Research paper : On a class of time-fractional differential equations

Fractional Calculus and Applied Analysis

Volumn 15, Issue 4, 2012, Pages 639-668

  Li, Cheng Gang ^a   Kostić, Marko ^b   Li, Miao ^a   Piskarev, Sergey ^c  

^a SICHUAN UNIVERSITY   (China)

^b UNIVERSITY OF NOVI SAD   (Serbia)

^c MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

A times resolvent families; Generators; Resolvent families; Time fractional differential equations

Indexed keywords

EID: 84869148635     PISSN: 13110454     EISSN: 13142444     Source Type: Journal    
DOI: 10.2478/s13540-012-0044-x     Document Type: Article

References (38)

1. W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems. Monographs in Mathematics 96, Birkhauser Verlag, Basel (2001).
2. A. Ashyralyev, H.O. Fattorini, On uniform difference schemes for second-order singular perturbation problems in Banach spaces. SIAM J. Math. Anal. 23, No 1 (1992), 29-54.
3. A. Ashyralyev, F. Dal, Z. Pinar, A note on the fractional hyperbolic differential and difference equations. Applied Math. Comp. 217, No 9 (2011), 4642-4664.
4. E. Bajlekova, Fractional Evolution Equations in Banach Spaces. Ph.D. Dissertation, Einhoven University of Technology (2001).
5. E. Bazhlekova, Existence and uniqueness results for a fractional evolution equation in Hilbert space. Fract. Calc. Appl. Anal. 15, No 2 (2012), 232-243; DOI: 10.2478/s13540-012-0017-0; http://www.springerlink.com/content/ 1311-0454
6. Yu. V. Bogacheva, Resolution's Problems of Initial Problems for Abstact Differential Equations with Fractional Derivatives. Ph.D. Dissertation, Belgorod (2006).
7. Yu. V. Bogacheva, A.V. Glushak, About one abstract problem such as Cauchy with fractional derivatives. The International Conference on Differential Equations and Dynamical Systems. Abstracts, Suzdal, p. 244 (2006).
8. C. Chen, M. Li, On fractional resolvent operator functions. Semigroup Forum 80, No 1 (2010), 121-142.
9. C. Chen, M. Li, F.-B. Li, On boundary values of fractional resolvent families. J. Math. Anal. Appl. 384, No 2 (2011), 453-467.
10. I. Cioranescu, C. Lizama, On the inversion of the Laplace transform for resolvent famlies in UMD spaces. Arch. Math. 81, No 2 (2003), 182-192.
11. K. Diethelm, The Analysis of Fractional Differential Equations. Springer-Verlag, Berlin (2010).
12. H.O. Fattorini, The hyperbolic singular perturbation problem: An operator theoretic approach. J. Diff. Equ. 70, No 1 (1987), 1-41.
13. Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. Osaka J. Math. 27, No 2 (1990), 309-321.
14. Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation II. Osaka J. Math. 27, No 4 (1990), 797-804.
15. D. Guidetti, B. Karasozen, S. Piskarev. Approximation of abstract differential equations. J. of Math. Sci. 122, No 2 (2004), 3013-3054.
16. M. Haase, The complex inversion formula revisited. J. Aust. Math. Soc. 84, No 1 (2008), 73-83.
17. R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific Publ. Co. (2000).
18. J. Kemppainen, Existence and uniqueness of the solution for a timefractional diffusion equation. Fract. Calc. Appl. Anal. 14, No 3 (2011), 411-418; DOI: 10.2478/s13540-011-0025-5; http://www.springerlink.com/content/ 1311-0454
19. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Netherlands (2006).
20. M. Kosti'c, Generalized Semigroups and Cosine Functions. Mathematical Institute Belgrade (2011).
21. M. Kosti'c, Abstract time-fractional equations: existence and growth of solutions. Frac. Calc. Appl. Anal. 14, No 2(2011), 301-316; DOI: 10.2478/s13540-011-0018-4; http://www.springerlink.com/content/1311-0454
22. M. Kostíc, Systems of abstract time-fractional equations. Publ. Inst. Math. (Beograd) (N.S.), to appear.
23. M. Kosti'c, Abstract differential operators generating fractional regularized resolvent families. Submitted to Acta. Math. Sci. (English Ed.).
24. M. Kosti'c, C.-G. Li, M. Li, On a class of abstract time-fractional equations on locally convex spaces. Abstr. Appl. Anal., To appear.
25. F.-B. Li, M. Li, Q. Zheng, Fractional evolution equations governed by coercive differential operators. Abstr. Appl. Anal. 2009 (2009), Article ID 438690, 14 pages.
26. M. Li, C. Chen, F.-B. Li, On the fractional powers of generators of fractional resolvent families. J. Funct. Anal. 259, No 10 (2010), 2702-2726.
27. C. Lizama, On the convergence and approximation of integrated semigroups. J. Math. Anal. Appl. 181, No 1 (1994), 89-103.
28. Y. Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Frac. Calc. Appl. Anal. 15, No 1 (2012), 141-160; DOI: DOI: 10.2478/s13540-012-0010-7; http://www.springerlink.com/ content/1311-0454
29. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press & World Scientific, London-Singapore (2010).
30. F. Mainardi, R. Gorenflo, On Mittag-Leffler-Type functions in fractional evolution processes. J. Comp. Appl. Math. 118, No 1-2 (2000), 283-299.
31. E. Orsingher, L. Beghin, Time-fractional telegraph equations and telegrph processes with brownian time. Probab. Theory Relat. Fields 128, No 1 (2004), 141-160.
32. I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).
33. J. Pruss, Evolutionary Integral Equations and Applications. Monographs in Math. 87, Birkhauser Verlag, Basel (1993).
34. A.V. Pskhu, Partial Differential Equations of Fractional Order. Nauka, Moscow (2005) (in Russian).
35. R.K. Saxena, A.M. Mathai, H.J. Haubold, Reaction-diffusion systems and nonlinear waves. Astrophys. Space Sci. 305, No 3 (2006), 297-303.
36. W.R. Schneider, W. Wyss, Fractional diffusion and wave equations. J. Math. Phys. 30, No 1 (1989), 134-144.
37. M. Sova, Perturbations numeriques des evolutions parabolique et hyperbolique (In French). Cas. Pest. Mat. 96, No 4 (1971), 406-425.
38. W. Wyss, Fractional diffusion equation. J. Math. Phys. 27 (1986), 2782-2785.