ON HADAMARD FRACTIONAL CALCULUS

This paper is devoted to the investigation of the Hadamard fractional calculus in three aspects. First, we study the semigroup and reciprocal properties of the Hadamard-type fractional operators. Then, the definite conditions of certain class of Hadamard-type fractional differential equations (HTFDEs) are proposed through the Banach contraction mapping principle. Finally, we prove a novel Gronwall inequality with weak singularity and analyze the dependence of solutions of HTFDEs on the derivative order and the perturbation terms along with the proposed initial value conditions. The illustrative examples are presented as well.

[1]  Lyapunov–Schmidt Reduction for Fractional Differential Systems , 2016 .

[2]  D. Baleanu,et al.  Lattice fractional diffusion equation in terms of a Riesz–Caputo difference , 2015 .

[3]  Shuqin Zhang,et al.  Monotone iterative method for initial value problem involving Riemann–Liouville fractional derivatives , 2009 .

[4]  SEPARABLE LOCAL FRACTIONAL DIFFERENTIAL EQUATIONS , 2015, 1511.06203.

[5]  Hong Wang,et al.  Wellposedness of Variable-Coefficient Conservative Fractional Elliptic Differential Equations , 2013, SIAM J. Numer. Anal..

[6]  R. Hilfer.  FRACTIONAL TIME EVOLUTION , 2000 .

[7]  Guanrong Chen,et al.  SCALING ATTRACTORS OF FRACTIONAL DIFFERENTIAL SYSTEMS , 2006 .

[8]  Changpin Li,et al.  On the Hadamard Type Fractional Differential System , 2012 .

[9]  Luigi Rodino,et al.  Existence and Uniqueness for a Nonlinear Fractional Differential Equation , 1996 .

[10]  Changpin Li,et al.  Center Manifold of Fractional Dynamical System , 2016 .

[11]  Jia Yuan,et al.  Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces , 2008 .

[12]  Kai Diethelm.  An extension of the well-posedness concept for fractional differential equations of Caputo’s type , 2014 .

[13]  Changpin Li,et al.  Existence and Continuation Theorems of Riemann-Liouville Type fractional differential equations , 2012, Int. J. Bifurc. Chaos.

[14]  Maria da Graça Marcos,et al.  Some Applications of Fractional Calculus in Engineering , 2010 .

[15]  Richard L. Magin,et al.  Fractional calculus models of complex dynamics in biological tissues , 2010, Comput. Math. Appl..

[16]  YangQuan Chen,et al.  Fractional-order Systems and Controls , 2010 .

[17]  Gabriela Planas,et al.  Mild solutions to the time fractional Navier-Stokes equations in R-N , 2015 .

[18]  Anatoly A. Kilbas,et al.  HADAMARD-TYPE FRACTIONAL CALCULUS , 2001 .

[19]  Shihui Zhu.  On the blow-up solutions for the nonlinear fractional Schrödinger equation , 2016 .

[20]  N. Tatar,et al.  Well-Posedness and Stability for a Differential Problem with Hilfer-Hadamard Fractional Derivative , 2013 .

[21]  S. Bhalekar,et al.  Solving Fractional Delay Differential Equations: A New Approach , 2015 .

[22]  Roberto Garrappa,et al.  Stability-Preserving High-Order Methods for Multiterm fractional differential equations , 2012, Int. J. Bifurc. Chaos.

[23]  I. Podlubny,et al.  Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives , 2005, math-ph/0512028.

[24]  Changpin Li,et al.  Introduction to fractional integrability and differentiability , 2011 .

[25]  Weihua Deng,et al.  ANALYSIS OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH MULTI-ORDERS , 2007 .

[26]  V. Uchaikin.  Fractional Derivatives for Physicists and Engineers , 2013 .

[27]  Paul L. Butzer,et al.  Fractional calculus in the Mellin setting and Hadamard-type fractional integrals , 2002 .