I am searching for a new research application to fractional differential equations (fractional calculus). It could be in the field of science, engineering, finance or the like.
Any recommendations?
Google "Fractance" which is an application of fractional derivative to electrical circuits.
As far as I know, fractional calculus can be applied to some control engineering problems, system modeling and identification.
See if this helps: http://abhayparvate.wordpress.com/more/, http://scholar.google.com/citations?user=C3PR11gAAAAJ&hl=en
Google "Fractance" which is an application of fractional derivative to electrical circuits.
I've recently applied fractional derivatives of moment generating function of a PDF to write closures for non-integer moments encountered in Eulerian-Eulerian multiphase flows:
Closure of non-integer moments arising in multiphase flow phenomena
Chemical Engineering Science. 06/2012; 75(18):424–434.
and for more information on the Eulerian-Eulerian formulation of poly dispersed turbulent flows see:
A mesoscopic description of polydispersed particle laden turbulent flows
Progress in Energy and Combustion Science. 12/2011; 37(6):716-740
Hope this helps
Fractional behaviour is everywhere. I suggest you signals biomedical signals like ECG or EEG.
This is the branch of mathematics which study the differential and integral equations of real order. Which has lot application in physics, chemistry, ....
Such equations arises
It might be helpful also to look for "fractional stochastic calculus". In stochastic calculus, the deterministic fractional calculus is used to define stochastic integrals with respect to the fractional Brownian motion. This has nice applications in financial mathematics since fractional stochastic calculus may be used to model, for instance, interest rates having long-range dependence.
Yes. The interpretation in terms of systems leads to obtain more simple results. I used the two-sided derivatives to define fractional Brownian motion as an integral of fractional noise that is the fractional derivative of white noise. In this case the stochastic case is not essentially different from the deterministic.
You can try modeling the stock market with fractional differential equations.
Research --both theoretical and applied-- on the so-called fractional Brownian process (fBm) involves fractional calculus. Modelling by means of the fBm has been done in many areas, in particular in mathematical finance.
Machado, J. Tenreiro, Francesco Mainardi, and Virginia Kiryakova. "Fractional Calculus: Quo Vadimus?(Where are we Going?)." Fractional Calculus and Applied Analysis 18.2 (2015): 495-526.
I hope all the real time dynamical system applications which works with ordinary differential equations will also work with Caputo fractional differential equations as the fractional derivatives are the interpolated values of ordinary derivatives.
Dear Sarah
We can separate the applications of fractional differental equations int 3 fields
1 - Theoretical - for each definition of derivative you have many differential equations. The most interesting in applications are the linear.
2 - Practical realization - electronic implementations are appearing and need correct modelling. There are many in other areas.
3 - Modelling of natural and man made systems. Most of models are created from frequency measurements. It is very important to state simple rules for designing good filters.
The discrete-version of fractional calculus open new perspectives into applications.
M. Krishna Sastry refferred Caputo derivative. I am convinced that it is the worst fractional derivative, for several reasons.
I can justify my affirmations with papers: [email protected]
Dear Sarah,
If you need as references in the book of Das he gave the applications of fractional differential equations.
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