Recent content by Irid

Hello, I've come across equations where people use the approximation \int_0^1 \exp(f(x))\, dx \approx \exp \left( \int_0^1 f(x)\, dx\right) I can see that this is correct if f(x) is small, one just uses exp(x) = 1+x+... However, it appears that this approximation has a broader validity...

Oops, sorry, I did. Actually I think I figured out how to find the solution. It's by the Method of Characteristics. No more help needed...

Hello, I am doing some physics and I end up with this PDE: \frac{\partial q(x,y,t)}{\partial t} = -(x^2 + y^2)q(x,y,t) + ax\frac{\partial q(x,y,t)}{\partial y} where q(x,y,t) is the scalar field to determine and a is a parameter. I need to consider two types of initial conditions...

Hmm.. seems to make sense. Why is there a minus sign popping up?

Hello, I was wondering if one can give meaning to the Fourier transform of the linear function: \int_{-\infty}^{+\infty} x e^{ikx}\, dx I found that it is \frac{\delta(k)}{ik} , does this make sense?

I'm trying to solve a non-linear stochastic PDE of the type dx/dt = d2x/ds2 + F[x] + noise(t) and I would really benefit if it could be done with the Fourier transform because then I only need the first few terms of the Fourier expansion to have enough information about the x(s).. of course...

Well I'm trying to keep things simple here by requiring dx/ds = 0 at the boundaries, hence all the sine terms are zero. I don't know what to make of the integration by parts. Here's what I get \int_0^1 x^2 cos(ps\pi)\, ds = -2\int_0^1 x\frac{dx}{ds}\sin(ps\pi)\, ds I don't see how...

I think what you have observed is called "envelope": http://mathworld.wolfram.com/Envelope.html

Hello, I was wondering if such a thing even exists, so here it goes... Let's say I have a function x(s) (it is real, smooth, differentiable, etc.) defined on (0,1). In addition, dx/ds = 0 on the boundary (s=0 and s=1). I can compute its Fourier transform (?) as a_p = \int_0^1 x(s)...

Hmmm... If my initial guess is normalized, all I need to do is to ensure that the norm of Δψ is zero at each iteration. This can be naturally done by choosing a suitable log(C) = int f1/int f2. I've just tried this, and the whole thing converges within 10 iterations max haha :D But apparently...

OK, great, that's some fancy math right there! Indeed, a>0 always. The f(x) will typically be known only numerically on the grid points and is not an elementary function, but generally it will resemble something between a Gaussian and an exponential decay, so it will have a maximum near the...

Hello, I'm not really sure where does this question fit and what title should it bear, but here is my problem: \psi(x) \exp (a\psi(x)^2) = C f(x) given a positive definite f(x), find ψ(x) and the constant C, subject to the condition \int \psi(x)\, dx = 1 I want to solve this numerically...

Hmm.. I'll give it a try. Right now I'm looking forward to learn some FreeFEM, because it's going to be too time consuming to write all the scripts myself..

How easy would that be? I can't find examples of something similar having been done. I get two coupled 3rd order differential equations if I introduce vorticity.

The equations are \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial z^2} = k(x,z) u \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial z^2} = k(x,z) v \frac{\partial u}{\partial x} + \frac{\partial v}{\partial z} = 0