Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives.

The main goal of this work is to find the solutions of linear and nonlinear fractional differential equations with the Mittag-Leffler nonsingular kernel. An accurate numerical method to search this problem has been constructed. The theoretical results are proved by utilizing two experiments.

[1]  KM Saad,et al.  Analytical study for time and time-space fractional Burgers’ equation , 2017 .

[2]  H. Yépez-Martínez,et al.  Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel , 2016, Advances in Difference Equations.

[3]  Dumitru Baleanu,et al.  Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions , 2018, Commun. Nonlinear Sci. Numer. Simul..

[4]  K. Saad.  A reliable analytical algorithm for space–time fractional cubic isothermal autocatalytic chemical system , 2018, Pramana.

[5]  José Francisco Gómez-Aguilar,et al.  A new modified definition of Caputo-Fabrizio fractional-order derivative and their applications to the Multi Step Homotopy Analysis Method (MHAM) , 2019, J. Comput. Appl. Math..

[6]  J. F. Gómez‐Aguilar,et al.  A mathematical analysis of a circular pipe in rate type fluid via Hankel transform , 2018, The European Physical Journal Plus.

[7]  Mohammed Al-Smadi,et al.  Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates , 2019, Appl. Math. Comput..

[8]  J. F. Gómez‐Aguilar,et al.  Analysis of reaction–diffusion system via a new fractional derivative with non-singular kernel , 2018, Physica A: Statistical Mechanics and its Applications.

[9]  Hamzeh Agahi,et al.  Mittag-Leffler-Gaussian distribution: Theory and application to real data , 2019, Math. Comput. Simul..

[10]  K. M. Owolabi.  Numerical patterns in reaction–diffusion system with the Caputo and Atangana–Baleanu fractional derivatives , 2018, Chaos, Solitons & Fractals.

[11]  José Francisco Gómez-Aguilar,et al.  Chaos in a nonlinear Bloch system with Atangana–Baleanu fractional derivatives , 2018 .

[12]  P. Yadav.  Motion through a non-homogeneous porous medium: Hydrodynamic permeability of a membrane composed of cylindrical particles , 2018 .

[13]  J. F. Gómez‐Aguilar,et al.  Fractional Transmission Line with Losses , 2014 .

[14]  J. F. Gómez‐Aguilar,et al.  A new derivative with normal distribution kernel: Theory, methods and applications , 2017 .

[15]  Ilyas Khan,et al.  A theoretical study on the performance of a solar collector using CeO2 and Al2O3 water based nanofluids with inclined plate: Atangana–Baleanu fractional model , 2018, Chaos, Solitons & Fractals.

[16]  K. Saad,et al.  Comparative study of a cubic autocatalytic reaction via different analysis methods , 2019, Discrete & Continuous Dynamical Systems - S.

[17]  Y. Chatibi,et al.  Variational calculus involving nonlocal fractional derivative with Mittag–Leffler kernel , 2019, Chaos, Solitons & Fractals.

[18]  Omar Abu Arqub,et al.  The RKHS method for numerical treatment for integrodifferential algebraic systems of temporal two-point BVPs , 2017, Neural Computing and Applications.

[19]  Ilyas Khan,et al.  Engine oil based generalized brinkman‐type nano‐liquid with molybdenum disulphide nanoparticles of spherical shape: Atangana‐Baleanu fractional model , 2018 .

[20]  Abdon Atangana,et al.  New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications , 2017 .

[21]  J. F. Gómez‐Aguilar,et al.  On the solutions of fractional order of evolution equations , 2017 .

[22]  Abdon Atangana,et al.  Fractional derivatives with no-index law property: Application to chaos and statistics , 2018, Chaos, Solitons & Fractals.

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

[24]  Mohamed Mbehou,et al.  A peculiar application of Atangana–Baleanu fractional derivative in neuroscience: Chaotic burst dynamics , 2018, Chaos, Solitons & Fractals.

[25]  K. Saad,et al.  On the numerical evaluation for studying the fractional KdV, KdV-Burgers and Burgers equations , 2018, The European Physical Journal Plus.

[26]  Boying Wu,et al.  A new reproducing kernel collocation method for nonlocal fractional boundary value problems with non-smooth solutions , 2018, Appl. Math. Lett..

[27]  Saeid Abbasbandy,et al.  Reproducing kernel method for the numerical solution of the 1D Swift-Hohenberg equation , 2018, Appl. Math. Comput..

[28]  K. Saad,et al.  A numerical approach for solving the fractional Fisher equation using Chebyshev spectral collocation method , 2018 .

[29]  Maysaa Mohamed Al Qurashi,et al.  Bateman-Feshbach Tikochinsky and Caldirola-Kanai Oscillators with New Fractional Differentiation , 2017, Entropy.

[30]  Modeling the transmission dynamics of flagellated protozoan parasite with Atangana–Baleanu derivative: Application of 3/8 Simpson and Boole’s numerical rules for fractional integral , 2018, Chaos, Solitons & Fractals.

[31]  M. G. López-López,et al.  Triple pendulum model involving fractional derivatives with different kernels , 2016 .

[32]  K. M. Owolabi.  Analysis and numerical simulation of multicomponent system with Atangana–Baleanu fractional derivative , 2018, Chaos, Solitons & Fractals.

[33]  Abdon Atangana,et al.  Numerical approximation of Riemann‐Liouville definition of fractional derivative: From Riemann‐Liouville to Atangana‐Baleanu , 2018 .

[34]  D. Baleanu,et al.  On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel , 2018 .

[35]  F. Gómez-Aguilar,et al.  The Feng’s first integral method applied to the nonlinear mKdV space-time fractional partial differential equation , 2016 .

[36]  Marco Pasetto,et al.  A reproducing kernel enhanced approach for peridynamic solutions , 2018, Computer Methods in Applied Mechanics and Engineering.

[37]  H. M. Srivastava,et al.  Fractional and operational calculus with generalized fractional derivative operators and Mittag–Leffler type functions , 2010 .

[38]  Hari M. Srivastava,et al.  Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel , 2009, Appl. Math. Comput..

[39]  J. F. Gómez‐Aguilar,et al.  Numerical simulations of multilingual competition dynamics with nonlocal derivative , 2018, Chaos, Solitons & Fractals.

[40]  J. F. Gómez‐Aguilar,et al.  Stability analysis and optimal control of a fractional human African trypanosomiasis model , 2018, Chaos, Solitons & Fractals.

[41]  R. F. Escobar-Jiménez,et al.  Analytical and numerical solutions of electrical circuits described by fractional derivatives , 2016 .