Abstract
In this paper, the Chebyshev polynomials method is applied to solve a space-time variable fractional order integro-differential equation. Using operational matrices of Chebyshev polynomials furnished from the Caputo–Prabhakar sense and also suitable collocation points, the variable fractional order integro-differential equation would be converted to the system of algebraic equations. The main aim of the Chebyshev polynomials method is to derive four kinds of operational matrices of Chebyshev polynomials. With such operational matrices, an equation is transformed into the products of several dependent matrices, which can also be viewed as the system of linear equations after dispersing the variables. An error bound is proved for the approximate solution obtained by the proposed method. Finally, some numerical examples are presented to demonstrate the accuracy of the proposed method.
Similar content being viewed by others
References
Z. M. Odibat, “A study on the convergence of variational iteration method,” Math. Comput. Modelling 51 (9-10), 1181–1192 (2010).
I. L. El-Kalla, “Convergence of the Adomian method applied to a class of nonlinear integral equations,” Appl. Math. Lett. 21 (4), 372–376 (2008).
M. M. Hosseini, “Adomian decomposition method for solution of nonlinear differential algebraic equations,” Appl. Math. Comput. 181 (2), 1737–1744 (2006).
S. Momani, Z. Odibat, and V. S. Erturk, “Generalized differential transform method for solving a space and time fractional diffusion wave equation,” Phys. Lett. A 370 (5–6), 379–387 (2007).
Y. Chen, M. Yi, and C. Yu, “Error analysis for numerical solution of fractional differential equation by Haar wavelets method,” J. Comput. Sci. 3 (5), 367–373 (2012).
Y. Xu and V. Ertürk, “A finite difference technique for solving variable order fractional integro–differential equations,” Bull. Iranian Math. Soc. 40 (3), 699–712 (2014).
M. Zayernouri and G. E. Karniadakis, “Fractional spectral collocation methods for linear and nonlinear variable order FPDEs,” J. Comput. Phys. 293, 312–338 (2015).
A. Mohebbi and M. Saffarian, “Implicit RBF Meshless method for the solution of two-dimensional variable order fractional cable equation,” J. Appl. Comput. Mech. 6 (2), 235–247 (2020).
H. Aminikhah, A. R. Sheikhani, and H. Rezazadeh, “Exact solutions for the fractional differential equations by using the first integral method,” Nonlinear Engineering 4 (1), 15–22 (2015).
M. Mashoof and A. R. Sheikhani, “Numerical solution of fractional control system by Haar-wavelet operational matrix method,” Int. J. Industrial Math. 8 (3), 303–312 (2016).
M. Mashoof, A. R. Sheikhani, and H. S. Najafi, “Stability analysis of distributed-order Hilfer–Prabhakar systems based on inertia theory,” Math. Notes 104 (1), 74–85 (2018).
M. Mashoof, A. R. Sheikhani, and H. S. Naja, “Stability analysis of distributed order Hilfer-Prabhakar differential equations,” Hacet. J. Math. Stat. 47 (2), 299–315 (2018).
H. S. Najafi, S. A. Edalatpanah, and A. R. Sheikhani, “Convergence analysis of modified iterative methods to solve linear systems,” Mediterr. J. Math. 11 (3), 1019–1032 (2014).
F. Shariffar, A. Refahi Sheikhani, and HS. Najafi, “An efficient chebyshev semi-iterative method for the solution of large systems,” Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 80 (4), 239–252 (2018).
R. Garra, R. Gorenflo, F. Polito, and Ž. Tomovski, “Hilfer–Prabhakar derivatives and some applications,” Appl. Math. Comput. 242, 576–589 (2014).
T. R. Prabhakar, “A singular integral equation with a generalized Mittag Leffler function in the kernel,” Yokohama Math. J. 19, 7–15 (1971).
M. Ichise, Y. Nagayanagi, and T. Kojima, “An analog simulation of non-integer order transfer functions for analysis of electrode processes,” J. Electroanal. Chem. and Interfacial Electrochem. 33 (2), 253–265 (1971).
