Timeline for Generalized Fuchsian-type PDE
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 8 at 16:10 | comment | added | Math2024 | Thanks. If one focuses on solving the original PDE in a small $t$ expansion with the given boundary conditions (including $a_i(0)=0$ for all $i$), one can directly verify that the solutions are uniquely fixed. However, I do not have anything to say about non-perturbative solutions at the moment. I see your point about 𝜉 now, but I guess I am interested in the solution with a more general 𝑥. I am not sure about $M$ yet. | |
| Mar 8 at 15:42 | comment | added | Willie Wong | @Math2024 (two above) if you solve pertubatively, you have to deal with the fact that your ODE for $x$ has nontrivial kernel (e.g. the function $x$) that is not ruled out by the BC that you provided. I don't see how you managed to get a unique solution. (one above) for small $x$ you have $\ln(1-x) \approx -x \approx \frac{x}{x-1}$. For $x$ closer to $1$ I don't see your point as unless you can show convergence of the series expansion these kinds of differences are immaterial. | |
| Mar 7 at 15:17 | comment | added | Math2024 | For the original PDE, I am not sure if $\xi^{\alpha}$ is the best form – in the small $t$ expansion, the solutions $a_i(x)$ have $ln (1-x)$ structure. | |
| Mar 7 at 15:17 | comment | added | Math2024 | Thanks, I agree $T$ and $X$ make the PDE more compact. The original way I obtained the solution (to the simpler PDE): I started with the original PDE and solved for the perturbative solutions $a_i (x)$ (say $i=1$ to $i=10$) using the BCs described in the post. In this process I did not need any additional conditions. Then, I focused on the leading small-$x$ contributions from $a_i(x)$ to find a pattern allowing me to resum to get the hypergeometric function. After that, I found the hypergeometric function satisfies the simpler PDE. | |
| Mar 7 at 4:43 | history | edited | Willie Wong | CC BY-SA 4.0 |
Improved exposition a bit.
|
| Mar 7 at 4:34 | history | answered | Willie Wong | CC BY-SA 4.0 |