I'm trying to solve the following PDE:
$$u_t+yu_x=-y-\mathbb{H}(y_x)$$
where y satisfies the inviscid Burgers equation
$$y_t+yy_x=0$$
and the Hilbert Transform is defined as
$$\mathbb{H}(f) = PV \int_{-\infty}^{\infty} \frac{f(x')}{x-x'} \ dx',$$
where PV means principal value.
The solutions...
Hi,
Given a holomorphic function u(x,y) defined in the half plane ( x\in (-\infty,\infty), y\in (-\infty,0)), with boundary value u(x,0) = f(x) , the solution to this equation (known as the Poisson integral formula) is
u(x,y) = \int_{-\infty}^{\infty} \frac{y\ f(t) }{(t-x)^2 +y^2} \...
Hi,
I'm solving a problem numerically that takes the form
Q_{ij} \ddot{y}_j +S_{ijk}\dot{y}_j\dot{y}_k +V_i=0,
where (Q_{ij},S_{ijk},V_i) are all functions of the dependent variables y_i. The dependent variables are all functions of the variable t. The resolution of this spectral...
Hi,
First off, apologies if this is in the wrong place - any directions on where this is more appropriate are appreciated.
I am trying to figure out how to convert a smoothing operation in physical space into one in Fourier space.
In physical space, with equally spaced points x_j, (j...
Hi,
I'm trying to solve an algebraic-integral equation (I don't know if this is the proper description of this class of equations, it just seems like the least wrong way to describe them) and have run into several issues that I'll describe below, but first I'll outline the problem.
I'm...
Hm. I suspected as much. I'm going to end up differentiating this term w.r.t \alpha_n as it is part of the potential term in a Lagrangian. I don't see how that would help solve this, but I note it for completeness.
Hi,
I am trying to make progress on the following integral
I = \int_0^{2\pi} \sqrt{(1+\sum_{n=1}^N \alpha_n e^{-inx})(1+\sum_{n=1}^N \alpha_n^* e^{inx})} \ dx
where * denotes complex conjugate and the Fourier coefficients \alpha_n are constant complex coefficients, and unspecified...
Hi,
I'm a bit uncertain about the validity of my argument/approach to the following:
I'm trying to prove that the solution to a partial differential equation
\frac{\partial u(x,t)}{\partial t} + N[u(x,t)] = 0, where N is some nonlinear operator, CAN BE (not necessarily is) asymmetric...
Hi,
I'm trying to find
\iint_S \sqrt{1-\left(\frac{x}{a}\right)^2 -\left(\frac{y}{b}\right)^2} dS
where S is the surface of an ellipse with boundary given by \left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 = 1 .
Any suggestions are appreciated!
Thanks,
Nick
On second thought, the Joukowski map seems inappropriate here. I think the map I want is
z\to a \cosh(\xi + i \eta) so that
x=a\sinh (\xi) \cos(\eta) and y = a\cosh (\xi) \sin(\eta).
This will effectively give me the change in functional form that I expect; however, I still don't see...
Hi,
Given that the flow normal to a thin disk or radius r is given by
\phi = -\frac{2rU}{\pi}\sqrt{1-\frac{x^2+y^2}{r^2}}
where U is the speed of the flow normal to the disk, find the flow normal to an ellipse of major axis a and minor axis b.
I can only find the answer in the...
Hi,
I'm trying to find a toy (i.e. analytic) example of a nonlinear system that has very different behavior for two different types of forcing:
1) \frac{\partial u(x,t)}{\partial t}+ N(u(x,t)) = F(x)
where u(x,t) is the dependent variable, N represents some nonlinear operator with only...
Hi,
I'm trying to make headway on the following ghastly integral:
\int_0^{\infty} x^{\frac{3}{2}}e^{-xd} J_o(rx) \frac{\sin (\gamma \sqrt{x}\sqrt{x^2+\alpha^2}t)}{\sqrt{x^2+\alpha^2}}\ dx
where d,r, \alpha, \gamma ,t \in \mathbb{R}^+ and J_o is the zeroth order Bessel function of...