Solving fractional variable-order differential equations of the non-singular derivative using Jacobi operational matrix
Keywords:FDEs; Non-singular derivative; Variable order; Operational matrix.
This research derives the shifted Jacobi operational matrix (JOM) with respect to fractional derivatives, implemented with the spectral tau method for the numerical solution of the Atangana-Baleanu Caputo (ABC) derivative. The major aspect of this method is that it considerably simplifies problems by reducing them to ones that can be solved by solving a set of algebraic equations. The main advantage of this method is its high robustness and accuracy gained by a small number of Jacobi functions. The suggested approaches are applied in solving non-linear and linear ABC problems according to initial conditions, and the efficiency and applicability of the proposed method are proved by several test examples. A lot of focus is placed on contrasting the numerical outcomes discovered by the new algorithm together with those discovered by previously well-known methods.
K. S. Miller, & B. Ross, “An introduction to the fractional calculus and fractional differential equations”, Wiley, (1993).
K. Oldham, & J. Spanier, “The fractional calculus theory and applications of differentiation and integration to arbitrary order”, Elsevier, (1974).
M. Amairi, M. Aoun, S. Najar & M. N. Abdelkrim, “A constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation”, Applied Mathematics and Computation 217 (2010) 2162. DOI: https://doi.org/10.1016/j.amc.2010.07.015
J. Deng, & L. Ma, “Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations”, Applied Mathematics Letters 23 (2010) 676. DOI: https://doi.org/10.1016/j.aml.2010.02.007
L. K. Alzaki, & H. K. Jassim, “Time-Fractional Differential Equations with an Approximate Solution”, Journal of the Nigerian Society of Physical Sciences (2022) 818. DOI: https://doi.org/10.46481/jnsps.2022.818
O. A. Uwaheren, A. F. Adebisi, & O. A. Taiwo, “Perturbed collocation method for solving singular multi-order fractional differential equations of Lane-Emden type”, Journal of the Nigerian Society of Physical Sciences (2020) 141. DOI: https://doi.org/10.46481/jnsps.2020.69
K. M. Owolabi, & A. Atangana, “On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems”, Chaos: An Interdisciplinary Journal of Nonlinear Science 29 (2019) 023111. DOI: https://doi.org/10.1063/1.5085490
S. Arshad, I. Saleem, O. Defterli, Y. Tang, & D. Baleanu, “Simpson’s method for fractional differential equations with a non-singular kernel applied to a chaotic tumor model”, Physica Scripta 96 (2021) 124019. DOI: https://doi.org/10.1088/1402-4896/ac1e5a
S. Qureshi, & A. Yusuf, “Modeling chickenpox disease with fractional derivatives: From caputo to atangana-baleanu.” Chaos, Solitons & Fractals 122 (2019) 111. DOI: https://doi.org/10.1016/j.chaos.2019.03.020
I. Ahmed, E. F. D. Goufo, A. Yusuf, P. Kumam, P. Chaipanya, & K. Nonlaopon, “An epidemic prediction from analysis of a combined HIVCOVID-19 co-infection model via ABC-fractional operator”, Alexandria Engineering Journal 60 (2021) 2979. DOI: https://doi.org/10.1016/j.aej.2021.01.041
S. Djennadi, N. Shawagfeh, M. S. Osman, J. F. Gomez-Aguilar, & O.´ A. Arqub, “The Tikhonov regularization method for the inverse source problem of time fractional heat equation in the view of ABC-fractional technique”, Physica Scripta 96 (2021) 094006. DOI: https://doi.org/10.1088/1402-4896/ac0867
A. Saadatmandi, & M. Dehghan, “A new operational matrix for solving fractional-order differential equations”, Computers & mathematics with applications 59 (2010) 1326. DOI: https://doi.org/10.1016/j.camwa.2009.07.006
E. H. Doha, A. H. Bhrawy, & S. S. Ezz-Eldien, “Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations”, Applied Mathematical Modelling 35 (2011) 5662. DOI: https://doi.org/10.1016/j.apm.2011.05.011
A. H. Bhrawy, A. S. Alofi, & S. S. Ezz-Eldien, “A quadrature tau method for fractional differential equations with variable coefficients”, Applied Mathematics Letters 24 (2011) 2146. DOI: https://doi.org/10.1016/j.aml.2011.06.016
E. H. Doha, A. H. Bhrawy, & S. S. Ezz-Eldien, “A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order”, Computers & Mathematics with Applications 62 (2011) 2364. DOI: https://doi.org/10.1016/j.camwa.2011.07.024
F. Ghoreishi, & S. Yazdani, “An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis”, Computers & Mathematics with Applications 61 (2011) 30. DOI: https://doi.org/10.1016/j.camwa.2010.10.027
S. K. Vanani, & A. Aminataei, “Tau approximate solution of fractional partial differential equations”, Computers & Mathematics with Applications 62 (2011) 1075. DOI: https://doi.org/10.1016/j.camwa.2011.03.013
A. Pedas, & E. Tamme, “On the convergence of spline collocation methods for solving fractional differential equations”, Journal of Computational and Applied Mathematics 235 (2011) 3502. DOI: https://doi.org/10.1016/j.cam.2010.10.054
S. Esmaeili, & M. Shamsi, “A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations”, Communications in Nonlinear Science and Numerical Simulation 16 (2011) 3646. DOI: https://doi.org/10.1016/j.cnsns.2010.12.008
S. Esmaeili, M. Shamsi, & Y. Luchko, “Numerical solution of fractional differential equations with a collocation method based on Muntz polyno-¨ mials”, Computers & Mathematics with Applications 62 (2011) 918. DOI: https://doi.org/10.1016/j.camwa.2011.04.023
