×

zbMATH — the first resource for mathematics

Novel orthogonal functions for solving differential equations of arbitrary order. (English) Zbl 1362.34017
Summary: 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.

MSC:
34A08 Fractional ordinary differential equations
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
PDF BibTeX Cite
Full Text: DOI
References:
[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. · Zbl 0924.34008
[3] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2002) 367-386. · Zbl 1042.26003
[4] K.M. Kolwankar, Studies of fractal structures and processes using methods of the fractional calculus, www.arxiv:chaodyn/9811008V14Nov 1998 <http://www.arxiv:chaodyn/9811008V14Nov 1998>.
[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. · Zbl 1215.34001
[7] K.S. Miller, B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. · Zbl 0789.26002
[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: · Zbl 1373.49003
[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. · Zbl 1271.65105
[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:
[12] A. Lotfi, M. Dehghan, S.A. Yousefi, A numerical technique for solving fractional optimal control problems, Comput. Math. Appl., 62 (2011) 1055-1067. · Zbl 1228.65109
[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. · Zbl 1217.65174
[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. · Zbl 1107.65121
[17] Q. Wang, Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method, Appl. Math. Comput. 182 (2006) 1048-1055. · Zbl 1107.65124
[18] S. Kazem, S. Abbasbandy, S. Kumar, Fractional-order Legendre functions for solving fractional-order differential equations, Appl. Math. Model., 37 (2013) 54985510 · Zbl 1449.33012
[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. · Zbl 1309.65080
[20] I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 674-684. · Zbl 1221.65277
[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. · Zbl 1290.26008
[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. · Zbl 1306.65294
[25] M. Kolk, A. Pedas, E. Tamme, Modified spline collocation for linear fractional differential equations, J. Comput. Appl. Math., 283 (2015) 28-40. · Zbl 1311.65094
[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. · Zbl 1262.65091
[27] S.B. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216 (2006) 264-274. · Zbl 1094.65085
[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. · Zbl 1351.34007
[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. · Zbl 1247.34001
[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. · Zbl 1250.65094
[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. · Zbl 1250.65121
[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. · Zbl 1355.65180
[34] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, 2006. · Zbl 1092.45003
[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. · Zbl 1133.65116
[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. · Zbl 1323.41027
[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. · Zbl 1231.65126
[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. · Zbl 1292.05032
[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. · Zbl 1186.65136
[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. · Zbl 1221.76157
[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. · Zbl 1334.76134
[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. · Zbl 1204.65159
[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. · Zbl 1334.76134
[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. · Zbl 1218.85008
[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. · Zbl 1303.92105
[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. · Zbl 0802.65122
[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. · Zbl 1222.65063
[54] A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl. 59 (2010) 1326-1336. · Zbl 1189.65151
[55] K. Diethelm, N.J. Ford, Numerical solution of the Bagley-Torvik equation, BIT 42 (2002) 490-507. · Zbl 1035.65067
[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. · Zbl 1180.65178
[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. · Zbl 1065.70002
[58] Y. Xu, Z. He., The short memory principle for solving abel differential equation of fractional order, Comput. Math. Appl. 64 (2011) 4796-4805. · Zbl 1236.34008
[59] M. Razzaghi, S. Yousefi, The Legendre wavelets operational matrix of integration, Int. J. Sys. Sci., 32(4) (2001) 495-502. · Zbl 1006.65151
[60] S.A. Yousefi, M. Behroozifar, Operational matrices of Bernstein polynomials and their applications, Int. J. Sys. Sci., 41:6 (2010) 709-716. · Zbl 1195.65061
[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. · Zbl 1283.65078
[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. · Zbl 1177.65100
[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. · Zbl 1326.76110
[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).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.