New results on fractional power series: Theories and applications

Entropy

Volumn 15, Issue 12, 2013, Pages 5305-5323

  El Ajou, Ahmad ^a   Arqub, Omar Abu ^a   Al Zhour, Zeyad ^b   Momani, Shaher ^c  

^a AL BALQA APPLIED UNIVERSITY   (Jordan)

^b IMAM ABDULRAHMAN BIN FAISAL UNIVERSITY   (Saudi Arabia)

^c UNIVERSITY OF JORDAN   (Jordan)

Author keywords

Caputo fractional derivative; Fractional differential equations; Fractional power series

Indexed keywords

EID: 84891671008     PISSN: None     EISSN: 10994300     Source Type: Journal    
DOI: 10.3390/e15125305     Document Type: Article

References (48)

1. Mainardi, F. Fractional calculus: Some basic problems in continuum and statistical mechanics. In Fractals and Fractional Calculus in Continuum Mechanics; Carpinteri, A., Mainardi, F., Eds.; Springer-Verlag: Wien, Austria, 1997; pp. 291-348.
2. Miller, K.S.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; John Willy and Sons, Inc.: New York, NY, USA, 1993.
3. Oldham, K.B.; Spanier, J. The Fractional Calculus; Academic Press: New York, NY, USA, 1974.
4. Podlubny, I. Fractional Differential Equations; Academic Press: New York, NY, USA, 1999.
5. Beyer, H.; Kempfle, S. Definition of physical consistent damping laws with fractional derivatives. Z. Angew. Math. Mech. 1995, 75, 623-635.
6. He, J.H. Some applications of nonlinear fractional differential equations and their approximations. Sci. Technol. Soc. 1999, 15, 86-90.
7. He, J.H. Approximate analytic solution for seepage flow with fractional derivatives in porous media. Comput. Method. Appl. M. 1998, 167, 57-68.
8. Caputo, M. Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. Int. 1967, 13, 529-539.
9. Yan, J.P.; Li, C.P. On chaos synchronization of fractional differential equations. Chaos, Solitons Fractals 2007, 32, 725-735.
10. Sommacal, L.; Melchior, P.; Dossat, A.; Petit, J.; Cabelguen, J.M.; Oustaloup, A.; Ijspeert, A.J. Improvement of the muscle fractional multimodel for low-rate stimulation. Biomed. Signal Process. Control 2007, 2, 226-233.
11. Silva, M.F.; Machado, J.A.T.; Lopes, A.M. Fractional order control of a hexapod robot. Nonlinear Dyn. 2004, 38, 417-433.
12. Mathieu, B.; Melchior, P.; Oustaloup, A.; Ceyral, C. Fractional differentiation for edge detection. Signal Process. 2003, 83, 2421-2432.
13. Zunino, L.; Pérez, D.G.; Martín, M.T.; Garavaglia, M.; Plastino, A.; Rosso, O.A. Permutation entropy of fractional Brownian motion and fractional Gaussian noise. Phys. Lett. A 2008, 372, 4768-4774.
14. Jumarie, G. Path probability of random fractional systems defined by white noises in coarse-grained time applications of fractional entropy. Frac. Diff. Eq. 2011, 1, 45-87.
15. Ubriaco, M.R. Entropies based on fractional calculus. Phys. Lett. A 2009, 373, 2516-2519.
16. Li, H.; Haldane, F.D.M. Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states. Phys. Rev. Lett. 2008, 101, 010504.
17. Hoffmann, K.H.; Essex, C.; Schulzky, C. Fractional diffusion and entropy production. J. Non-Equilib. Thermodyn. 1998, 23, 166-175.
18. Machado, J.A.T. Entropy analysis of integer and fractional dynamical systems. Nonlinear Dyn. 2010, 62, 371-378.
19. Essex, C.; Schulzky, C.; Franz, A.; Hoffmann, K.H. Tsallis and Rényi entropies in fractional diffusion and entropy production. Physica A 2000, 284, 299-308.
20. Cifani, S.; Jakobsen, E.R. Entropy solution theory for fractional degenerate convection-diffusion equations. Ann. Inst. Henri Poincare C, Non Linear Anal. 2011, 28, 413-441.
21. Prehl, J.; Essex, C.; Hoffmann, K.H. Tsallis relative entropy and anomalous diffusion. Entropy 2012, 14, 701-706.
22. Prehl, J.; Boldt, F.; Essex, C.; Hoffmann, K.H. Time evolution of relative entropies for anomalous diffusion. Entropy 2013, 15, 2989-3006.