G. Szego, “Am. Math. Soc. Colloq. Pub. 23”, Orthogonal Polynomials¨ (1985).
E. H. Doha, A. H. Bhrawy, & R. M. Hafez “A Jacobi dual-Petrov-Galerkin method for solving some odd-order ordinary differential equations”, Abstract and Applied Analysis 2011 (2011). DOI: https://doi.org/10.1155/2011/947230
E. H. Doha, A. H. Bhrawy, & S. S. Ezz-Eldien, “A new Jacobi operational matrix: an application for solving fractional differential equations”, Applied Mathematical Modelling 36 (2012) 4931. DOI: https://doi.org/10.1016/j.apm.2011.12.031
E. H. Doha, “On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials”, Journal of Physics A: Mathematical and General 37 (2004) 657. DOI: https://doi.org/10.1088/0305-4470/37/3/010
Y. L. Luke, “The special functions and their approximations”, 53 (1969).
M. Basim, N. Senu, Z. B. Ibrahim, A. Ahmadian, & S. Salahshour, “a Robust Operational Matrix of Nonsingular Derivative to Solve Fractional Variable-Order Differential Equations”, Fractals 30 (2022) 2240041. DOI: https://doi.org/10.1142/S0218348X22400412
J. D.Djida, A. Atangana, & I. Area, “Numerical computation of a fractional derivative with non-local and non-singular kernel”, Mathematical Modelling of Natural Phenomena 12 (2017) 4. DOI: https://doi.org/10.1051/mmnp/201712302
M. Toufik, & A. Atangana, “New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models”, The European Physical Journal Plus 132 (2017) 1. DOI: https://doi.org/10.1140/epjp/i2017-11717-0
M.Caputo, & M. Fabrizio, “A new definition of fractional derivative without singular kernel”, Progress in Fractional Differentiation & Applications 1 (2015) 73.
A. Atangana, & D. Baleanu, “New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model”, arXiv preprint arXiv:1602.03408 (2016). DOI: https://doi.org/10.2298/TSCI160111018A
X. Li, Y. Gao, & B. Wu, “Approximate solutions of Atangana-Baleanu variable order fractional problems”, AIMS Mathematics 5 (2020) 2285.
L. J. Rong, & P. Chang, “Jacobi wavelet operational matrix of fractional integration for solving fractional integro-differential equation”, Journal of Physics: Conference Series 693 (2016) 012002. DOI: https://doi.org/10.1088/1742-6596/693/1/012002
C. Phang, Y. T. Toh, & F. S. Md Nasrudin, “An operational matrix method based on poly-Bernoulli polynomials for solving fractional delay differential equations”, Computation 8 (2020) 82. DOI: https://doi.org/10.3390/computation8030082
S. Salahshour, A. Ahmadian, M. Salimi, M. Ferrara, & D. Baleanu, “Asymptotic solutions of fractional interval differential equations with nonsingular kernel derivative”, Chaos: An Interdisciplinary Journal of Nonlinear Science 29 (2019) 083110. DOI: https://doi.org/10.1063/1.5096022
A. Al-Rabtah, S. Momani, & M. A. Ramadan, “Solving linear and nonlinear fractional differential equations using spline functions”, Abstract and Applied Analysis 2012 (2012).
Z. M. Odibat, & S. Momani, “An algorithm for the numerical solution of differential equations of fractional order”, Journal of Applied Mathematics & Informatics 26 (2008) 15.
A. Al-Rabtah, S. Momani, & M. A. Ramadan, “Solving linear and nonlinear fractional differential equations using spline functions”, Abstract and Applied Analysis 2012 (2012). DOI: https://doi.org/10.1155/2012/426514
X. Li, Y. Gao, & B. Wu, “Approximate solutions of Atangana-Baleanu variable order fractional problems”, AIMS Mathematics 5 (2020) 2285. DOI: https://doi.org/10.3934/math.2020151
D. Baleanu, “Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations”, Inst Mathematics & Informatics (2019).
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