Title | ON L-FRACTIONAL DERIVATIVES AND L-FRACTIONAL HOMOGENEOUS EQUATIONS |

Publication Type | Journal Article |

Year of Publication | 2017 |

Authors | LAZOPOULOS, AK, KARAOULANIS, D |

Secondary Title | Communications in Applied Analysis |

Volume | 21 |

Issue | 2 |

Start Page | 249 |

Pagination | 20 |

Date Published | 03/2017 |

Type of Work | scientific: mathematics |

ISSN | 1083-2564 |

AMS | 26A33, 34A08, 34K37 |

Abstract | Many different fractional derivatives exist that serve different aspects of fractional calculus. Nevertheless, they have failed to correspond to a reliable fractional differential. Lazopoulos [22] has introduced the L-fractional derivative having meaningful geometrical configuration. Moreover, the L-fractional derivative has a meaningful fractional differential as well. Hence, Fractional Differential Geometry could be established. The solutions of the linear homogenous L-fractional differential equation with constant coefficients will be studied in the present work. The solutions will be obtained with the help of power series expansions. Numerical solutions are presented for various fractional differentiation orders. Finally, comparison is presented between the cases of fractional differential homogeneous equations with Riemann-Liouville derivatives and Leibnitz derivatives. The similarities and differences are spotted. Those differences support that the solutions of the equations with L-fractional derivatives lie between the conventional (where $\alpha =1$) and the corresponding ones with Riemann-Liouville derivatives. |

URL | http://www.acadsol.eu/en/articles/21/2/6.pdf |

DOI | 10.12732/caa.v21i2.6 |

Short Title | L-Fractional Derivatives and L-Fractional Equations |

Alternate Journal | CAA |

Refereed Designation | Refereed |

Full Text | ## REFERENCES[1] G.W. Leibnitz, Letter to G.A. L’Hospital, Leibnitzen Mathematishe Schriften, 2 (1849), 301-302. L-Fractional Derivatives and L-Fractional Equations 265 [2] J. Liouville, Sur le calcul des differentielles a indices quelconques, J. Ec. Polytech., 13 (1832), 71-162. [3] B. Riemann, Versuch einer allgemeinen Auffassung der Integration and Differentiation, In: Gesammelte Werke (1876), 62. [4] T.M. Atanackovic, B. Stankovic, Dynamics of a viscoelastic rod of fractional derivative type, ZAMM, 82, no. 5 (2002), 377-386. [5] H. Beyer, S. Kempfle, Definition of physically consistent damping laws with fractional derivatives, ZAMM, 75, no. 7 (1995), 623-635. [6] K.A. Lazopoulos, Nonlocal continuum mechanics and fractional calculus, Mechanics Research Communications, 33 (2006), 753-757. [7] F.B. Adda, Interpretation geometrique de la differentiabilite et du gradient d’ordre reel, C.R. Acad. Sci. Paris, 326, Serie I (1998), 931-934. [8] A. Carpinteri, P. Cornetti, A. Sapora, A fractional calculus approach to non-local elasticity, Eur. Phys. J. Special Topics, 193 (2011), 193-204. [9] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [10] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam, 1993. [11] I. Podlubny, Fractional Differential Equations (An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications), Academic Press, San Diego-Boston-New York-London-Tokyo-Toronto, 1999. [12] K.B. Oldham, J. Spanier, The fractional calculus, Academic Press, New York and London, 1974. [13] V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer-Verlag, Berlin, 2010. [14] G. Calcani, Geometry of fractional spaces, Adv. Theor. Math. Phys., 16 (2012), 549-644. 266 A.K. Lazopoulos, D. Karaoulanis [15] K.A. Lazopoulos, Fractional vector calculus and fractional continuum mechanics, Prog. Fract. Differ. Appl., 2, no. 1 (2016), 67-86. [16] V.E. Tarasov, Fractional vector calculus and fractional Maxwell’s equations, Annals of Physics, 323 (2008), 2756-2778. [17] F.B. Adda, The differentiability in the fractional calculus, Nonlinear Analysis, 47 (2001), 5423-5428. [18] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Physical Review E, 53, no. 2 (1996), 1890-1899. [19] K.M. Konwalkar, A.D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos, 6, no. 3 (1996), 505-513. [20] C.S. Drapaca, Sivaloganathan, A fractional model of continuum mechanics, Journal of Elasticity, 107 (2012), 107-123. [21] A. Balankin, B. Elizarrataz, Hydrodynamics of fractal continuum flow, Physical Review E, 85, (2012) 025302(R). [22] K.A Lazopoulos, Fractional vector calculus and fractional continuum mechanics, Abstracts, Conference, Mechanics Through Mathematical Modelling, MTMM2015, In honour of Academician T. Atanackovic, Novi Sad, 7-10 Sept. 2015, 40. [23] S.A. Silling, M. Zimmermann, R. Abeyaratne, Deformation of a peridynamic bar, Journal of Elasticity, 73 (2003), 173-190. [24] X.J. Yang, Generalized local fractional Taylor’s formula with local fractional derivative, Journal of Expert Systems, 1, no. 1 (2012), 26-30. [25] Y.S. Liang, W.Y. Su, The relationship between the fractal dimensions of the type of fractal functions and the order of the fractal calculus, Chaos, Solitons and Fractals, 34 (2007), 682-692. [26] S.A. Silling, R.B. Lehoucq, Peridynamic theory of solid mechanics, Advances in Applied Mechanics, 44 (2010), 73-168. L-Fractional Derivatives and L-Fractional Equations 267 [27] K. Yao, W.Y. Su, S.P. Zhou, On the connection between the order of fractional calculus and the dimensions of a fractal function, Chaos, Solitons and Fractals, 23 (2005), 621-629. [28] A. Carpinteri, P. Cornetti, A. Sapora, Static-kinematic fractional operator for fractal and non-local solids, ZAMM, 89, no. 2 (2009), 207-217. [29] A.K. Goldmankhaneh, D. Baleanu, Lagrangian and Hamiltonian mechanics, Int. Jnl. Theor. Rhys., 52 (2013), 4210-4217. [30] Y. Liang, W. Su, Connection between the order of fractional calculus and fractional dimensions of a type of fractal functions, Analysis in Theory and Applications, 23, no. 3 (2007), 354-362. [31] F. Riewe, Mechanics with fractional derivatives, Physical Review E, 55, no. 2 (1997), 3581-3592. [32] X.J. Yang, Local Fractional Functional Analysis and its Applications, Asian Academic Publisher Limited, Hong Kong, 2011. [33] X.J. Yang, Expression of generalized Newton iteration method via generalized local fractional Taylor series, Advances in Computer Science and its Applications, 1, no. 2 (2012), 89-92. [34] J. Sabatier, O. Agrawal, J.A. Tenreiro, Machado, Advances in Fractional Calculus (Theoretical Applications and Development in Physics and Engineering), Springer, Netherlands, 2007. |