Asia Mathematika
Published May 21, 2021 | Version v1
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On Lambda-Fractional Maxwell equations

Creators

  • 1. School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Rafina, Greece.
  • 2. NTUA External Science Collaborator, Athens, Greece.
  • 3. Mathematical Sciences Department Hellenic Army Academy Vari, 16673, Athens, Greece.
  • 4. Civil Engineering Department, National Technical University of Athens, Trikala, Greece.

Description

Adapting the \(\Lambda\)-fractional derivative, in fact the unique fractional derivative corresponding to a differential and able to generate fractional differential geometry, fractional Maxwell’s equations are defined for Electric and Magnetic fields. That fractional theory concerning Maxwell’s equations may be used in complex systems exhibiting memory effects. Further, those equations concern micro or nano-scale where the various principles demand non-local derivatives. 

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Journal article: 2457-0834 (ISSN)

References

  • G.W. Leibnitz: 1849, Letter to G. A. L'Hospital, Leibnitzen Mathematische Schriften, 2, 301-302.
  • J.Liouville: 1832, Sur le calcul des differentielles a indices quelconques. J. Ec. Polytech., 13, 71-162.
  • B.Riemann: 1876, Versuch einer allgemeinen Auffassung der Integration and Differentiation. In: Gesammelte Werke, 62.
  • J.T. Machado, V. Kiryakova, F. Mainardi, :2011, Comm. In Nonlin. Sci. $\&$ Num. Sim., 16(3),1940-1153.
  • A.Carpinteri, P.Cornetti, A.Sapora: 2011, A fractional calculus approach to non-local elasticity, Eur. Phys. J. Special Topics, 193, 193-204.
  • P.B. Beda: 2017, Dynamical systems approach of internal length in Fractional calculus, Engineering Transactions, 65(1), pp.209-215.
  • H. Beyer, S. Kempfle: 1995, Definition of Physically Consistent Damping Laws with Fractional Derivatives, ZAMM, 75(8), 623-635.
  • I. Tejado, E.Pérez, D. Valéri; 2020, Fractional Derivatives for Economic Growth Modelling of the Group of Twenty: Application to Prediction. Mathematics, 8, 50.
  • A.A.Kilbas, H.M.Srivastava, and J.J.Trujillo: 2006, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.
  • S.G. Samko, A.A. Kilbas, and O.I. Marichev: 1993, Fractional integrals and derivatives: theory and applications. Gordon and Breach, Amsterdam.
  • I.Podlubny: 1999, 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.
  • K. B.Oldham $\&$ J.Spanier: 1974, The fractional calculus, Academic Press, New York, and London.
  • T.M. Atanackovic, B. Stankovic: 2002, Dynamics of a viscoelastic rod of fractional derivative type. ZAMM, 82 (6), 377-386.
  • R. Gorenflo and F. Mainardi, 2003, Invited chapter to the book Processes with Long Range Correlations, edited by G.Rangarajan and M. Ding, Springer-Verlag, Berlin 2003, pp. 148-166. [Lecture Notes in Physics, No. 621]
  • J. A. Ochoa-Tapia, F. J. Valdes-Parada, J. Alvarez-Ramirez,2007, A fractional-order Darcy's law, Physica A. 374, pp.1-14
  • F. Riewe, 1997, Mechanics with fractional derivatives, Physical Review E, 55(3), pp. 3581-3592.
  • K.A. Lazopoulos, A.K. Lazopoulos: 2019, On the Mathematical Formulation of Fractional Derivatives.Prog. Fract. Diff. Appl. 5(4), pp.261-267.
  • K.A. Lazopoulos, A.K. Lazopoulos: 2020, On plane $\Lambda$ fractional linear elasticity theory, Theoretical $\&$ Applied Mechanics Letters, 10, pp.270-275.
  • D. Baleanu, Ali K. Golmankhaneh, Alireza K. Golmankhaneh, M. C.Baleanu, 2009, Fractional Electromagnetic Equations Using Fractional Forms, Int. J. Theor. Phys. 48: 3114–3123.
  • M. D. Ortigueira, M. Rivero,J.J. Trujillo, 2015, From a generalised Helmholtz decomposition theorem to fractional Maxwell equations, Commun Nonlinear Sci Numer Simulat, 22, 1036–1049.
  • Dineshkumar C, Udhayakumar R., 2020, Results on approximate controllability of nondensely defined fractional neutralstochastic diferential systems. Numer Methods Partial Diferential Eq., pp.1-27.
  • Dineshkumar C, Udhayakumar R.,2021, New results concerning toapproximate controllability of Hilfer fractional neutral stochastic delayintegro-diferential systems. Numer Methods Partial Diferential Eq., 37, pp.1072-1090.
  • Dineshkumar C, Udhayakumar R, Vijayakumar V, Kottakkaran Sooppy Nisar, 2021, A discussion on the approximate controllability of Hilfer fractional neutral stochastic integro-diferential systems, Chaos, Solitons Fractals, 142, 110472.
  • Dineshkumar C, Udhayakumar R, Vijayakumar V, Sooppy Nisar K., 2020, Results on approximate controllability of neutral integro-diferential stochastic system with state-dependent delay. Numer Methods Partial Diferential Eq.,1-15.
  • Vijayakumar V, Udhayakumar R, Dineshkumar C, 2021, Approximate controllability of second order nonlocal neutral diferential evolution inclusions, IMA Journal of Mathematical Control and Information, 38, Issue 1, ,pp. 192-210.