Calculus on fractal curves in Rn

Fractals

Volumn 19, Issue 1, 2011, Pages 15-27

  Parvate, Abhay ^a   Satin, Seema ^a   Gangal, A D ^a,b  

^a UNIVERSITY OF PUNE   (India)

^b Sai Trinity Building   (India)

Author keywords

Calculus; Fractal Curves; Fractal Derivatives; Fractal Dimension; Fractal Integrals; Fractal Taylor Series

Indexed keywords

CALCULATIONS; FUNCTIONS; INTEGRAL EQUATIONS; SOBOLEV SPACES;

ALGORITHMICS; CALCULUS; CONJUGACY; FRACTAL CURVES; FRACTAL DERIVATIVE; FRACTAL INTEGRAL; FRACTAL TAYLOR SERIES; KOCH CURVES; MASS FUNCTIONS; TAYLOR-SERIES;

FRACTAL DIMENSION;

EID: 79952610289     PISSN: 0218348X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218348X1100518X     Document Type: Article

References (29)

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. K. Falconer, Fractal Geometry: Mathematical Foundations and Application (John Wiley and Sons, 1990).
4. K. Falconer, The Geometry of Fractal Sets (Cambridge University Press, 1985).
5. K. Falconer, Techniques in Fractal Geometry (John Wiley and Sons, 1997).
6. C. Tricot, Curves and Fractal Dimensions (Springer-Verlag, 1995).
7. G. A. Edgar, Integral, Probability and Fractal Measures (Springer-Verlag, New York, 1998).
8. L. F. Abbott andM. B.Wise, Dimension of quantum mechanical path, Am. J. Phys. 49(1) (1981) 37.
9. D. Ben-Avraham and S. Havlin, Diffusion and Reaction in Fractals and Disordered Systems (Cambridge University Press, 2000).
10. A. Adrover, W. A. Schwalm, M. Giona and D. Bachand, scaling and scaling crossover for transport on anisotropic fractal structures, Phys. Rev. E 55 (1997) 7304-7312. (Pubitemid 127736428)
11. W. A. Schwalm, B. Moritz, M. Giona and M. K. Schwalm, Vector difference calculus for physical lattice models, Phys. Rev. E 59 (1999) 1217-1233. (Pubitemid 129693998)
12. J. Kigami and M. L. Lapidus, Weyl's Problem for the Spectral Distribution of Laplacians on P.C.F. Self-Similar Fractals, Comm. Math. Phys. 158 (1993) 93-125.
13. M. Giona, Contour integrals and vector calculus on fractal curves and interfaces, Chaos Solitons Fractals 10 (1999) 1349-1370.
14. F. Mendivil and E. R. Vrscay, Fractal vector measures and vector calculus on planar fractal domains, Chaos Solitons Fractals 14 (2002) 1239-1254. (Pubitemid 34863127)
15. K. M. Kolwankar and A. D. Gangal, Local fractional Fokker-Planck equation, Phys. Rev. Lett. 80 (1998) 214-217. (Pubitemid 128621921)
16. A. Parvate and A. D. Gangal, Calculus on fractal subsets of real line-I: formulation, Fractals 17(1) (2009) 53-81.
17. A. Parvate and A. D. Gangal, Calculus on fractal subsets of real line-II: Conjugacy with integer order calculus, Pune University pre-print.
18. A. Parvate and A. D. Gangal, Fractal differential equations and fractal-time dynamical systems, Pramana 64(3) (2005) 389-409. (Pubitemid 40407933)
19. R. R. Goldberg, Methods of Real Analysis (Oxford and IBH, 1970).
20. L. Nottale and J. Schnieder, Fractals and nonstandard analysis, J. Math. Phys. 25(5) (1984) 1296-1300.
21. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives - Theory and Applications (Gordon and Breach Science Publshers, 1993).
22. R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000).
23. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (John Wiley, New York, 1993).
24. K. B. Oldham and J. Spanier, The Fractional Calculus (Academic Press, New York, 1974).
25. K. M. Kolwankar and A. D. Gangal, Fractional differentiability of nowhere differetiable functions and dimension, Chaos 6 (1996) 505-513.
26. G. Jumarie, Modified Riemann-Liounlle derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl. 51 (2006) 1367-1376.
27. B. Adda and F. Cresson, About non-differentiable functions, J. Math. Anal. Appl. 263 (2001) 721-737.
28. X. Li, C. Essex and M. Davison, A local fractional denvative, in Proceedings of the First Symposium on fractional Derivatives and Their Applications, American Society of Mechanical Engineering, September 2-6 (2003).
29. A. Kufner, O. John and S. Fucik, Function Spaces (Noordhoff International Publishing, Leyden, 1977).