A Chebyshev functions method for solving linear and nonlinear fractional differential equations based on Hilfer fractional derivative
Subject Areas : Numerical analysisM. H. Derakhshan 1 , A. Aminataei 2
1 - Faculty of Mathematics, K. N. Toosi University of Technology, P. O. Box: 16765-3381, Tehran, Iran
2 - Faculty of Mathematics, K. N. Toosi University of Technology, P. O. Box: 16765-3381, Tehran, Iran
Keywords: Operational matrix, Chebyshev polynomials, fractional-order differential equation, Hilfer fractional derivative,
Abstract :
The theory of derivatives and integrals of fractional in fractional calculus have found enormousapplications in mathematics, physics and engineering so for that reason we needan efficient and accurate computational method for the solution of fractional differential equations.This paper presents a numerical method for solving a class of linear and nonlinear multi-order fractional differential equations with constant coefficients subject to initial conditions based on the fractional order Chebyshev functions that this function is defined as follows:\begin{equation*}\overline{T}_{i+1}^{\alpha}(x)=(4x^{\alpha}-2)\overline{T}_{i}^{\alpha}(x)\overline{T}_{i-1}^{\alpha}(x),\,i=0,1,2,\ldots,\end{equation*}where $\overline{T}_{i+1}^{\alpha}(x)$ can be defined by introducing the change of variable $x^{\alpha},\,\alpha>0$, on the shifted Chebyshevpolynomials of the first kind. This new method is an adaptation of collocationmethod in terms of truncated fractional order Chebyshev Series. To do this method, a new operational matrix of fractional order differential in the Hilfer sense for the fractional order Chebyshev functions is derived. By using this method we reduces such problems to those ofsolving a system of algebraic equations thus greatly simplifying the problem. At the end of this paper, several numerical experiments are given to demonstrate the efficiency and accuracy of the proposed method.
[1] M. Ahmadi Darani, A. Saadatmandi, The operational matrix of fractional derivative of the fractional-order Chebyshev functions and its applications, Comput. Methods. Differ. Equ. 5 (1) (2017), 67-87.
[2] A. Abedian, A. Düster, Equivalent Legendre polynomials: Numerical integration of discontinuous functions in the finite element methods, Comput. Methods. Appl. Mech. Eng. 343 (2019), 690-720.
[3] M. A. Abdelkawy, A collocation method based on Jacobi and fractional order Jacobi basis functions for multi-dimensional distributed-order diffusion equations, Int. J. Nonlinear. Sci. Numer. Simul. 19 (7-8) (2018), 781-792.
[4] A. Ansari, A. Refahi Sheikhani, H. Saberi Najafi, Solution to system of partial fractional differential equations using the fractional exponential operators, Math. Methods. Appl. Sci. 35 (1) (2012), 119-123.
[5] D. Baleanu, B. Shiri, H. M. Srivastava, M. Al Qurashi, A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel, Adv. Differ. Equ. 2018, 2018:353.
[6] M. Behroozifar, N. Habibi, A numerical approach for solving a class of fractional optimal control problems via operational matrix Bernoulli polynomials, J. Vib. Control. 24 (12) (2018), 2494-2511.
[7] A. H. Bhrawy, M. A. Zaky, J. A. T. Machado, Numerical solution of the two-sided space-time fractional
telegraph equation via Chebyshev tau approximation, J. Optim. Theory. Appl. 174 (1) (2017), 321-341.
[8] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods, Springer-Verlag, Berlin, 2006.
[9] 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, Comput. Math. Appl. 62 (5) (2011), 2364-2373.
[10] M. R. Eslahchi, M. Dehghan, S. Amani, The third and fourth kinds of Chebyshev polynomials and best uniform approximation, Math. Comput. Model. 55 (5-6) (2012), 1746-1762.
[11] C. M. Fan, C. N. Chu, B. Sarler, T. H. Li, Numerical solutions of waves-current interactions by generalized finite difference method, Eng. Anal. Bound. Elem. 100 (2019), 150-163.
