Hi. These are good questions. I encountered a similar problem once of having to estimate derivates from noisy data (for a commercial venture). One issue there was that we did not really know how noisy the data was. Based on the things you've suggested, I can share some of my own experience...
Hi Smallphi,
I'm glad you got Mathematica to produce an "answer," but do you have any way of checking it, aside from the fact that its first and second derivatives look smooth? I ask this because I've seen Mathematica produce wrong solutions to problems. Once I worked with a couple physicists...
Hey Davin,
I'll bet you're still online, so I'll respond to your response before I go to bed. My suggestion of inviting someone to coffee to talk about math was meant exactly how you took it: someone closer to your own age will always give you the most honest advice. That's simply a social...
Hi Davin,
It sounds like you've already made a decision, but I feel like jumping into the conversation anyway. I take you're about 17 years old? At that point in your career, it's impossible to tell which academic subjects will be relevant for interests you plan to pursue later -- and by that...
One more thing: you never said anything about the functions \alpha, \beta, or the initial condition R_0. Presumably they're such that, say, the right hand side remains bounded away from zero and \sqrt{\beta + \alpha/R} remains bounded for all time, preventing the solution from blowing up to...
Hi. You're probably confusing yourself too much by wanting to think of this as a PDE. It really is just an ODE in disguise. That being said, Matlab should be able to solve it using ode45 or whatever solver provides a Runge-Kutta method. (Sorry, my Matlab is a bit rusty.) You just have to look at...
problem evidently resolved
It looks like I've found the problem, indeed a bug: a typo among variable names in initializing the system. Ludicrous how something like that can "emulate" a numerical error, especially ludicrous when considering how much time I spent looking for it! Thanks anyway to...
Hi Emreth,
I said in my last post that the initial condition u_0 is "basically flat at the endpoints." Thinking that that may be the problem, I modified it to satisfy the Neumann BC's exactly: to no avail. The same problem persists.
My starting guess for finding U^{n+1} with Newton's...
To the moderator: please move this to the section on differential equations if you think it would be better there.
I'm looking at a reaction-diffusion equation of the form
\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + f(u),
where, e.g., f(u) = u(1-u) for the...
Maybe this thread is helping you talk yourself through the points until you locate the source of the problem. I really should read up on the Kronig Penny model myself. Do the parameters a, V_0 appear in the constants k_i in your expression for f(\epsilon), or do their values "fall out" in the...
Hi Maverick,
Thanks for the clarification. If I understand it right, the pairs (\chi, \epsilon) satisfying the equation
\cos \chi = f(\epsilon)
should lie on the parabola \epsilon = C\chi^2. So far so good? The jpg you attached indeed shows a discrepancy much higher than one would...
Hi Vivek,
I'd love to help but need a bit more information about how you're calculating energy from the wavenumber. Are you using an explicit formula? I'm not a physicist by training, so please excuse me if there is a closed form for the energy as a function of parameters that I just don't...
Instead of using a plane wave ansatz, why not "integrate" the equations of motion directly? I've written some code in C for precisely this kind of problem. I'll give it to you if you're interested. It's base on the ODE solvers from the GSL:
http://www.gnu.org/software/gsl/
You only have...