# Novel orthogonal functions for solving differential equations of arbitrary order

Open access

## Abstract

Fractional calculus and the fractional differential equations have appeared in many physical and engineering processes. Therefore, an efficient and suitable method to solve them is very important. In this paper, novel numerical methods are introduced based on the fractional order of the Chebyshev orthogonal functions (FCF) with Tau and collocation methods to solve differential equations of the arbitrary (integer or fractional) order. The FCFs are obtained from the classical Chebyshev polynomials of the first kind. Also, the operational matrices of the fractional derivative and the product for the FCFs have been constructed. To show the efficiency and capability of these methods we have solved some well-known problems: the momentum, the Bagley-Torvik, and the Lane-Emden differential equations, then have compared our results with the famous methods in other papers.

If the inline PDF is not rendering correctly, you can download the PDF file here.

• [1] G.W. Leibniz Letter from Hanover Germany to G.F.A. L'Hopital September 30; 1695 in Mathematische Schriften 1849; reprinted 1962 Olms verlag; Hidesheim Germany Vol. 2 PP. 301-302 1965.

• [2] I. Podlubny Fractional Differential Equations Academic Press San Diego 1999.

• [3] I. Podlubny Geometric and physical interpretation of fractional integration and fractional differentiation Fract. Calc. Appl. Anal. 5 (2002) 367-386.

• [4] K.M. Kolwankar Studies of fractal structures and processes using methods of the fractional calculus www.arxiv:chaodyn/9811008V14Nov1998

• [5] M. Delkhosh Introduction of Derivatives and Integrals of Fractional order and Its Applications Appl. Math. Phys. 1 (4) (2013) 103-119.

• [6] K. Diethelm The analysis of fractional differential equations Berlin: Springer-Verlag 2010.

• [7] K.S. Miller B. Ross An Introduction to The Fractional Calculus and Fractional Differential Equations Wiley New York 1993.

• [8] J. He Nonlinear oscillation with fractional derivative and its applications in: International Conference on Vibrating Engineering'98 Dalian China 1998 pp. 288-291.

• [9] E. Keshavarz Y. Ordokhani M. Razzaghi A numerical solution for fractional optimal control problems via Bernoulli polynomials J. Vibr. Contr. (2015) doi:

• Crossref
• Export Citation
• [10] S.A. Yousefi A. Lotfi M. Dehghan The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems J. Vibr. Contr. 17(13) (2011) 2059-2065.

• [11] M. Dehghan E. Hamedi H. Khosravian-Arab A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials J. Vibr. Contr. (2014) doi:

• Crossref
• Export Citation
• [12] A. Lotfi M. Dehghan S.A. Yousefi A numerical technique for solving fractional optimal control problems Comput. Math. Appl. 62 (2011) 1055-1067.

• [13] K. Moaddy S. Momani I. Hashim The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics Comput. Math. Appl. 61 (2011) 1209-1216.

• [14] J. He Some applications of nonlinear fractional differential equations and their approximations Bull. Sci. Technol. 15 (1999) 86-90.

• [15] I. Grigorenko E. Grigorenko Chaotic dynamics of the fractional lorenz system Phys. Rev. Lett. 91 (2003) 034101034104.

• [16] S. Momani N.T. Shawagfeh Decomposition method for solving fractional Riccati differential equations Appl. Math. Comput. 182 (2006) 1083-1092.

• [17] Q. Wang Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method Appl. Math. Comput. 182 (2006) 1048-1055.

• [18] S. Kazem S. Abbasbandy S. Kumar Fractional-order Legendre functions for solving fractional-order differential equations Appl. Math. Model. 37 (2013) 54985510

• [19] M.A. Darani M. Nasiri A fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations Comput. Method. Diff. Equ. 1 (2013) 96-107.

• [20] I. Hashim O. Abdulaziz S. Momani Homotopy analysis method for fractional IVPs Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 674-684.

• [21] K. Parand M. Nikarya Application of Bessel functions for solving differential and integro-differential equations of the fractional order Appl. Math. Model. 38 (2014) 4137-4147.

• [22] J.A. Rad S. Kazem M. Shaban K. Parand A. Yildirim Numerical solution of fractional differential equations with a Tau method based on Legendre and Bernstein polynomials Math. Method. Appl. Sci. 37 (2014) 329-342.

