I have been trying to solve a nonlinear PDE by using the Laplace transform.
After all the procedure, the ressult I got for F(s) ended up with
$2 F(s)[2Em+s^2\hbar ^2]=F(s-4) [2Em+(s-4)^2\hbar ^2]+F(s+4) [2Em+(s+4)^2\hbar ^2]+(F(s-2)+F(s+2))$
I have seen some people doing tricky transformations and converting the original PDE so it can be seen as the result of the hipergeometric series.
To be honest I find myself more ingulfed about the form of the Laplace transform of this PDE (and how it seems to repeat)
Is it possible to infere somehow from this Laplace Transform (and its really particular form) that it is related to the hypergeometric series?
I think this is a dead end for this method. Is it? Any suggestion would be appreciated. Thanks
JMtz.
I doubt if it is possible to directly use the Laplace transform method in the case of nonlinear PDEs. As an illustrative example, consider a simple nonlinear term $u^2$ in a PDE (where $u$ denotes the solution), and apply a Laplace transform to it. There is no known general formula expressing a connection between $L[u^2]$ and $L[u]$, so that you should not be able to obtain an ODE from a Laplace transformed PDE. You might possibly precede the application of the Laplace transform by some sort of perturbation analysis, to linearize the problem, but this leads to considerable complications, and to limitations on parameter values. You may want to have a look at the recent papers of mine and colleagues [SIAM J.Appl.Math. 81(2021)208, Electrochim.Acta 428(2022)140896] which discussed a related problem of converting some nonlinear PDEs to integral equations with the help of Laplace transforms.