega t)[/tex]
[itex]A_0[/itex], [itex]v_{th}[/itex], and [itex]\tau[/itex] are all positive real constants.
And the solution [itex]N[/itex] must be normalizable over [itex]u[/itex] and [itex]v[/itex]
[tex]\frac{\partial}{\partial t}N + \frac{1}{\tau}N = -a(t)\frac{\partial}{\partial u}N - u\frac{\partial}{\partial x}N -v\frac{\partial}{\partial y}N + \frac{1}{\tau}D(u,v)\int_{-\infty}^\infty\int_{-\infty}^\infty Ndudv[/tex]
Where
[tex]N = N(u,v,x,y,t)[/tex]
[itex]D(u,v)[/itex] Is a given normalized distribution function. I get to choose this, if you need a particular example, use:
[tex]D(u,v) = \frac{1}{2\pi v_{th}^2}e^{-(u^2+v^2)/2v_{th}^2}[/tex]
[tex]a(t) = A_0\cos(\omega t)[/tex]
[itex]A_0[/itex], [itex]v_{th}[/itex], and [itex]\tau[/itex] are all positive real constants.
And the solution [itex]N[/itex] must be normalizable over [itex]u[/itex] and [itex]v[/itex]