# Fractal differential equations and fractal-time dynamical systems

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## Abstract

Differential equations and maps are the most frequently studied examples of dynamical systems and may be considered as continuous and discrete time-evolution processes respectively. The processes in which time evolution takes place on Cantor-like fractal subsets of the real line may be termed as fractal-time dynamical systems. Formulation of these systems requires an appropriate framework. A new calculus calledF α-calculus, is a natural calculus on subsetsF⊂ R of dimension α,0 < α ≤ 1. It involves integral and derivative of order α, calledF α-integral andF α-derivative respectively. TheF α-integral is suitable for integrating functions with fractal support of dimension α, while theF α-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function occur naturally as solutions ofF α-differential equations. Hence the latter can be used to model fractal-time processes or sublinear dynamical systems.

We discuss construction and solutions of some fractal differential equations of the form

$$D_{F,t}^\alpha x = h(x,t),$$

whereh is a vector field andD α F,t is a fractal differential operator of order α in timet. We also consider some equations of the form

$$D_{F,t}^\alpha W(x,t) = L[W(x,t)],$$

whereL is an ordinary differential operator in the real variablex, and(t,x)F × Rn whereF is a Cantor-like set of dimension α.

Further, we discuss a method of finding solutions toF α-differential equations: They can be mapped to ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a couple of examples.

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## References

1. B B Mandelbrot,The fractal geometry of nature (Freeman and Company, 1977)

2. A Bunde and S Havlin (Eds),Fractals in science (Springer, 1995)

3. B J West, M Bologna and P Grinolini,Physics of fractal operators (Springer Verlag, New York, 2003)

4. K Falconer,The geometry of fractal sets (Cambridge University Press, 1985)

5. K Falconer,Fractal geometry: Mathematical foundations and applications (John Wiley and Sons, 1990)

6. K Falconer,Techniques in fractal geometry (John Wiley and Sons, 1997)

7. G A Edgar,Integral, probability and fractal measures (Springer-Verlag, New York, 1998)

8. S G Samko, A A Kilbas and O I Marichev,Fractional integrals and derivatives — Theory and applications (Gordon and Breach Science Publishers, 1993)

9. R Hilfer,Applications of fractional calculus in physics (World Scientific Publ. Co., Singapore, 2000)

10. K S Miller and B Ross,An introduction to the fractional calculus and fractional differential equations (John Wiley, New York, 1993)

11. K B Oldham and J Spanier,The fractional calculus (Academic Press, New York, 1974)

12. R Metzler, W G Glöckle and T F Nonnenmacher,Physica A211, 13 (1994)

13. R Metzler, E Barkai and J Klafter,Phys. Rev. Lett. 82, 3563 (1999)

14. R Hilfer and L Anton,Phys. Rev. E51, R848 (1995)

15. A Compte,Phys. Rev. E53, 4191 (1996)

16. G M Zaslavsky,Physica D76, 110 (1994)

17. R Metzler, E Barkai and J Klafter,Physica A266, 343 (1999)

18. R Hilfer,J. Phys. Chem. B104, 3914 (2000)

19. K M Kolwankar and A D Gangal,Chaos 6, 505 (1996)

20. J Levy Vehel and K M Kolwankar,Fract. Calc. Appl. Anal. 4, 285 (2001)

21. K M Kolwankar and A D Gangal,Pramana — J. Phys. 48, 49 (1997)

22. K M Kolwankar and A D Gangal,Phys. Rev. Lett. 80, 214 (1998)

23. K M Kolwankar and A D Gangal, Local Fractional Calculus: A Calculus for Fractal Space-Time, in:Fractals: Theory and applications in engineering edited by M Dekking, J Levy Vehelet al (Springer, London, 1999)

24. F B Adda and J Cresson,J. Math. Anal. Appl. 263, 721 (2001)

25. A Babakhani and V Daftardar-Gejji,J. Math. Anal. Appl. 270, 66 (2002)

26. M T Barlow,Diffusion on fractals, Lecture notes (Math. Vol. 1690, Springer, 1998)

27. J Kigami,Analysis on fractals (Cambridge University Press, 2000)

28. K Dalrymple, R S Strichartz and J P Vinson,J. Fourier Anal. Appl. 5, 205 (1999)

29. R S Strichartz,J. Funct. Anal. 174, 76 (2000)

30. U Freiberg and M Zähle,Potential Anal. 16, 265 (2002)

31. U Freiberg and M Zähle, Harmonic calculus on fractals — A measure geometric approach II (2000) Preprint

32. A Parvate and A D Gangal, math-ph/0310047 (2003)

33. A Parvate and A D Gangal,Calculus on fractal subsets of real line — II: Conjugacy with ordinary calculus, Pune University Preprint (2004)

34. R R Goldberg,Methods of real analysis (Oxford and IBH Publishing Co. Pvt. Ltd., 1970)

35. E Hille and J D Tamarkin,American Mathematics Monthly 36, 255 (1929)

36. M F Shlesinger,Ann. Rev. Phys. Chem. 39, 269 (1988)

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Parvate, A., Gangal, A.D. Fractal differential equations and fractal-time dynamical systems. Pramana - J Phys 64, 389–409 (2005). https://doi.org/10.1007/BF02704566