On some new properties of fractional derivatives with Mittag-Leffler kernel

Abstract We establish a new formula for the fractional derivative with Mittag-Leffler kernel, in the form of a series of Riemann–Liouville fractional integrals, which brings out more clearly the non-locality of fractional derivatives and is easier to handle for certain computational purposes. We also prove existence and uniqueness results for certain families of linear and nonlinear fractional ODEs defined using this fractional derivative. We consider the possibility of a semigroup property for these derivatives, and establish extensions of the product rule and chain rule, with an application to fractional mechanics.

[1]  Dumitru Baleanu,et al.  Fractional Order Models of Industrial Pneumatic Controllers , 2014 .

[2]  Abdon Atangana,et al.  Electrical circuits RC, LC, and RL described by Atangana–Baleanu fractional derivatives , 2017, Int. J. Circuit Theory Appl..

[3]  M. T. Cicero.  FRACTIONAL CALCULUS AND WAVES IN LINEAR VISCOELASTICITY , 2012 .

[4]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[5]  José António Tenreiro Machado,et al.  Some pioneers of the applications of fractional calculus , 2013 .

[6]  Thomas J. Osler,et al.  An integral analogue of Taylor’s series and its use in computing Fourier transforms , 1972 .

[7]  M. Al-Refai,et al.  Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications , 2014 .

[8]  A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel , 2017, Journal of inequalities and applications.

[9]  V. E. Tarasov.  Fractional Vector Calculus and Fractional Maxwell's Equations , 2008, 0907.2363.

[10]  I. Petráš.  Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation , 2011 .

[11]  B. West.  Fractional Calculus in Bioengineering , 2007 .

[12]  Vasily E. Tarasov,et al.  No violation of the Leibniz rule. No fractional derivative , 2013, Commun. Nonlinear Sci. Numer. Simul..

[13]  H. Qi,et al.  An inverse problem to estimate an unknown order of a Riemann–Liouville fractional derivative for a fractional Stokes’ first problem for a heated generalized second grade fluid , 2015 .

[14]  José Francisco Gómez-Aguilar,et al.  Irving–Mullineux oscillator via fractional derivatives with Mittag-Leffler kernel , 2017 .

[15]  A. Atangana,et al.  Numerical computation of a fractional derivative with non-local and non-singular kernel , 2016, 1610.07171.

[16]  Jordan Hristov,et al.  Transient heat diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffrey’s Kernel to the Caputo-Fabrizio time-fractional derivative , 2016 .

[17]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[18]  Thomas J. Osler,et al.  The Integral Analog of the Leibniz Rule , 1972 .

[19]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[20]  Dumitru Baleanu,et al.  Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel , 2016, 1607.00262.

[21]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[22]  Alain Oustaloup,et al.  The infinite state approach: Origin and necessity , 2013, Comput. Math. Appl..

[23]  Jean-Claude Trigeassou,et al.  Initial conditions and initialization of linear fractional differential equations , 2011, Signal Process..

[24]  M. Ezzat,et al.  On fractional thermoelasticity , 2011 .

[25]  Badr Saad T. Alkahtani,et al.  Chua's circuit model with Atangana–Baleanu derivative with fractional order , 2016 .

[26]  A. Elsaid,et al.  SERIES SOLUTION FOR FRACTIONAL RICCATI DIFFERENTIAL EQUATION AND ITS CONVERGENCE , 2015 .

[27]  Stevan Pilipović,et al.  Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles , 2014 .

[28]  Roberto Garrappa,et al.  Exponential Quadrature Rules for Linear Fractional Differential Equations , 2013, 1311.1388.

[29]  Can Li,et al.  A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative , 2011, 1109.2345.

[30]  Thomas J. Osler,et al.  The Fractional Derivative of a Composite Function , 1970 .

[31]  T. Hartley,et al.  Initialization of Fractional-Order Operators and Fractional Differential Equations , 2008 .

[32]  Xiuwen Li,et al.  Approximate Controllability of Fractional Evolution Systems with Riemann-Liouville Fractional Derivatives , 2015, SIAM J. Control. Optim..

[33]  Eqab M. Rabei,et al.  On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative , 2007, 0708.1690.

[34]  K. Sadarangani,et al.  On a differential equation with Caputo-Fabrizio fractional derivative of order $1<\beta\leq 2$ and application to mass-spring-damper system , 2016, 1605.07381.

[35]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[36]  Alain Oustaloup,et al.  State variables and transients of fractional order differential systems , 2012, Comput. Math. Appl..

[37]  ON THE SOLUTION OF NONLINEAR DIFFERENTIAL EQUATIONS WITH CONVOLUTION PRODUCT NONLINEARITIES , 1986 .

[38]  I. Bazhlekov,et al.  Viscoelastic flows with fractional derivative models: Computational approach by convolutional calculus of Dimovski , 2014 .

[39]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .

[40]  Arak M. Mathai,et al.  Special Functions for Applied Scientists , 2008 .

[41]  Thomas J. Osler,et al.  Leibniz Rule for Fractional Derivatives Generalized and an Application to Infinite Series , 1970 .

[42]  A. Pskhu.  On the Theory of the Continual Integro-Differentiation Operator , 2004 .

[43]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[44]  Dumitru Baleanu,et al.  Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels , 2016 .

[45]  Ji-Huan He.  Approximate solution of nonlinear differential equations with convolution product nonlinearities , 1998 .