Bernstein Operational Matrix for Solving Variable Order Fractional Differential Equations with Generalized Caputo-Type
DOI:
https://doi.org/10.31185/wjps.348Abstract
In this research, we present novel generalized derivative founded on the newly constructed (NGCFVODs) novel generalized Caputo Fractional variable order derivatives. Utilizing these operators, a numerical methodology has been formulated to resolve the Fractional Variable Order Differential Equations (FVODEs). We estimate the solution for (FVODEs) by employing bernstein polynomials as foundational vectors. We have further expanded the derivative operational matrix of bernstein Polynomials (bPs) to generalized derivative an operational matrix in sense of (NGCFVODs). The efficiency of developed numerical methodology is tested by a taking a various test examples. We also a compare results of our suggested approach with the methodologies existing in academic papers. In this study the Fractional variable order differential operator of new generalized Caputo was described by three categories:
(i) various value in ρ and Fractional variable order a parameter, (ii) various value in a fractional parameters while Fractional variable order and ρ parameter are fixed, and (iii) various value in Fractional variable order parameter controlling fractional and ρ parameter.
References
Anatolii Aleksandrovich Kilbas, Hari M Srivastava, Juan J Trujillo, Theory and Applications of Fractional Differential Equations, Vol. 204, elsevier, 2006.
J. Tenreiro Machado, Virginia Kiryakova, Francesco Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer.Simul. 16 (3) (2011) 1140–1153.
Nadeem Ahmad Sheikh, Dennis Ling Chuan Ching, Ilyas Khan, Devendra Kumar, Kottakkaran Sooppy Nisar, A new model of fractional Casson fluid based on generalized Fick’s and Fourier’s laws together with heat and mass transfer, Alex. Eng. J. 59 (5) (2020a) 2865–2876.
Nadeem Ahmad Sheikh, Dennis Ling Chuan Ching, Sami Ullah, Ilyas Khan, Mathematical and statistical analysis of RL and RC fractional-order circuits, Fractals 28 (8) (2020b) 2040030–2040210.
Zaid Odibat, Dumitru Baleanu, Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives, Appl. Numer. Math. 156 (2020) 94–105.
Vedat Suat Erturk, Pushpendra Kumar, Solution of a COVID-19 model via new generalized Caputo-type fractional derivatives, Chaos Solitons Fractals 139 (2020) 110280.
Abbas Saadatmandi, Mehdi Dehghan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl. 59 (3) (2010) 1326–1336.
Abdon Atangana, Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system, Chaos Solitons Fractals 102 (2017) 396–406.
Abdon Atangana, Muhammad Altaf Khan, et al., Modeling and analysis of competition model of bank data with fractal-fractional Caputo-Fabrizio operator, Alex. Eng. J. 59 (4) (2020) 1985–1998.
JF Gomez-Aguilar, T Cordova-Fraga, Thabet Abdeljawad, Aziz Khan, Hasib Khan, Analysis of fractal–fractional malaria transmission model, Fractals (2020) 2040041.
Pandey, V. (2016). Physical and Geometrical Interpretation of Fractional Derivatives in Viscoelasticity and Transport Phenomena. (Doctoral disser- tation, University of Oslo).
Zaid Odibat, Dumitru Baleanu, Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives, Appl. Numer. Math. 156 (2020) 94–105.
K. Diethelm, N.J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput. 154 (2004) 621–640.
X. Li et al.,A new numerical method for variable order fractional functional differential equations ,Applied Mathematics Letters 68 (2017) 80–86.
J.X. Cao, Y.N. Qiu, A high order numerical scheme for variable order fractional ordinary differential equation, Appl. Math.Lett. 61 (2016) 88–94
S. Shen, F. Liu, J. Chen, I. Turner, V. Anh, Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput. 218 (2012) 10861–10870
A.M. Shloof, N. Senu, A. Ahmadian et al.,An efficient operation matrix method for solving fractal–fractional differential equations with generalized Caputo-type fractional–fractal derivative,Mathematics and Computers in Simulation 188 (2021) 415–435.
Downloads
Published
Issue
Section
License
Copyright (c) 2024 alaa abbas
This work is licensed under a Creative Commons Attribution 4.0 International License.
