On shifted Jacobi spectral approximations for solving fractional differential equations. (English) Zbl 1291.65207

Summary: A new formula of Caputo fractional-order derivatives of shifted Jacobi polynomials of any degree in terms of shifted Jacobi polynomials themselves is proved. We discuss a direct solution technique for linear multi-order fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using a shifted Jacobi tau approximation. A quadrature shifted Jacobi tau (Q-SJT) approximation is introduced for the solution of linear multi-order FDEs with variable coefficients. We also propose a shifted Jacobi collocation technique for solving nonlinear multi-order fractional initial value problems. The advantages of using the proposed techniques are discussed and we compare them with other existing methods. We investigate some illustrative examples of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.


65L05 Numerical methods for initial value problems involving ordinary differential equations
34A08 Fractional ordinary differential equations
34A30 Linear ordinary differential equations and systems
34A34 Nonlinear ordinary differential equations and systems
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations


DFOC; sysdfod
Full Text: DOI


[1] Ansari, A.; Refahi Sheikhani, A.; Saberi Najafi, H., Solution to system of partial fractional differential equations using the fractional exponential operators, Math. Methods Appl. Sci., 35, 119-123 (2012) · Zbl 1234.26013
[2] Bagley, R. L.; Torvik, P. J., On the appearance of the fractional derivative in the behaviour of real materials, J. Appl. Mech., 51, 294-298 (1984) · Zbl 1203.74022
[3] Baleanu, D.; Mustafa, O. G.; Agarwal, R. P., An existence result for a superlinear fractional differential equation, Appl. Math. Lett., 23, 1129-1132 (2010) · Zbl 1200.34004
[4] Baleanu, D.; Mustafa, O. G.; Agarwal, R. P., On the solution set for a class of sequential fractional differential equations, J. Phys. A: Math. Theory, 43, 385209 (2010) · Zbl 1216.34004
[5] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional calculus models and numerical methods, (Series on Complexity, Nonlinearity and Chaos (2012), World Scientific: World Scientific Singapore) · Zbl 1248.26011
[6] Bhrawy, A. H.; Alofi, A. S.; Ezz-Eldien, S. S., A quadrature tau method for variable coefficients fractional differential equations, Appl. Math. Lett., 24, 2146-2152 (2011) · Zbl 1269.65068
[7] Bhrawy, A. H.; Alofi, A. S., A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations, Commun. Nonlinear Sci. Numer. Simul., 17, 62-70 (2012) · Zbl 1244.65099
[8] Bhrawy, A. H.; Alghamdi, M. A., A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals, Boundary Value Prob., 2012, 62 (2012) · Zbl 1280.65079
[9] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods in Fluid Dynamics (1988), Springer: Springer New York · Zbl 0658.76001
[10] Deng, J.; Ma, L., Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, Appl. Math. Lett., 23, 676-680 (2010) · Zbl 1201.34008
[11] Diethelm, K.; Ford, N. J., Multi-order fractional differential equations and their numerical solutions, Appl. Math. Comput., 154, 621-640 (2004) · Zbl 1060.65070
[12] Doha, E. H., On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, J. Phys. A: Math. Gen., 37, 657-675 (2004) · Zbl 1055.33007
[13] Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S., Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Modell., 35, 5662-5672 (2011) · Zbl 1228.65126
[14] Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S., A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl., 62, 2364-2373 (2011) · Zbl 1231.65126
[15] Doha, E. H.; Bhrawy, A. H.; Ezzeldeen, S. S., A new Jacobi operational matrix: an application for solving fractional differential equation, Appl. Math. Modell., 36, 4931-4943 (2012) · Zbl 1252.34019
[16] Fornberg, B., A Practical Guide to Pseudospectral Methods (1998), Cambridge University Press: Cambridge University Press Cambridge, MA · Zbl 0912.65091
[17] Guo, B.-Y., Spectral Methods and Their Applications (1998), World Scientific: World Scientific River Edge, NJ · Zbl 0906.65110
[18] Guo, B.-Y.; Wang, L.-L., Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approximation Theory, 128, 1-41 (2004) · Zbl 1057.41003
[19] Gupta, P. K.; Singh, M., Homotopy perturbation method for fractional Fornberg-Whitham equation, Comput. Math. Appl., 61, 250-254 (2011) · Zbl 1211.65138
