Homework Statement
I want to invert a function from Laplace transform space to normal space.
Homework Equations
In Laplace transform space, the function takes the form $$ \bar f (s) = \frac{\exp\left[ x (-a +\sqrt{a^2+ b +c s} )\right]}{-a +\sqrt{a^2+ b +c s}}.
$$
Here, ##s## is the Laplace...
Hi everyone,
I've been looking at a problem that seems simple at first, but appears to be deceptively difficult (unless I'm missing something).
1. Homework Statement
I've been looking at a problem that involves the diffusion of a scalar quantity, ##q(x)##, on the semi-infinite domain, ##\leq...
I'm looking to model a system in which events are nearly perfectly randomly distributed but with a slight tendency for events to avoid each other. As you know, if the system were perfectly random, I could use a Poisson distribution. The probability distribution for the number of events would...
I have a complicated recursion replation, which I'm sure is unsolvable. (By "unsolvable" I mean that there is no closed form solution expressing \xi_1, \xi_2, \xi_3, etc. in terms of \xi_0.) It goes
\frac{(k+4)!}{k!}\xi_{k+4} +K_1 (k+2)(k+1)\xi_{k+2}+ [ K_2 k(k-1) +K_3] \xi_{k} +K_4...
I have been working on a derivation in which the following simultateous ordinary differential equations have appeared:
f^{(4)}(x)-2 a^2 f''(x)+a^4 f(x)+b(g''(x)-a^2 g(x))=0,
g^{(4)}(x)-2 a^2 g''(x)+a^4 g(x)-b(f''(x)-a^2 f(x))=0,
where a and b are constants. I figured that I could solve...
Say we have a 3D function, p(x,y,z) and we define it in terms of another function f(x,y,z) via,
\nabla ^2 p = f.
I know that if we are working in R^3 space (with no boundaries) we can say that,
p= \frac{-1}{4\pi}\iiint \limits_R \frac{f(x',y',z')}{\sqrt{(x-x')^2 +(y-y')^2+(z-z')^2}} dx'...
Homework Statement
I'm looking to determine the Fourier sine transfom of 1.
Homework Equations
One this site http://mechse.illinois.edu/research/dstn/teaching_files2/fouriertransforms.pdf [Broken] (page 2) it gives the sine transform as
\frac{2}{\pi \omega}
The Attempt at a...
What is a "lifting function"?
Hi,
I was reading a journal article and they mentioned something called a "lifting function". It was apparently used with the Navier-Stokes equation to translate the boundary conditions (which were complicated, and NOT non-slip), into a body force.
It looks...
I have a series of large 2x2 matricies, each of which is stored inside a .mat file. These files have the names data1.mat, data2.mat, data3.mat,..., data60.mat. I have sucessfully loaded each of these .mat files.
I want to create a 1x60 array whose entires are the average values of the...
I'm reading a journal article at the moment which uses a piece of notation which they dont actually define. It looks like this:
\nabla_{\vec Q}
(As it happens, \vec Q is an ordinary vector indicating the orientation of a polymer.)
I've never seen vector subscript on the gradient symbol...
I've got a problem. I'm trying to design a reactor with internal cooling provided by water flowing through tubes directly within the reactor. Basically, it's like a shell and tube heat exchanger with an internal heat source.
The problem is that the cooling water is to be evaporated within...
I need a method for estimating the liquid flow rate required by a gas scrubber (I know the amount of gas that needs to be scrubbed and the amount of particulates with it, but I don't know about the nature of the particles, except that they are organic).
It doesn't need to be particularly...
I have a question relating to a particle rotating around a point with velocity u = \Omega \times r, where \Omega is the angular velocity and r is the position relative to the pivot point.
I need to prove that the acceleration is given by,
a = -\frac{1}{2} \nabla [(\Omega \times r)^2]
I...
I have the problem,
ty(t)= \int_{0}^{t}\tau^{\alpha-1}y(t-\tau)d\tau
subject to the constraint that \int_{0}^{\infty}y(t)dt=1.
In need to get the answer in the form of, Y(p)=something (where Y(p) is the Laplace transform of y(t)).
I can see that the right hand side is...
To invert the Mellin transformed function F(s), the equation is,
f(x) = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} x^{-s}F(s)ds
What is the rule the value of c? I know that in inverse Laplace Transforms, c is any real number large enough that all the residues of F(s) lie to the left of the...
I have the question,
\int_{-\pi/2}^{\frac{\pi}{2}} e^{-ilk}cos^n kdk
It says, "Set t=ik". So,
-i\int_{-i\pi/2}^{i\pi/2}e^{-lt} cosh^n tdt
But then it says, "Use the method of steepest descent to show that as n \rightarrow \infty with r = l/n."
I'm supposed to get:
\sim...
I've been looking for steam tables on the internet, but all that I can find are those steam table computer programs.
So... Does anyone know a site which has the old fashioned steam tables?
And which of those steam table computer programs would be worth getting?
Hi everybody, this is my first post here.
I got this question, but I don't know what it means:
Fix n ≥ 1. If the nth roots of 1 are w0, w1, w2, . . . , wn−1, show that they satisfy:
(z − w0)(z − w1)(z - w2) · · · (z − wn−1) = z^n − 1
(z and wn are all complex numbers)
What I don't...