Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel

In this manuscript we define the right fractional derivative and its corresponding right fractional integral for the recently introduced nonlocal fractional derivative with Mittag-Leffler kernel. Then, we obtain the related integration by parts formula. We use the $Q-$operator to confirm our results. The corresponding Euler-Lagrange equations are obtained and one illustrative example is discussed.

[1]  Feng Gao,et al.  Fractional Maxwell fluid with fractional derivative without singular kernel , 2016 .

[2]  Dumitru Baleanu,et al.  On exact solutions of a class of fractional Euler–Lagrange equations , 2007, 0708.1433.

[3]  D. Baleanu,et al.  Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives , 2008 .

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

[5]  Carl F. Lorenzo,et al.  Variable Order and Distributed Order Fractional Operators , 2002 .

[6]  T. Abdeljawad.  On Delta and Nabla Caputo Fractional Differences and Dual Identities , 2011, 1102.1625.

[7]  F. Mainardi,et al.  Recent history of fractional calculus , 2011 .

[8]  H. Srivastava,et al.  THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS. NORTH-HOLLAND MATHEMATICS STUDIES , 2006 .

[9]  I. Podlubny.  Fractional differential equations , 1998 .

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

[11]  M. Sababheh,et al.  A new definition of fractional derivative , 2014, J. Comput. Appl. Math..

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

[13]  H. Srivastava,et al.  Local Fractional Integral Transforms and Their Applications , 2015 .

[14]  R. K. Saxena,et al.  Generalized mittag-leffler function and generalized fractional calculus operators , 2004 .

[15]  Thabet Abdeljawad,et al.  On conformable fractional calculus , 2015, J. Comput. Appl. Math..

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

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

[18]  Om P. Agrawal,et al.  Formulation of Euler–Lagrange equations for fractional variational problems , 2002 .

[19]  S. Arabia,et al.  Properties of a New Fractional Derivative without Singular Kernel , 2015 .

[20]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[21]  J. A. Tenreiro Machado,et al.  A NEW FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL Application to the Modelling of the Steady Heat Flow , 2015, 1601.01623.

[22]  T. Abdeljawad.  Dual identities in fractional difference calculus within Riemann , 2011, 1112.5795.

[23]  Ilknur Koca,et al.  Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order , 2016 .

[24]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[25]  Thabet Abdeljawad,et al.  On the Definitions of Nabla Fractional Operators , 2012 .

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

[27]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[28]  Aimin Yang,et al.  On steady heat flow problem involving Yang-Srivastava-Machado fractional derivative without singular kernel , 2016 .