An efficient operation matrix method for solving fractal-fractional differential equations with generalized Caputo-type fractional-fractal derivative

Abstract In this study, we present the new generalized derivative and integral operators which are based on the newly constructed new generalized Caputo fractal-fractional derivatives (NGCFFDs). Based on these operators, a numerical method is developed to solve the fractal-fractional differential equations (FFDEs). We approximate the solution of the FFDEs as basis vectors of shifted Legendre polynomials (SLPs). We also extend the derivative operational matrix of SLPs to the generalized derivative operational matrix in the sense of NGCFFDs. The efficiency of the developed numerical method is tested by taking various test examples. We also compare the results of our proposed method with the methods existed in the literature In this paper, we specified the fractal–fractional differential operator of new generalized Caputo in three categories: (i) different values in ρ and fractal parameters, (ii) different values in fractional parameter while fractal and ρ parameters are fixed, and (iii) different values in fractal parameter controlling fractional and ρ parameters.

[1]  Aydın Seçer,et al.  A New Operational Matrix of Fractional Derivatives to Solve Systems of Fractional Differential Equations via Legendre Wavelets , 2018 .

[2]  Phang Chang,et al.  Legendre Wavelet Operational Matrix of fractional Derivative through wavelet-polynomial transformation and its Applications in Solving Fractional Order Brusselator system , 2016 .

[3]  C. Tunç,et al.  New operational matrices of orthogonal Legendre polynomials and their operational , 2019, Journal of Taibah University for Science.

[4]  K. Shah,et al.  Fractal-Fractional Mathematical Model Addressing the Situation of Corona Virus in Pakistan , 2020, Results in Physics.

[5]  Mehdi Dehghan,et al.  A new operational matrix for solving fractional-order differential equations , 2010, Comput. Math. Appl..

[6]  Dumitru Baleanu,et al.  On solutions of fractional Riccati differential equations , 2017, Advances in Difference Equations.

[7]  Constantin Bota,et al.  Approximate Analytical Solutions of the Fractional-Order Brusselator System Using the Polynomial Least Squares Method , 2015 .

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

[9]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[10]  Abdon Atangana,et al.  Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system , 2017 .

[11]  Abdon Atangana,et al.  A comparative study of convective fluid motion in rotating cavity via Atangana–Baleanu and Caputo–Fabrizio fractal–fractional differentiations , 2020 .

[12]  Fatmawati,et al.  Modeling and analysis of competition model of bank data with fractal-fractional Caputo-Fabrizio operator , 2020, Alexandria Engineering Journal.

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

[14]  Dumitru Baleanu,et al.  A Jacobi operational matrix for solving a fuzzy linear fractional differential equation , 2013 .

[15]  J. F. Gómez‐Aguilar,et al.  ANALYSIS OF FRACTAL–FRACTIONAL MALARIA TRANSMISSION MODEL , 2020 .

[16]  Fei Peng,et al.  Discrimination of natural images and computer generated graphics based on multi-fractal and regression analysis , 2017 .

[17]  Dumitru Baleanu,et al.  On the generalized fractional derivatives and their Caputo modification , 2017 .

[18]  E. H. Doha,et al.  A NEW JACOBI OPERATIONAL MATRIX: AN APPLICATION FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS , 2012 .

[19]  Chee Seng Chan,et al.  Fractional Differential Systems: A Fuzzy Solution Based on Operational Matrix of Shifted Chebyshev Polynomials and Its Applications , 2017, IEEE Transactions on Fuzzy Systems.

[20]  Y. Yang,et al.  NUMERICAL TREATMENT OF THE SPACE–TIME FRACTAL–FRACTIONAL MODEL OF NONLINEAR ADVECTION–DIFFUSION–REACTION EQUATION THROUGH THE BERNSTEIN POLYNOMIALS , 2020 .

[21]  D. Baleanu,et al.  Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives , 2020 .

[22]  Soheil Salahshour,et al.  Uncertain inverse problem for fractional dynamical systems using perturbed collage theorem , 2021, Commun. Nonlinear Sci. Numer. Simul..

[24]  D. L. C. Ching,et al.  A generalized model for quantitative analysis of sediments loss: A Caputo time fractional model , 2020 .

[25]  Dumitru Baleanu,et al.  Uncertain viscoelastic models with fractional order: A new spectral tau method to study the numerical simulations of the solution , 2017, Commun. Nonlinear Sci. Numer. Simul..

[26]  Roberto Garrappa,et al.  Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial , 2018 .

[27]  S. Salahshour,et al.  Tau method for the numerical solution of a fuzzy fractional kinetic model and its application to the oil palm frond as a promising source of xylose , 2015, J. Comput. Phys..

[28]  S. Momani,et al.  Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order , 2008 .

[29]  V. S. Erturk,et al.  Solution of a COVID-19 model via new generalized Caputo-type fractional derivatives , 2020, Chaos, Solitons & Fractals.

[30]  Chang Phang,et al.  New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials , 2017, Journal of King Saud University - Science.

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

[32]  S. Saratha,et al.  Fractional generalised homotopy analysis method for solving nonlinear fractional differential equations , 2020, Comput. Appl. Math..

[33]  Devendra Kumar,et al.  A new model of fractional Casson fluid based on generalized Fick’s and Fourier’s laws together with heat and mass transfer , 2020 .

[34]  D. L. C. Ching,et al.  MATHEMATICAL AND STATISTICAL ANALYSIS OF RL AND RC FRACTIONAL-ORDER CIRCUITS , 2020 .

[35]  Udita N. Katugampola.  Existence and Uniqueness results for a class of Generalized Fractional Differential Equations , 2014, 1411.5229.