23. Prehl, J.; Essex, C.; Hoffmann, K.H. The superdiffusion entropy production paradox in the space-fractional case for extended entropies. Physica A 2010, 389, 215-224.
24. Magin, R.L.; Ingo, C.; Colon-Perez, L.; Triplett, W.; Mareci, T.H. Characterization of anomalous diffusion in porous biological tissues using fractional order derivatives and entropy. Microporous Mesoporous Mater. 2013, 178, 39-43.
25. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies; Elsevier Science B.V.: Amsterdam, The Netherlands, 2006.
26. Lakshmikantham, V.; Leela, S.; Vasundhara Devi, J. Theory of Fractional Dynamic Systems; Cambridge Academic: Cambridge, UK, 2009.
27. Apostol, T. Calculus; Blaisdell Publishing: Waltham, MA, USA, 1990.
28. Chang, Y.F.; Corliss, G. ATOMFT: Solving ODE's and DAE's using Taylor series. Comput. Math. Appl. 1994, 28, 209-233.
29. Sezer, M.; Dascioglu, A.A. A Taylor method for numerical solution of generalized pantograph equations with linear functional argument. J. Comput. Appl. Math. 2007, 200, 217-225.
30. Yalçinbaş, S.; Sezer, M. The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials. Appl. Math. Comput. 2000, 112, 291-308.
31. Abu Arqub, O.; Abo-Hammour, Z.; Al-badarneh, R.; Momani, S. A reliable analytical method for solving higher-order initial value problems. Discrete Dyn. Nat. Soc. 2014, doi:10.1155/2013/673829.
32. Abu Arqub, O.; El-Ajou, A.; Bataineh, A.; Hashim, I. A representation of the exact solution of generalized Lane-Emden equations using a new analytical method. Abstr. Appl. Anal. 2013, doi:10.1155/2013/378593.
33. Abu Arqub, O. Series solution of fuzzy differential equations under strongly generalized differentiability. J. Adv. Res. Appl. Math. 2013, 5, 31-52.
34. El-Ajou, A.; Abu Arqub, O.; Momani, S. Homotopy analysis method for second-order boundary value problems of integrodifferential equations. Discrete Dyn. Nat. Soc. 2012, doi:10.1155/2012/365792.
35. Abu Arqub, O.; El-Ajou, A.; Momani, S.; Shawagfeh, N. Analytical solutions of fuzzy initial value problems by HAM. Appl. Math. Inf. Sci. 2013, 7, 1903-1919.
36. El-Ajou, A.; Odibat, Z.; Momani, S.; Alawneh, A. Construction of analytical solutions to fractional differential equations using homotopy analysis method. Int. J. Appl. Math. 2010, 40: 43-51.
37. El-Ajou, A.; Abu Arqub, O.; Momani, S. Solving fractional two-point boundary value problems using continuous analytic method. Ain Shams Eng. J. 2013, 4, 539-547.
38. Abu Arqub, O.; El-Ajou, A. Solution of the fractional epidemic model by homotopy analysis method. J. King Saud Univ. Sci. 2013, 25, 73-81.
39. Momani, S.; Odibat, Z. Numerical comparison of methods for solving linear differential equations of fractional order. Chaos, Solitons Fractals 2007, 31, 1248-1255.
40. Shawagfeh, N. Analytical approximate solutions for nonlinear fractional differential equations. Appl. Math. Comput. 2002, 131, 517-529.
41. Odibat, Z.; Momani, S. Application of variation iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 2006, 7, 27-34.
42. He, J.H. Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlinear Sci. Numer. Simul. 2005, 6, 207-208.
43. Hardy, G. Riemann's form of Taylor series. J. London Math. Soc. 1945, 20, 48-57.
44. Wantanable, Y. Notes on the generalized derivatives of Riemann-Liouville and its application to Leibntz's formula. Tohoku Math. J. 1931, 24, 8-41.
45. Truilljo, J.; Rivero, M.; Bonilla, B. On a Riemann-Liouville generalized Taylor's formula. J. Math. Anal. Appl. 1999, 231, 255-265.
46. Odibat, Z.; Shawagfeh, N. Generalized Taylor's formula. Appl. Math. Comput. 2007, 186, 286-293.
47. Schneider, W.R. Completely monotone generalized Mittag-Leffler functions. Expo. Math. 1996, 14, 3-16.
48. Turchetti, G.; Usero, D.; Vazquez, L. Fractional derivative and Hamiltonian systems. Tamsui Oxford J. Math. Sci. 2002, 18, 45-56.