There are many different approaches to discretization of fractional derivatives, but many of them do not preserve the fundamental properties of the systems described by continuous-time equations such as stability of the system. For example, the asymptotically stable continuous-time fractional system after discretization using shifted Grunwald-Letnikov formula becomes unstable.
Is there any known method of discretization that preserves the properties of the original system?
Dear Rogowski
The best way to expand the fractional derivative given by the Caputo definition is to use the orthogonal polynomials. The numerical solution of FDEs by using spectral method based on any kind of classical orthogonal polynomials gives more accurate approximate solution. Please take a look of my articles in this field.
Regards
A.H. Bhrawy
Hello Krzysztof,
we can hardly say that there is the best way for such discretization in general. There are simply to many aspects (stability, accuracy, time efficiency, simplicity etc.). Nevertheless, you have mentioned the stability property. If you are looking for a simple method preserving stability of the underlying continuous system, maybe a discretization derived from implicit Euler method and Grunwald-Letnikov formulas could be of your interest. We have studied stability of this discretization in some of our papers for the case of Riemann-Liouville fractional derivative, but it can be utilized also for the Caputo one. Write me a message if you need help.
Best regards,
Tomas
Hi Krysztof and hi everyone,
I suggest you to read the paper "Fractional differential equations solved by using Mellin transform" by myself anf prof. Mario Di Paola. You can find it at the following wesites:
http://arxiv.org/pdf/1402.5949.pdf
http://dx.doi.org/10.1016/j.cnsns.2013.11.022
It is about a very powerful and versatile technique to solve fractional differential equations with constant coefficients by using the mellin transform. Very breafly, what we do is to transform the equation in the complex (mellin) domain, solve for a discretization of the transformed unkown function, and recover the solution in the time domain (or in whatever other physical space) by applying the discretized version of the inverse Mellin transform.
I hope to be helpful
Salvatore
The best way to expand the fractional derivative given by the Caputo definition is to use the Muntz polynomials. Since Caputo derivative of a Muntz oolynomial is a Muntz polynomial then we can preserve the stability. recently, I prepare a article about the using of these polynomials in the numerical solution of fractional differential equations with prof. H. M. Srivastava. Our obtained results have the powerful superiority in compared with the known methods in the literature.
I would suggest you to see this link:
On fractional derivatives fractional-order dynamic systems ... - TUKE
http://people.tuke.sk/igor.podlubny/pspdf/cdc97_fp16_5.pdf
Thank you for your answers.
Qefsere Doko Gjonbalaj, I know matrix approach method introduced by Igor Podlubny and other his results, but this is not useful in my considerations.
I will skim the other approaches that you have suggested . My aim is to prove that the discrete-time system obtained by the use of this discrete-time fractional derivatives has the same properties as the original continuous-time fractional system.
Thank you once again.
The discrete sum like this is very effective, yet I do not know whether it is the best...
I'll give you two suggestions
1 - Leave the C derivative
2 - Reformulate your problem directly as discrete. Send me a mail
and I'll send you two papers presented at FDA 2013 concerning the discrete-time fractional differential systems.
Approximations of the Caputo derivative which have order between one and two work well in practice.
In the last 20 years I've tried to show that, from application point of view, mainly engineering, Caputo and Riemann-Liouville are useless derivatives (I can send papers on the subject) for several reasons (for example, the initial conditions). How must we proceed?
1 - Work on R, not on R+
2 - Use the two-sided Laplace transform
3 - Use a derivative such that LT[ D^a f(t) ] = s^a F(s). The forward Grunwald-Letnikov (defined with summation till infinite) is suitable, but it can be the Liouville derivative.
4 - This can lead to the transfer function. In the transfer function you can use the bilinear transformation and work on the unit circle. The fft can be used to return to time.
5 - There is an alternative way: to formulate the problem directly in discrete-time. You can approach the continuous system with small sampling instants. See for example my paper:
Discrete-time differential systems - Signal Processing 107 (2015) 198–217
6 - For me, RL and C derivatives are "walking dead" derivatives. Don't loose time with them.
Please check our last work and the freely availble toolbox
https://www.researchgate.net/project/Spectral-Methods-for-numerical-solution-of-fractional-delay-differential-equations-MATLAB-Toolbox?_esc=profileProjectCards&_sg=HH0vLbLUpJ1h6CkkjvyyFyPS8UZ8YZfU7-oGifPDNLfn6M-6Bqk-oDYNSHlXMzpA1GyzYIkfmcP1112PselccA.ixprc3GBj13QRNlW-dc_JI1GEwutD4JQrYdGdgvZIBr6ua5G5xere5J0EwPNZm73.zvIQqdWSeqjfpANBQxoYHj0TPXTFRoNLk4ZGfTRb3pUKuYTAysPWzzjG5RmtfLv5&_viewIds%5B0%5D=Iri7uPrseF3TPkNJZ52nScEi&_viewIds%5B1%5D=MOczh1TTTSkTvWOJb81J86sY
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