Introduction to fractional integrability and differentiability

European Physical Journal: Special Topics

Volumn 193, Issue 1, 2011, Pages 5-26

  Li, C P ^a   Zhao, Z G ^a  

^a SHANGHAI UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 79953697124     PISSN: 19516355     EISSN: 19516401     Source Type: Journal    
DOI: 10.1140/epjst/e2011-01378-2     Document Type: Review

References (62)

1. G. W. Leibniz, Leibnizens Mathematische Schriften (Olms, Hildesheim, 1962).
2. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theorey and applications of Fractional Differential equations, edited by J. V. Mill (Elsevier, Amsterdam, 2006).
3. F. Mainardi, E. Bonetti, Rheol. Acta 26, 64 (1988).
4. C. P. Li, G. J. Peng, Chaos, Soli. Frac. 22, 443 (2004).
5. T. S. Zhou, C. P. Li, Phys. D 212, 111 (2005).
6. C. P. Li, W. H. Deng, D. Xu, Phys. A 360, 171 (2006).
7. C. P. Li, W. H. Deng, Int. J. Modern Phys. B 20, 791 (2006).
8. R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000).
9. B. Mandelbrot, IEEE Trans. Inform. Theory 13, 289 (1967).
10. M. Ichise, Y. Nagayanagi, T. Kojima, J. Electroanal. Chem. Interfacial Electrochem. 33, 253 (1971).
11. H. H. Sun, A. A. Abdelwahab, B. Onaral, IEEE Trans. Auto. Contr. 29, 441 (1984).
12. H. L. Li, Adv. Math. (China) 6, 187 (1963).
13. S. L. Wang, Adv. Math. (China) 3, 346 (1964).
14. C. P. Li, W. H. Deng, G. R. Chen, Fractals 14, 303 (2006).
15. W. H. Deng, C. P. Li, J. H. Lu, Nonl. Dyn. 48, 409 (2007).
16. C. P. Li, C. X. Tao, Comput. Math. Appl. 58, 1573 (2009).
17. D. L. Qian, C. P. Li, R. P. Agarwal, et al., Math. Comput. Model. 52, 862 (2010).
18. C. P. Li, Z. Q. Gong, D. L. Qian, et al., Chaos 20, 013127 (2010).
19. Y. Y. Zheng, C. P. Li, Z. G. Zhao, Comput. Math. Appl. 59, 1718 (2010).
20. Y. Y. Zheng, C. P. Li, Z. G. Zhao, Math. Problems in Engineering (2010), Article ID 279038 doi: 10. 1155/2010/279038.
21. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, Amsterdam, 1993).
22. N. H. Abel, Mag. Naturvidenskaberne 1, 11 (1823).
23. N. H. Abel, J. reine angew. Math. 1, 153 (1826).
24. R. Gorenflo, S. Vessella, Abel Integral Equations: Analysis and Applications, Lecture Notes in Mathematics (Springer, Berlin Heidelberg, 1991).
25. L. Tonelli, Anna. Math. 99, 183 (1928).
26. J. D. Tamarkin, Anna. Math. 31, 219 (1930).
27. J. Liouville, J. l'Ecole Roy. Polytéchn. 13, 1 (1832).
28. J. Liouville, J. l'Ecole Roy. Polytéchn. 13, 71 (1832).
29. J. Liouville, J. l'Ecole Roy. Polytéchn. 13, 163 (1832).
30. J. Liouville, Reine Angew. Math. 11, 1 (1834).
31. J. Liouville, Reine Angew. Math. 12, 273 (1834).
32. J. Liouville, Reine Angew. Math. 13, 219 (1835).
33. J. Liouville, J. l'Ecole Roy. Polytéchn. 15, 17 (1835).
34. J. Liouville, J. l'Ecole Roy. Polytéchn. 15, 58 (1837).
35. B. Riemann, Gesamm. Math. Werke Wissensch. 331(1876).
36. E. M. Stein, A. Zygmund, Proc. London Math. Soc. 14A, 249 (1965).
37. A. Zygmund, Duke Math. J. 12, 455 (1945).
38. G. V. Welland, Trans. Amer. Math. Soc. 132, 487 (1968).
39. C. P. Li, D. L. Qian, Y. Q. Chen, On the Riemann-Liouville derivative (submitted).
40. K. M. Kolwankar, A. D. Gangal, Chaos 6, 505 (1996).
41. A. K. Grünwald, Z. Angew. Math. Phys. 12, 441 (1867).
42. A. V. Letnikov, Mat. Sb. 3, 1 (1868).
43. M. Caputo, Geophys., J. R. Astr. Soc. 13, 529 (1967).
44. M. Caputo, Elasticità e Dissipazione (Zanichelli, Bologna, 1969).
45. A. M. A. El-Sayed, Appl. Math. Comput. 80, 1 (1994).
46. A. M. A. El-Sayed, J. Frac. Calc. 7, 89 (1995).
47. I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, CA, 1999).
48. F. Mainardi, Chaos, Soli. Frac. 7, 1461 (1996).
49. R. Gorenflo, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, edited by A. Carpinteri, F. Mainardi (Springer, Wien, 1997), p. 223.
50. R. Gorenflo, F. Mainardi, Maphysto Center (2000) http://www. maphysto. dk/oldpages/events/LevyCAC2000/MainardiNotes/fm2k0a. ps.
51. F. Mainardi, Y. Luchko, G. Pagnini, Frac. Calc. Appl. Anal. 4, 153 (2001).
52. F. Mainardi, R. Gorenflo, Frac. Calc. Appl. Anal. 10, 269 (2007).
53. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity (Imperial College Press, London, 2010).
54. C. P. Li, W. H. Deng, Appl. Math. Comp. 187, 777 (2007).
55. C. P. Li, X. H. Dao, P. Guo, Nonl. Anal. TMA 71, 1857 (2009).
56. M. Meerschaert, D. Benson, B. Baeumer, Phys. Rev. E 59, 5026 (1999).
57. V. J. Ervin, J. P. Roop, Num. Meth. Part. Diff. Eq. 22, 558 (2006).
58. K. M. Kolwankar, A. D. Gangal, Pramana 48, 49 (1997).
59. K. M. Kolwankar, A. D. Gangal, in Proceedings of the Conference on Fractals in Engineering (Archanon, 1997).
60. X. Li, M. Davison, C. Essex (preprint) http://www. apmaths. uwo. ca/mdavison/-library/preprints/lfd2. pdf.
61. Y. Chen, Y. Yan, K. Zhang, J. Math. Anal. Appl. 362, 17 (2010).
62. G. A. Anastassiou, Fractional Differentiation Inequalities (Springer, New York, 2009).