[20] Hashim, I.; Abdulaziz, O.; Momani, S., Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul., 14, 674-684 (2009) · Zbl 1221.65277
[21] Heydari, M. H.; Hooshmandasl, M. R.; Maalek Ghaini, F. M.; Mohammadi, F., Wavelet collocation method for solving multi-order fractional differential equations, J. Appl. Math., 2012, 19 (2012) · Zbl 1235.42034
[22] Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific: World Scientific Singapore · Zbl 0998.26002
[23] Jaradat, H. M.; Awawdeh, Fadi; Rawashdeh, E. A., An analytical scheme for multi-order fractional differential equations, Tamsui Oxford J. Math. Sci., 26, 305-320 (2010) · Zbl 1216.65088
[24] Jiang, Y.-L.; Ding, X.-L., Waveform relaxation methods for fractional differential equations with the Caputo derivatives, J. Comput. Appl. Math., 238, 51-67 (2013) · Zbl 1259.65113
[25] Jiang, H.; Liu, F.; Turner, I.; Burrage, K., Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain, J. Math. Anal. Appl., 389, 1117-1127 (2012) · Zbl 1234.35300
[26] Kadem, A.; Baleanu, D., Fractional radiative transfer equation within Chebyshev spectral approach, Comput. Math. Appl., 59, 1865-1873 (2010) · Zbl 1189.35359
[27] Khan, Y.; Diblik, J.; Faraz, N.; Smarda, Z., An efficient new perturbative Laplace method for space-time fractional telegraph equations, Adv. Difference Equ., 2012, 204 (2012) · Zbl 1377.35269
[28] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier: Elsevier San Diego · Zbl 1092.45003
[29] Kumer, P.; Agrawal, O. P., An approximate method for numerical solution of fractional differential equations, Signal Process., 84, 2602-2610 (2006) · Zbl 1172.94436
[30] Li, Y.; Zhao, W., Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput., 216, 2276-2285 (2010) · Zbl 1193.65114
[31] Li, C.; Zeng, F.; Liu, F., Spectral approximations to the fractional integral and derivative, Fract. Calc. Appl. Anal., 15, 383-406 (2012) · Zbl 1276.26016
[32] Liu, F.; Meerschaert, M. M.; McGough, R. J.; Zhuang, P.; Liu, Q., Numerical methods for solving the multi-term time-fractional wave-diffusion equations, Fract. Calc. Appl. Anal., 16, 9-25 (2013) · Zbl 1312.65138
[34] Luke, Y., The Special Functions and Their Approximations, vol. 2 (1969), Academic Press: Academic Press New York · Zbl 0193.01701
[35] Magin, R. L., Fractional Calculus in Bioengineering (2006), Begell House Inc.: Begell House Inc. Redding, CT
[36] Miller, K.; Ross, B., An Introduction to the Fractional Calaulus and Fractional Differential Equations (1993), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York · Zbl 0789.26002
[37] Odibat, Z.; Momani, S., Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fract., 36, 167-174 (2008) · Zbl 1152.34311
[38] Oldham, K.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York-London · Zbl 0428.26004
[39] Ortigueira, M., Introduction to fraction linear systems. Part 2: discrete-time case, IEE Proceedings Vision, Image, Signal Processing, 147, 71-78 (2000)
[40] Ortiz, E. L.; Samara, H., Numerical solutions of differential eigenvalues problems with with an operational approach to the Tau method, Computing, 31, 95-103 (1983) · Zbl 0508.65045
[41] Petras, I., Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (2011), Springer: Springer New York · Zbl 1228.34002
[42] Podlubny, I., Fractional Differential Equations (1999), Academic Press Inc.: Academic Press Inc. San Diego, CA · Zbl 0918.34010
[43] Saadatmandi, A.; Dehghan, M., A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59, 1326-1336 (2010) · Zbl 1189.65151
[44] Shawagfeh, N. T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput., 131, 517-529 (2002) · Zbl 1029.34003
[45] Szegö, G., Orthogonal polynomials, Am. Math. Soc. Colloq. Pub., 23 (1985) · JFM 61.0386.03
[46] Yanga, S.; Xiao, A.; Su, H., Convergence of the variational iteration method for solving multi-order fractional differential equations, Comput. Math. Appl., 60, 2871-2879 (2010) · Zbl 1207.65109
[47] Sabatier, J.; Nguyen, H.; Farges, C.; Deletage, J. Y.; Moreau, X.; Guillemard, F.; Bavoux, B., Fractional models for thermal modeling and temperature estimation of a transistor junction, Adv. Difference Equ., 2011 (2011) · Zbl 1344.93016
[48] Yuzbasi, S., Numerical solution of the Bagley-Torvik equation by the Bessel collocation method, Math. Methods Appl. Sci. (2012)
[49] Zhang, X.; Liu, L.; Wu, Y.; Lu, Y., The iterative solutions of nonlinear fractional differential equations, Appl. Math. Comput., 219, 4680-4691 (2013) · Zbl 06447274
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