Is anyone interested in or looking for further collaboration on approximating the solution of nonlinear PDEs (of the diffusion-type) using the Laplace transform? In essence, its derived results evidently have issues with simple problems, yet its application in PDEs can align quite well with numerical solutions..
Dear Professor Lin,
The Laplace transform method for solving nonlinear PDE has always been one of my considerations. I am sincerely happy to see that you are also considering this issue.
Here are a few of the latest literature on numerical Laplace methods that I like to share with you, hoping to be of some help to your research.
The most effective numerical Laplace transform method and inverse Laplace transform method at present should be the Contour Integrated Method (CIM). For nonlinear problems, Prof. Li has provided an efficient solution method by linearizing the nonlinear term and based on the idea of linear multi-step method. For details, please refer to
Article A High-order Exponential Integrator for Nonlinear Parabolic ...
For the numerical Laplace transform method, Prof J. A. C. Weideman & Bengt Fornberg recently also provided a type of method
Article Fully numerical Laplace transform methods
In addition, we have also extended some new applications and improvements to the Laplace transform method for linear problems, as shown in
Article Analyses of the Contour Integral Method for Time Fractional ...
and
Article The contour integral method for Feynman-Kac equation with tw...
Sincerely wish you better research results.
Best wishes
Fugui
Dear Prof. Lin,
I have also been looking for collaboration on this topic, as l have some articles that provide approximate and numerical solutions to PDEs and fractional PDEs. Hence, it would be great if we work together on this topic. You can see my papers in my profile. If you want, l can send them to you.
Sincerely,
Hami GÜNDOĞDU.
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