S. Khubalkar, A. Junghare, M. Aware, and S. Das, “Unique fractional calculus engineering laboratory for learning and research,” Int. J. Electrical Engineering Education (2018).
J. Lai, S. Mao, J. Qiu, H. Fan, Q. Zhang, Z. Hu, and J. Chen, “Investigation progress and applications of fractional derivative model in geotechnical engineering,” Math. Problems in Eng. 2016, Art. ID 9183296 (2016).
M. D. Ortigueira, Fractional Calculus for Scientists and Engineers, in Lect. Notes Electr. Eng. (Springer, Dordrecht, 2011), Vol. 84.
F. Shariffar and R. Sheikhani, “A new two-stage iterative method for linear systems and its application in solving Poissons equation,” Int. J. Industrial Math. 11 (4), 283–291 (2019).
A. H. R. Sheikhani and M. Mashoof, “A Collocation Method for Solving Fractional Order Linear System,” J. Indones. Math. Soc. 23 (1), 27–42 (2017).
A. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in North-Holland Math. Stud. (Elsevier Sci., Amsterdam, 2006), Vol. 204.
R. Garra and R. Garrappa, “The Prabhakar or three parameters Mittag-Leffler function: theory and application,” Commun. Nonlinear Sci. Numer. Simul. 56, 314–329 (2018).
R. Garrappa, “Grünwald–Letnikov operators for fractional relaxation in Havri\=liak-Negamimodels,” Commun. Nonlinear Sci. Numer. Simul. 38, 178–191 (2016).
A. Giusti, I. Colombaro, R. Garra, R. Garrappa, F. Polito, M. Popolizio, and F. Mainardi, “A practical guide to Prabhakar fractional calculus,” Fract. Calc. Appl. Anal. 23 (1), 9–54 (2020).
F. Mainardi and R. Garrappa, “On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics,” J. Comput. Phys. 293, 70–80 (2015).
S. C. Pandey, “The Lorenzo–Hartley’s function for fractional calculus and its applications pertaining to fractional order modelling of anomalous relaxation in dielectrics,” Comput. Appl. Math. 37 (3), 2648–2666 (2018).
R. K. Gupta, B. S. Shaktawat, and D. Kumar, “Certain relation of generalized fractional calculus associated with the generalized Mittag-Leffler function,” J. Rajasthan Acad. Phys. Sci. 15 (3), 117–126 (2016).
A. Giusti and I. Colombaro, “Prabhakar-like fractional viscoelasticity,” Commun. Nonlinear Sci. Numer. Simul. 56, 138–143 (2018).
R. Agarwal, S. O. N. A. L. Jain, and R. P. Agarwal, “Analytic solution of generalized space-time fractional reaction-diffusion equation,” Fract. Differ. Calc. 7, 169–84 (2017).
E. H. Doha, M. A. Abdelkawy, A. Z. M. Amin, and A. M. Lopes, “On spectral methods for solving variable-order fractional integro-differential equations,” Comput. Appl. Math. 37 (3), 3937–3950 (2018).
M. A. Snyder, Chebyshev Methods in Numerical Approximation (Prentice-Hall, Englewood Cliffs, NJ, 1966).
K. Sun and M. Zhu, “Numerical algorithm to solve a class of variable order fractional integral-differential equation based on Chebyshev polynomials,” Math. Probl. Eng. 2015, Art. ID 902161 (2015).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, 2022, Vol. 111, pp. 676–691 https://doi.org/10.4213/mzm13511.
Rights and permissions
About this article
Cite this article
Bagherzadeh Tavasani, B., Refahi Sheikhani, A.H. & Aminikhah, H. Numerical Simulation of the Variable Order Fractional Integro-Differential Equation via Chebyshev Polynomials. Math Notes 111, 688–700 (2022). https://doi.org/10.1134/S0001434622050030
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434622050030