• [23] J. Ma J. Liub Z. Zhouc Convergence analysis of moving finite element methods for space fractional differential equations J. Comput. Appl. Math. 255 (2014) 661-670.

• [24] P. Mokhtary Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations J. Comput. Appl. Math. 279 (2015) 145-158.

• [25] M. Kolk A. Pedas E. Tamme Modified spline collocation for linear fractional differential equations J. Comput. Appl. Math. 283 (2015) 28-40.

• [26] B. Fakhr Kazemi F. Ghoreishi Error estimate in fractional differential equations using multiquadratic radial basis functions J. Comput. Appl. Math. 245 (2013) 133-147.

• [27] S.B. Yuste Weighted average finite difference methods for fractional diffusion equations J. Comput. Phys. 216 (2006) 264-274.

• [28] S. Kazem An integral operational matrix based on jacobi polynomials for solving fractional-order differential equations Appl. Math. Model. (2012) http://dx.doi.org/10.1016/j.apm.2012.03.033.

• [29] S. Kazem J.A. Rad K. Parand S. Abbasbandy A new method for solving steady ow of a third-grade fluid in a porous half space based on radial basis functions Z. Naturforschung A. 66(10) (2011) 591-598.

• [30] M. Delkhosh M. Delkhosh Analytic solutions of some self-adjoint equations by using variable change method and its applications J. Appl. Math. (2012) Article ID 180806 7 pages.

• [31] X. Li Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method Commun. Nonlinear Sci. Numer. Simulat. 17 (2012) 3934-3946.

• [32] A. Saadatmandi M. Dehghan M. R. Azizi The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients Commun. Nonlinear Sci. Numer. Simulat. 17 (2012) 4125-4136.

• [33] K. Parand M. Delkhosh Solving Volterra's population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions Ricerche Mat. 65(1) (2016) 307-328.

• [34] A.A. Kilbas H.M. Srivastava J.J. Trujillo Theory and Applications of Fractional Differential Equations Elsevier San Diego 2006.

• [35] Z. Odibat S. Momani An algorithm for the numerical solution of differential equations of fractional order J. Appl. Math. Inform. 26 (2008) 15-27.

• [36] G. Szego orthogonal polynomials American Mathematical Society Providence Rhode Island 1975.

• [37] M.R. Eslahchi M. Dehghan S. Amani Chebyshev polynomials and best approximation of some classes of functions J. Numer. Math. 23 (1) (2015) 41-50.

• [38] 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 (2011) 2364-2373.

• [39] A. Nkwanta E.R. Barnes Two Catalan-type Riordan arrays and their connections to the Chebyshev polynomials of the first kind J. Integer Seq. 15 (2012) 1-19.

• [40] A. Saadatmandi M. Dehghan Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method Numer. Method. Part. D. E. 26 (1) (2010) 239-252.

• [41] K. Parand M. Shahini M. Dehghan Solution of a laminar boundary layer ow via a numerical method Commun. Nonlinear Sci. Numer. Simulat. 15(2) (2010) 360-367.

• [42] K. Parand A. Taghavi M. Shahini Comparison between rational Chebyshev and modified generalized Laguerre functions pseudospectral methods for solving Lane-Emden and unsteady gas equations Acta Phys. Pol. B 40(12) (2009) 1749-1763.

• [43] K. Parand A.R. Rezaei A. Taghavi Numerical approximations for population growth model by rational Chebyshev and Hermite functions collocation approach: a comparison Math. Method. Appl. Sci. 33(17) (2010) 2076-2086.

• [44] K. Parand M. Shahini A. Taghavi Generalized Laguerre polynomials and rational Chebyshev collocation method for solving unsteady gas equation Int. J. Contemp. Math. Sci. 4(21) (2009) 1005-1011.

• [45] K. Parand S. Khaleqi The Rational Chebyshev of Second Kind Collocation Method for Solving a Class of Astrophysics Problems Euro. Phys. J. - Plus 131 (2016) 1-24.

• [46] K. Parand S. Abbasbandy S. Kazem A.R. Rezaei An improved numerical method for a class of astrophysics problems based on radial basis functions Phys. Scripta 83(1) (2011) 015011.

