Non-local fractional calculus from different viewpoint generated by truncated M-derivative

Abstract The core of this paper is to propose more favored novel non-local fractional integral and derivative operators with three parameters by using the iteration manner on truncated M -derivative involving truncated Mittag-Leffler function. This is related closely to the fact that, in a certain sense, to iterate a convenient local derivative for real phenomena under investigation is one of the best ways for obtaining more comprehensive fractional integration and differentiation. In order to gain a deeper grip, we address some significant theorems, reflected accomplishedly the features of the standard fractional calculus, about these aidful operators encompassing some other types of fractional operators. The presented new operators in Riemann–Liouville and Caputo sense are utilized to solve Sturm–Liouville problem subjected to the initial conditions. As a useful method, Laplace transform of the aforementioned fractional integral and derivative is introduced to solve anomalous dynamics of various complex systems. Moreover, we carry out comparison analysis by plotting governing Sturm–Liouville equation in order to grasp the advantages of these new fractional operators performing much better on account of the additional parameter γ .

[1]  A. Yusuf,et al.  Mathematical modeling for the impacts of deforestation on wildlife species using Caputo differential operator , 2019, Chaos, Solitons & Fractals.

[2]  J. F. Gómez‐Aguilar,et al.  Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense , 2018 .

[3]  T. Abdeljawad.  Fractional operators with boundary points dependent kernels and integration by parts , 2020, Discrete & Continuous Dynamical Systems - S.

[4]  N. Sene.  Mittag-Leffler input stability of fractional differential equations and its applications , 2020, Discrete & Continuous Dynamical Systems - S.

[5]  Udita N. Katugampola.  A New Fractional Derivative with Classical Properties , 2014, 1410.6535.

[6]  Dumitru Baleanu,et al.  New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator , 2019, Mathematics.

[7]  Sania Qureshi,et al.  Modeling chickenpox disease with fractional derivatives: From caputo to atangana-baleanu , 2019, Chaos, Solitons & Fractals.

[8]  J. Vanterler da C. Sousa,et al.  Mittag–Leffler Functions and the Truncated $${\mathcal {V}}$$V-fractional Derivative , 2017 .

[9]  J. Vanterler da C. Sousa,et al.  On the ψ-Hilfer fractional derivative , 2017, Commun. Nonlinear Sci. Numer. Simul..

[10]  T. Abdeljawad,et al.  Variational principles in the frame of certain generalized fractional derivatives , 2020, Discrete & Continuous Dynamical Systems - S.

[11]  T. Abdeljawad,et al.  Generalized fractional derivatives and Laplace transform , 2020, Discrete & Continuous Dynamical Systems - S.

[12]  N. Sene.  Stability analysis of the generalized fractional differential equations with and without exogenous inputs , 2019, Journal of Nonlinear Sciences and Applications.

[13]  E. Bas,et al.  The price adjustment equation with different types of conformable derivatives in market equilibrium , 2019, AIMS Mathematics.

[14]  Qasem M. Al-Mdallal,et al.  An efficient method for solving fractional Sturm–Liouville problems , 2009 .

[15]  S. Qureshi,et al.  Using Shehu integral transform to solve fractional order Caputo type initial value problems , 2019, Journal of Applied Mathematics and Computational Mechanics.

[16]  Mariusz Ciesielski,et al.  The fractional Sturm-Liouville problem - Numerical approximation and application in fractional diffusion , 2017, J. Comput. Appl. Math..

[17]  Abdon Atangana,et al.  Modeling attractors of chaotic dynamical systems with fractal–fractional operators , 2019, Chaos, Solitons & Fractals.

[18]  D. Anderson,et al.  Newly Defined Conformable Derivatives , 2015 .

[19]  E. Bas,et al.  Real world applications of fractional models by Atangana–Baleanu fractional derivative , 2018, Chaos, Solitons & Fractals.

[20]  Thabet Abdeljawad,et al.  Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality , 2017, J. Comput. Appl. Math..

[21]  J. Sousa,et al.  A new truncated $M$-fractional derivative type unifying some fractional derivative types with classical properties , 2017, 1704.08187.

[22]  Sturm-Liouville difference equations having Bessel and hydrogen atom potential type , 2018, Open Physics.

[23]  A. Peterson,et al.  Sturm Liouville Equations in the frame of fractional operators with exponential kernels and their discrete versions , 2018, Quaestiones Mathematicae.

[24]  Dumitru Baleanu,et al.  On a new class of fractional operators , 2017, Advances in Difference Equations.

[25]  Abdon Atangana,et al.  Mathematical analysis of dengue fever outbreak by novel fractional operators with field data , 2019, Physica A: Statistical Mechanics and its Applications.

[26]  Udita N. Katugampola.  New approach to a generalized fractional integral , 2010, Appl. Math. Comput..

[27]  Fathi M. Allan,et al.  An efficient algorithm for solving higher-order fractional Sturm-Liouville eigenvalue problems , 2014, J. Comput. Phys..

[28]  E. Bas,et al.  Fractional models with singular and non-singular kernels for energy efficient buildings. , 2019, Chaos.

[29]  Dumitru Baleanu,et al.  Fractional modeling of blood ethanol concentration system with real data application. , 2019, Chaos.

[30]  Sania Qureshi,et al.  Fractional derivatives applied to MSEIR problems: Comparative study with real world data , 2019, The European Physical Journal Plus.

[31]  J. Vanterler da C. Sousa,et al.  Leibniz type rule: ψ-Hilfer fractional operator , 2019, Commun. Nonlinear Sci. Numer. Simul..

[32]  Dumitru Baleanu,et al.  Two-strain epidemic model involving fractional derivative with Mittag-Leffler kernel. , 2018, Chaos.

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

[34]  A. Alsaedi,et al.  Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal , 2016 .

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

[36]  Mohammed Al-Refai,et al.  Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems , 2017, Complex..