Recent content by cg78ithaca

I am aware that hypergeometric type differential equations of the type: can be solved e.g. by means of Mellin transforms when σ(s) is at most a 2nd-degree polynomial and τ(s) is at most 1st-degree, and λ is a constant. I'm trying to reproduce the method for the case where λ is not constant...

I have thought about the problem, pursuing the sum of exponentials method above, and have realized that adding the faster and slower exponential terms towards log(t) -> ±∞ would not actually significantly affect the behavior of the sum of exponentials around s = 1. The very fast exponentials...

Hi Jason, I've posted the full problem at https://www.physicsforums.com/threads/modeling-diffusion-and-convection-in-a-complex-system.905722/ Cristian

continued below: Similarly for the time-dependent perfusate equation, writing the time-dependent solution as a component which converges to zero as t->∞ plus the steady-state: $$ K\left(x,z,t\right)=u\left(x,z,t\right)+K_{\infty{}}\left(x,z\right) $$ The function u satisfies a PDE of the type...

I am trying to come up with an analytical solution (even as a infinite series etc.) for the following diffusion-convection problem. A thin layer of gel (assumed rectangular) is in direct contact with a liquid layer (perfusate) flowing with velocity v in the x direction (left to right) just...

Hi Jason, I agree with you, I was also planning to just present the system and jump to the steady-state solutions and the differential equations that govern the time-dependent waves, and how that takes you to the Laplace transforms in question. I am going on vacation until next week so I will do...

Hi again Jason, I'll create a thread with the original problem as you advise, but that is a bit involved and relatively specific, so we'll have to see if people will have the patience to go through it. But to answer your other questions for now - F(s) is zero for s < 0. The equality in...

Hi Jason, thanks again for the detailed comments, I'll go in greater length tomorrow about the other issues that you have raised, but regarding the sign of f(t), the plot above is probably confusing because it shows log10(t) on the x-axis and the pre-exponential factor in the sum of exponentials...

I'm still not sure about how to arrive analytically at the inverse Fourier transform, but I have played around trying to fit a sum of exponentials to F(s), using F(s) values for s from 0 to 10 with a step size of 1e-3 and using 600 bins for the t from 0.005 to 30. The f(t) determined by...

Hi Jason, and thanks for the very detailed and helpful answer. You are correct I do not have much exposure to the theory of distributions, and to answer your question regarding what I'm trying to do - this is connected to another thread I have posted here at...

I understand the conditions for the existence of the inverse Laplace transforms are $$\lim_{s\to\infty}F(s) = 0$$ and $$ \lim_{s\to\infty}(sF(s))<\infty. $$ I am interested in finding the inverse Laplace transform of a piecewise defined function defined, such as $$F(s) =\begin{cases} 1-s...

This is mostly a procedural question regarding how to evaluate a Bromwich integral in a case that should be simple. I'm looking at determining the inverse Laplace transform of a simple exponential F(s)=exp(-as), a>0. It is known that in this case f(t) = delta(t-a). Using the Bromwich formula...

Thanks Mfb. That makes sense.

Thanks Mfb. How do you prove that statement, namely that a power series that converges to F(s) for the whole s > 0 does not in fact exist?

Thanks Mfb. Indeed I am aware of the existence of non-analytic functions, and regarding your point that F(s) is not differentiable at s_c, it is well-taken, but I am doing the expansion around s = 0 where it is infinitely differentiable. Regarding your other point which touches on the topic of...