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1. ### Clever coordinate substitution for linear PDE

Yes, it was actually the method that I used to solve it (before I knew there was a name for the method =P). The substitution $$r(\phi, v) = A\left(\frac{1}{3}v^3 - v_r^2 v\right) - B\sin(\phi)$$ is the characteristic of the ##\phi, v##-differential equation.
2. ### Clever coordinate substitution for linear PDE

Ah, yes, I see that.. =P But I think I actually have found a way to solve the problem. First to separate out the time dependence as you suggested. Then use the coordinate substitution ##r(\phi, v)## as I wrote in the first post, and simply put ##s(\phi, v) = v##. Then of course ##B\cos(\phi)##...
3. ### Clever coordinate substitution for linear PDE

Sorry, I forgort to mention also that v_r is constant.
4. ### Clever coordinate substitution for linear PDE

Hi! I am currently working with a linear PDE on the form \frac{\partial f}{\partial t} + A(v^2 - v_r^2)\frac{\partial f}{\partial \phi} + B\cos(\phi)\frac{\partial f}{\partial v} = 0. A and B are constants. I wish to find a clever coordinate substitution that simplifies, or maybe even...
5. ### Change of variables in a second order SDE

I have made some progress in the work. Treating H as constant \sigma_v can be found to be \sigma_v = \frac{\pi A R}{\sqrt{t_c}(v - v_r)}\cos(n\phi - \phi_w) by using Itô's lemma on the more simple form d v = \frac{d v}{d\phi}d\phi + \frac{1}{2}\frac{d^2 v}{d\phi^2}d[\phi,\phi]. I...
6. ### Change of variables in a second order SDE

Hello everyone! I am fairly new to SDE theory, so I'm sorry if my question may be a bit naive. I have the following coupled set of SDE:s d\phi = \frac{v - v_r}{R}d t + \frac{\pi}{\sqrt{t_c}}d W d v = A\cos(n\phi - \phi_w)d t + a_v d t + \sigma_v d W. W denotes a Wiener process, and the...
7. ### Nonlinear system of differential equations

Oh, this is really great! Thanks everyone for your help :D
8. ### Nonlinear system of differential equations

Hi! I'm working with my PhD thesis at the moment, and I've stumbled upon a pretty involved problem. What I have is a system of equations like this: \frac{dx}{dt} = A \cos(z) \frac{dy}{dt} = B x \frac{dx}{dt} \frac{dz}{dt} = y where A and B are constants. I also have a stochastic term to z...
9. ### Transmission Coefficient for two step potential

The velocity of the wave affects the probability flux of the wave. The transmission probability is properly defined as the ratio of the probability flux of the transmitted and the incident wave. In the potential step case the transmission probability is then T =...
10. ### How to define the b.c.s for the EM field of a perfectly contucting surface?

If you have a 3 dimensional perfectly conducting body the conditions at the boundary for the EM field is as follows: \boldsymbol{E}_{\parallel} = 0, B_{\perp} = 0, E_{\perp} = \frac{\sigma}{\epsilon_0}, \boldsymbol{B}_{\parallel} = \mu_0 \boldsymbol{j} \times \boldsymbol{\hat{n}} where \sigma...
11. ### Definition of time-ordered product for Dirac spinors

Sorry, I found what I did wrong in the derivations, and now I get it out right with the A definition. :)
12. ### Definition of time-ordered product for Dirac spinors

I guess the answer to this question actually should be pretty obvious, but I still have problems getting it right though. I wonder about the definition of the time ordered product for a pair of Dirac spinors. In all the books I've read it simply says: T\left\{\psi(x)\bar{\psi}(x')\right\} =...
13. ### Construction of an equivalent dielectric tensor

Btw, is it possible that you also take T. Hellsten's course at KTH and have this exercise as a deadline until next thursday? Just wondering. :)
14. ### Construction of an equivalent dielectric tensor

Hi! I actually did this one just a moment ago. I guess you are also solving exercise 6.1 in Melrose, McPhedran's book "Electromagnetic processes in dispersive media". :) You should use the following equations: K_{i,j}(\omega, \textbf{k}) = \delta_{ij} + \frac{i}{\omega...