Abstract
Recently, operational matrices were adapted for solving several kinds of fractional differential equations (FDEs). The use of numerical techniques in conjunction with operational matrices of some orthogonal polynomials, for the solution of FDEs on finite and infinite intervals, produced highly accurate solutions for such equations. This article discusses spectral techniques based on operational matrices of fractional derivatives and integrals for solving several kinds of linear and nonlinear FDEs. More precisely, we present the operational matrices of fractional derivatives and integrals, for several polynomials on bounded domains, such as the Legendre, Chebyshev, Jacobi and Bernstein polynomials, and we use them with different spectral techniques for solving the aforementioned equations on bounded domains. The operational matrices of fractional derivatives and integrals are also presented for orthogonal Laguerre and modified generalized Laguerre polynomials, and their use with numerical techniques for solving FDEs on a semi-infinite interval is discussed. Several examples are presented to illustrate the numerical and theoretical properties of various spectral techniques for solving FDEs on finite and semi-infinite intervals.
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Machado, J.A.T.: The effect of fractional order in variable structure control. Comput. Math. Appl. 64, 3340–3350 (2012)
Gutierrez, R.E., Rosario, J.M., Machado, J.A.T.: Fractional order calculus: basic concepts and engineering applications. Math. Prob. Eng. (2010). Article ID 375858, 19
Pinto, C.M.A., Tenreiro Machado, J.A.: Complex order van der Pol oscillator. Nonlinear Dyn. 65, 247–254 (2011)
Bota, C., Caruntu, B.: Approximate analytical solutions of the fractional-order brusselator system using the polynomial least squares method. Adv. Math. Phys. (2015). Article ID 450235
Liu, Y., Fang, Z., Li, H., He, S.: A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl. Math. Comput. 243, 703–717 (2014)
Merdan, M.: On the solutions of time-fractional generalized Hirota–Satsuma coupled-KDV equation with modified Riemann–Liouville derivative by an analytical technique. Proc. Rom. Acad. A 16, 3–10 (2015)
Inc, M., Kilic, B.: Classification of traveling wave solutions for time-fractional fifth-order KdV-like equation. Waves Random Complex Media 24, 393–403 (2014)
Katsikadelis, J.T.: Numerical solution of distributed order fractional differential equations. J. Comput. Phys. 259, 11–22 (2014)
Daftardar-Gejji, V., Sukale, Y., Bhalekar, S.: A new predictor–corrector method for fractional differential equations. Appl. Math. Comput. 244, 158–182 (2014)
Biswas, A., Bhrawy, A.H., Abdelkawy, M.A., Alshaery, A.A., Hilal, E.M.: Symbolic computation of some nonlinear fractional differential equations. Rom. J. Phys. 59(5–6), 433–442 (2014)
Parvizi, M., Eslahchi, M.R., Dehghan, M.: Numerical solution of fractional advection–diffusion equation with a nonlinear source term. Numer. Algorithms 68, 601–629 (2015)
Nagy, A.M., