http://mathworld.wolfram.com/GreensFunctionPoissonsEquation.html
In the context of electrostatics the Green's function of the Poisson equation is the electrc potential due to a point charge.
Objects at rest have a velocity of 0. If a net force acts on them their acceleration, the time-derivative of their velocity, will be non-zero making them move.
There is actually a mathematical anology between the vortex ring and a tokamak. The vorticity is the curl of the incompressible velocity, like the electric current is the curl of the divergence-free magnetic field. This analogy goes as far as providing analytical solutions that can be used both...
In my country and field, you get paid for four years of research and are sort-of expected to publish something like the same number papers (which may include conference preceedings). Of course things are not so strict, so that yoy may get away with less papers or you can stay a little bit...
What would you say is best for a career in science, provided that you have the choice:
- Do a PhD much faster than average (with an average number of publications), or
- Take the average time to produce a very extensive thesis (with more publications) ?
The component \sigma_{ij} of the stress tensor \sigma is the force per unit area in the i-direction on a surface with normal in the j-direction. When the indices are the same, the force is normal to the surface so that it represents a pressure. When the direction of the first index is normal to...
It does not necessarily have to be infinitely differentiable. I now matched two exponential functions: u1(x) = a1*exp(b1*x)+c1 and u1(x) = a2*exp(b2*x)+c2 at x=x0. The constants a1,a2 and c1,c2 follow from u(0)=0, u(1)=1, u1(x0) = u2(x0), u2(x0)=u2(x1). The three constants b1, b2 and x0 can...
With piecewise functions it should indeed be much more easy to escape from Runge's phenomenon, I would however be interested in finding a single (i.e. without Heaviside step functions) elementary function that is able to satisfy my demands.
You're right! So I am indeed only interested in functions with u'(0) and u'(1) between 0 and 1. I understand I change the problem all the time, apparently I didn't think it through all the way.
I was thinking that an error function erf((x-m)/s) does part of the trick, when properly shifted and...
Thank you both very much, and both solutions indeed do the job. However I must say I made a mistake in saying that I wanted the function to be single-valued.
What I actually want is that the function is monotonic, so that it only increases from x=0 to 1. Both proposed solutions do not always...
I look for a function u(x) with u(0)=0 and u(1)=1, which is single-valued and differentiable on the entire interval x= [0,1] and allows one to choose the derivatives u'(0) and u'(1) through two free parameters.
Seems simple enough, right?