Recently, I have learnt that many researchers are working on the theory of FDEs with $\psi-$Hilfer fractional derivative. The sufficient basic $\psi-$Hilfer fractional calculus is almost developed and corresponding IVPs and BVPs are being studied. But, is there any application done or earlier model improved by using this $\psi-$Hilfer fractional approach, or just the parallel theory development is going on...?
Dear Sandeep
I don't find any published paper about practical application of $\psi-$Hilfer fractional derivatives. Since there is no clear physical description for its initial condition or boundary condition, it can't be applied to engineering problems like Caputo derivative operator. However, I think there should be some statistical description and some applications in statistics and probability theory. I think the best starting point for you is Fox-H function and its relation with $\psi-$Hilfer fractional operator. There is some useful ideas in Mathai and Saxena's book, The H-Function (Theory and application).
https://www.springer.com/gp/book/9781441909152
Goodluck
Many thanks Prof Hazhir Aliahmadi and Maged Gumaan Bin-Saad for your answers. I would get some hints or ways to deal with practical applications of aforesaid operator. Thank you very much.
The theory of fractional calculus around the psi-Hilfer fractional operato has been well-defined in both the theoretical and practical sense. In the theoretical sense it is noteworthy that its construction and importance in the theory of fractional differential equations is important when investigating fundamental properties of their respective solutions, that is, existence, uniqueness, stability, attractivity, continuous dependence on data, among others. On the other hand, for the applicable sense, we are one step closer to making the theory well-grounded that will allow us to investigate practical problems and discuss which derivative best fits real data. In this sense, we believe that its importance has been observed throughout this period that was introduced.
For those who are interested on this topic, I have some applications with the psi-Caputo fractional derivative:
R. Almeida, A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. (44) 460-481 (2017)
R. Almeida, What is the best fractional derivative to fit data? Appl. Anal. Discrete Math. (11) 358-368 (2017)
R. Almeida, A. B. Malinowska and M. T. T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Meth. Appl. Sci. (41), 336-352 (2018)
Here some papers
Study of a boundary value problem for fractional order $$\psi $$ ψ -Hilfer fractional derivative
Harikrishnan, S., Shah, Kamal, Kanagarajan, K.Journal:Arabian Journal of Mathematics
A new representation formula for the Hilfer fractional derivative and its application
Kamocki, RafałJournal:Journal of Computational and Applied Mathematics
Here are some papers related to $\psi$-Hilfer derivative:
A study of ψ -Hilfer fractional differential system with application in financial crisis | Request PDF (researchgate.net)
(PDF) Existence results to a ψ - Hilfer neutral fractional evolution equation with infinite delay (researchgate.net)
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