[12] R. Garra, R. Gorenflo, F. Polito, Z. Tomovski, Hilfer-Prabhakar derivatives and some applications, Appl. Math. Momput. 242 (2014), 576-589.
[13] Y. Gu, W. Qu, W. Chen, L. Song, C. Zhang, The generalized finite difference method for long-time dynamic modeling of three-dimensional coupled thermoelasticity problems, J. Comput. Phys. 384 (2019), 42-59.
[14] O. E. Hepson, A. Korkmaz, I. Dag, Exponential B-spline collocation solutions to the Gardner equation, Int. J. Comput. Math. 97 (4) (2020), 837-850.
[15] M. H. Heydari, Z. Avazzadeh, A new wavelet method for variable-order fractional optimal control problems, Asian. J. Control. 20 (5) (2018), 1804-1817.
[16] E. Keshavarz, Y. Ordokhani, M. Razzaghi, Numerical solution of nonlinear mixed Fredholm-Volterra integro-differential equations of fractional order by Bernoulli wavelets, Comput. Methods. Differ. Equ. 7 (2) (2019), 163-176.
[17] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations (Vol. 204), Elsevier Science Limited, North-Holland Mathematics Studies, 2006.
[18] D. Mahapatra, S. Sanyal, S. Bhowmick, An approximate solution of functionally Graded Timoshenko beam using B-spline collocation method, J. Solid. Mech. 11 (2) (2019), 297-310.
[19] V. A. Manrique, M. Ramirez, Simple Approach to Special Polynomials: Laguerre, Hermite, Legendre, Tchebycheff and Gegenbauer, Appl. Math. IntechOpen. (2019), 1-18.
[20] M. Mashoof, A. Refahi Sheikhani, H. S. Naja, Stability analysis of distributed order Hilfer-Prabhakar differential equations, Hacettepe. J. Math. Stat. 47 (2) (2018), 299-315.
[21] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[22] P. Rana, N. Shukla, Y. Gupta, I. Pop, Homotopy analysis method for predicting multiple solutions in the channel flow with stability analysis, Commun. Nonlinear. Sci. Numer. Simul. 66 (2019), 183-193.
[23] K. Rabiei, Y. Ordokhani, Solving fractional pantograph delay differential equations via fractional order Boubaker polynomials, Eng. Comput. 35 (4) (2019), 1431-1441.
[24] H. Rezazadeh, H. Aminikhah, A. Refahi Sheikhani, Stability analysis of Hilfer fractional differential systems, Math. Commun. 21 (1) (2016), 45-64.
[25] B. Sahiner, M. Sezer, Determining constant breadth curve mate of a curve on a surface via Taylor collocation method, New Trends. Math. Sci. 6 (3) (2018), 103-115.
[26] M. R. M. Shabestari, R. Ezzati, T. Allahviranloo, Numerical solution of fuzzy fractional integro differential equation via two-dimensional Legendre wavelet method, J. Intell. Fuzzy. Syst. 34 (4) (2018), 2453-2465.
[27] A. Shojaei, U. Galvanetto, T. Rabczuk, A. Jenabi, M. Zaccariotto, A generalized finite difference method based on the Peridynamic differential operator for the solution of problems in bounded and unbounded domains, Comput. Methods. Appl. Mech. Eng. 343 (2019), 100-126.
[28] E. V. P. Spreafico, M. Rachidi, New approach of Bernoulli and Genocchi Numbers and their associated polynomials via generalized Fibonacci sequences of order, Proceeding Series. Braz. Soc. Comput. Appl. Math. 6 (2) (2018), 010436-(1-7).
[29] K. Yadav, J. P. Jaiswal, On the operational matrix for fractional integration and its application for solving Abel integral equation using Bernoulli wavelets, Glob. J. Pure. Appl. Math. 15 (1) (2019), 81-101.
[30] F. Yousefi, A. Rivaz, W. Chen, The construction of operational matrix of fractional integration for solving fractional differential and integro-differential equations, Neural. Comput. Appl. 31 (6) (2019), 1867-1878.