• [47] K Parand M Dehghan A Taghavi Modified generalized Laguerre function Tau method for solving laminar viscous ow: The Blasius equation Int. J. Numer. Method. H. 20(7) (2010) 728-743.

• [48] K. Parand J.A. Rad M. Nikarya A new numerical algorithm based on the first kind of modified Bessel function to solve population growth in a closed system Int. J. Comput. Math. 91(6) (2014) 1239-1254.

• [49] J.P. Boyd Chebyshev and Fourier Spectral Methods Second Edition DOVER Publications Mineola New York (2000).

• [50] G. Adomian Solving Frontier problems of Physics: The decomposition method Kluwer Academic Publishers 1994.

• [51] S.J. Liao The proposed homotopy analysis technique for the solution of nonlinear problems PhD thesis Shanghai Jiao Tong University 1992.

• [52] J.C. Mason D.C. Handscomb Chebyshev polynomials CRC Press Company ISBN 0-8493-0355-9.

• [53] M. Rehman R.A. Khan The Legendre wavelet method for solving fractional differential equations Commun. Nonlinear. Sci. Numer. Simulat. 16 (2011) 4163-4173.

• [54] A. Saadatmandi M. Dehghan A new operational matrix for solving fractional-order differential equations Comput. Math. Appl. 59 (2010) 1326-1336.

• [55] K. Diethelm N.J. Ford Numerical solution of the Bagley-Torvik equation BIT 42 (2002) 490-507.

• [56] Z. Odibat S. Momani Analytical comparison between the homotopy perturbation method and variational iteration method for differential equations of fractional order Int. J. Mod. Phys. B 22 (2008) 4041-4058.

• [57] E.A. Butcher H. Ma E. Bueler V. Averina Z. Szabo Stability of linear time-periodic delay-differential equations via Chebyshev polynomials Int. J. Numer. Meth. Engng. 59 (2004) 895-922.

• [58] Y. Xu Z. He. The short memory principle for solving abel differential equation of fractional order Comput. Math. Appl. 64 (2011) 4796-4805.

• [59] M. Razzaghi S. Yousefi The Legendre wavelets operational matrix of integration Int. J. Sys. Sci. 32(4) (2001) 495-502.

• [60] S.A. Yousefi M. Behroozifar Operational matrices of Bernstein polynomials and their applications Int. J. Sys. Sci. 41:6 (2010) 709-716.

• [61] A. Akgul M. Inc E. Karatas and D. Baleanu Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique Adv. Difference Equ. 2015: 220.

• [62] K. Parand M. Nikarya J.A. Rad Solving non-linear Lane-Emden type equations using Bessel orthogonal functions collocation method Celes. Mech. Dyn. Astr. 116 (1) (2013) 97-107.

• [63] M.S. Mechee and N. Senu Numerical Study of Fractional Differential Equations of Lane-Emden Type by Method of Collocation Appl. Math. 3 (2012) 851-856.

• [64] K. Parand M. Shahini M. Dehghan Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type J. Comput. Phys. 228 (23) (2009) 8830-8840.

• [65] K. Parand S. Khaleqi The rational Chebyshev of Second Kind Collocation Method for Solving a Class of Astrophysics Problems Eur. Phys. J. Plus 131 (2016) 24.

• [66] S.I. Khan N. Ahmed U. Khan S. U. Jan S.T. Mohyud-Din Heat transfer analysis for squeezing ow between parallel disks J. Egyptian Math. Soc. 23 (2015) 445-450.

• [67] M. Shaban E. Shivanian and S. Abbasbandy Analyzing magneto-hydrodynamic squeezing ow between two parallel disks with suction or injection by a new hybrid method based on the Tau method and the homotopy analysis method Eur. Phys. J. Plus 128 (2013) 133.

• [68] D.D. Ganji M. Abbasi J. Rahimi M. Gholami I. Rahimipetroudi On the MHD squeeze ow between two parallel disks with suction or injection via HAM and HPM Front. Mech. Eng. DOI 10.1007/s11465-014-0303-0 (2014).

### Tbilisi Mathematical Journal

Search
###### Journal information
Impact Factor

Mathematical Citation Quotient (MCQ) 2017: 0.11

Target audience:

researchers in all areas of mathematics

###### Metrics
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 390 139 5