3rd IFAC Workshop on Fractional Differentiation and its Applications 20:45-21:30 Round Table Unsolved research problems in FC

• Enrico Scalas • Complete characterization of the diffusive limit of continuous time random walks • What stochastic process do we have in the limit: Lévy Processes fully characterized  =2,  =1 (Bachelier-Wiener process)  <2,  =1 (Lévy flights)  ≤2,  <1

p

 , 

G

 ,   

F t

  1 1

E

  

G

  

t

,  

x

 

t

       

p

But what about other finite dimensional distributions?

• Francesco Mainardi

• The fundamental solution of the fractional diffusion equation of order  ≤2 in space and order 0<  <2 in time admits a probability interpretation as a spatial probability density function (pdf) evolving in time if: 1  ≤2 and 0<  ≤1 2 1  ≤  ≤2 For case 1) Gorenflo and Mainardi were able to provide random work models. For case 2) is a open problem.

• Possible use of FC to verify the foundation of the Riemann conjecture....

• Om Agrawal • Organization and its purpose and scope, Website • Applications – Should contain simple mathematics • Education – Courses at graduate and undergraduate level • Journal – Current problems and solutions • Specific Problems: • Analytical and numerical solution of fractional differential equations containing both forward and backward derivatives • Design of programmable fractional order controllers and their applications • Fractional optimal control • New fractional variational calculus and applications of new operators

• Tenreiro Machado • To find simple applications of FC • To have simple courses on FC • To include in the FC community new scientific areas and new members • Web repository of FC papers • To develop FC industrial products

• Raoul Nigmatullin • A possibility to derive Newton’s equation with memory?

m d

2

r dt

2 

t

0   

t

   

r

,

v

,  

d

 • Generalization of uncertainty principle containing an arbitrary moment • How to relate power – law exponent with thermodynamics  (p,T,v) - ?

• CRONE team • To create a board that will manage and synchronize FC activities in the IFAC working group • To create an

e

-letter on the FC activities • To extend FC concepts in the area on nonlinear systems • To establish a test benchmark • To develop FC industrial applications

• Dumitru Baleanu • The physical meaning of Fractional derivatives • Consistent construction of fractional quantum mechanics, fractional quantum field theory, fractional differential geometry • Fractional variational principles with delay • Experimental evidences and manifestations of fractional dynamics in complex systems

• Blas Vinagre • To find ways of collaboration between groups working on the same topics

Future research problems

Future research problems

Tadeusz Kaczorek Białystok Technical University Faculty of Electrical Engineering Białystok, Poland e-mail: [email protected]

Future research problems

Future research problems (2)

book: T. Kaczorek: Positive 1D and 2D systems. Springer Verlag 2004 3) Predictive control of fractional systems (chemical industry) 4) Adaptive control of fractional systems