On Caputo–Hadamard fractional differential equations

ABSTRACT In this paper, the existence and uniqueness of solution to Caputo–Hadamard fractional differential equation (FDE) are studied. The continuation theorem is established too. Then, Euler and predictor–corrector methods are built up to solve Caputo–Hadamard FDE. The stability and error analysis of the derived numerical schemes are investigated as well. At last, a numerical example is carried out to verify the numerical algorithm.

[1]  K. Diethelm.  AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER , 1997 .

[2]  W. Marsden.  I and J , 2012 .

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

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

[5]  Changpin Li,et al.  Finite difference methods with non-uniform meshes for nonlinear fractional differential equations , 2016, J. Comput. Phys..

[6]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[7]  Fanhai Zeng,et al.  The Finite Difference Methods for Fractional Ordinary Differential Equations , 2013 .

[8]  Yury F. Luchko,et al.  Algorithms for the fractional calculus: A selection of numerical methods , 2005 .

[9]  O. Agrawal,et al.  Advances in Fractional Calculus , 2007 .

[10]  J. Hadamard,et al.  Essai sur l'étude des fonctions données par leur développement de Taylor , 1892 .

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

[12]  Sachin Bhalekar,et al.  A new predictor-corrector method for fractional differential equations , 2014, Appl. Math. Comput..

[13]  Dumitru Baleanu,et al.  Caputo-type modification of the Hadamard fractional derivatives , 2012, Advances in Difference Equations.

[14]  K. Deimling.  Fixed Point Theory , 2008 .

[15]  R. Koeller.  Applications of Fractional Calculus to the Theory of Viscoelasticity , 1984 .

[16]  Juan J. Trujillo,et al.  Hadamard-type integrals as G-transforms , 2003 .

[17]  Changpin Li,et al.  On Finite Part Integrals and Hadamard-Type Fractional Derivatives , 2018, Journal of Computational and Nonlinear Dynamics.

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

[19]  Changpin Li,et al.  On the fractional Adams method , 2009, Comput. Math. Appl..

[20]  Alan D. Freed,et al.  Detailed Error Analysis for a Fractional Adams Method , 2004, Numerical Algorithms.

[21]  Changpin Li,et al.  ON HADAMARD FRACTIONAL CALCULUS , 2017 .

[22]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

[23]  Kai Diethelm.  Predictor-corrector strategies for single and multi-term fractional differential equations , 2001, HERCMA.

[24]  Changpin Li,et al.  Recent advances in applied nonlinear dynamics with numerical analysis : fractional dynamics, network dynamics, classical dynamics and fractal dynamics with their numerical simulations , 2013 .

[25]  Dumitru Baleanu,et al.  On cauchy problems with caputo hadamard fractional derivatives , 2016 .

[26]  Yaqing Ding,et al.  A method of approximate fractional order differentiation with noise immunity , 2015 .

[27]  Tiedong Ma,et al.  Dynamic analysis of a class of fractional-order neural networks with delay , 2013, Neurocomputing.

[28]  S. Ntouyas.  EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR CAPUTO-HADAMARD SEQUENTIAL FRACTIONAL ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS , 2017 .

[29]  Fanhai Zeng,et al.  Numerical Methods for Fractional Calculus , 2015 .