Fractional complex transforms for fractional differential equations

Advances in Difference Equations

Volumn 2012, Issue , 2012, Pages

  Ibrahim, Rabha W ^a  

^a UNIVERSITY OF MALAYA   (Malaysia)

Author keywords

analytic function; Cauchy differential equation; Fox Wright function; fractional calculus; fractional complex transform; fractional differential equations; Srivastava Owa fractional operators; unit disk

Indexed keywords

EID: 84873373013     PISSN: 16871839     EISSN: 16871847     Source Type: Journal    
DOI: 10.1186/1687-1847-2012-192     Document Type: Article

References (37)

1. Darus M, Ibrahim RW: Radius estimates of a subclass of univalent functions. Math. Vesnik 2011, 63:55-58.
2. Srivastava HM, Ling Y, Bao G: Some distortion inequalities associated with the fractional derivatives of analytic and univalent functions. JIPAM. J. Inequal. Pure Appl. Math. 2001, 2:1-6. (Pubitemid 39301049)
3. Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.
4. Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999. [Mathematics in Science and Engineering 198]
5. Hilfer R: Applications of Fractional Calculus in Physics. World Scientific Publishing, River Edge; 2000.
6. West BJ, Bologna M, Grigolini P: Physics of Fractal Operators. Springer, New York; 2003. [Institute for Nonlinear Science]
7. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006. [North-Holland Mathematics Studies 204]
8. Sabatier J, Agrawal OP, Machado JA: Advance in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht; 2007.
9. Baleanu D, Guvenc B, Tenreiro JA: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, New York; 2010.
10. Lakshmikantham V, Leela S, Vasundhara J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.
11. Bianca C: On set of nonlinearity in thermostatted active particles models for complex systems. Nonlinear Anal., Real World Appl. 2012, 13:2593-2608.
12. Mahmoud G, Mahmoud E: Modified projective Lag synchronization of two nonidentical hyperchaotic complex nonlinear systems. Int. J. Bifurc. Chaos 2011,2(8):2369-2379.
13. Mahmoud G, Mahmoud E, Ahmed M: On the hyperchaotic complex Lü system. Nonlinear Dyn. 2009, 58:725-738.
14. Fowler AC, Gibbon JD, McGuinness MJ: The complex Lorenz equations. Physica D 1982, 4:139-163.
15. Rauth A, Hannibal L, Abraham NB: Global stability properties of the complex Lorenz model. Physica D 1996, 99:45-58. (Pubitemid 126687486)
16. Sainty P: Contraction of a complex-valued fractional Brownian of order n. J. Math. Phys. 1992,33(9):3128-3149.
17. Jumarie G: Fractional Brownian motion with complex variance via random walk in the complex plane and applications. Chaos Solitons Fractals 2000, 11:1097-1111. (Pubitemid 30555663)
18. Jumarie G: Schrodinger equation for quantum fractal space-time of order n via the complex-valued fractional Brownian motion. Int. J. Mod. Phys. A 2001, 16:5061-5084.
19. Jumarie G: Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results. Chaos Solitons Fractals 2004, 22:907-925.
20. Jumarie G: Fractionalization of the complex-valued Brownian motion of order n using Riemann-Liouville derivative. Applications to mathematical finance and stochastic mechanics. Chaos Solitons Fractals 2006, 28:1285-1305. (Pubitemid 41815098)
21. Baleanu D, Diethelm K, Scalas E, Trujillo JJ: Fractional Calculus Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos). World Scientific, Singapore; 2012.
22. Baleanu D, Trujillo JJ: On exact solutions of a class of fractional Euler-Lagrange equations. Nonlinear Dyn. 2008, 52:331-335. (Pubitemid 351572392)
23. Magin R, Feng X, Baleanu D: Solving the fractional order Bloch equation. Concepts Magn. Reson., Part A 2009, 34A:16-23.
24. Ibrahim RW, Darus M: Subordination and superordination for univalent solutions for fractional differential equations. J. Math. Anal. Appl. 2008, 345:871-879.
25. Ibrahim RW: On solutions for fractional diffusion problems. Electron. J. Differ. Equ. 2010, 147:1-11.
26. Ibrahim RW: Existence and uniqueness of holomorphic solutions for fractional Cauchy problem. J. Math. Anal. Appl. 2011, 380:232-240.
27. Ibrahim RW: Ulam stability for fractional differential equation in complex domain. Abstr. Appl. Anal. 2012., 2012: Article ID 649517. doi:10.1155/2012/649517
28. Ibrahim RW: On holomorphic solution for space and time fractional telegraph equations in complex domain. J. Funct. Spaces Appl. 2012., 2012: Article ID 703681. doi:10.1155/2012/703681
29. Gordoa PR, Pickering A, Zhu ZN: Bücklund transformations for a matrix second Painlev equation. Phys. Lett. A 2010,374(34):3422-3424.
30. Molliq R, Batiha B: Approximate analytic solutions of fractional Zakharov-Kuznetsov equations by fractional complex transform. Int. J. Eng. Technol. 2012,1(1):1-13.
31. Srivastava HM, Owa S: Univalent Functions, Fractional Calculus, and Their Applications. Wiley, New York; 1989.
32. Ibrahim RW: On generalized Srivastava-Owa fractional operators in the unit disk. Adv. Differ. Equ. 2011, 55:1-10.
33. Li ZB, He JH: Fractional complex transform for fractional differential equations. Math. Comput. Appl. 2010, 15:970-973.
34. Li ZB: An extended fractional complex transform. Int. J. Nonlinear Sci. Numer. Simul. 2010, 11:0335-0337.
35. Li ZB, He JH: Application of the fractional complex transform to fractional differential equations. Nonlinear Sci. Lett. A 2011, 2:121-126.
36. He J-H, Elagan SK, Li ZB: Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A 2012, 376:257-259.
37. Srivastava HM, Karlsson PW: Multiple Gaussian Hypergeometric Series. Wiley, New York; 1985.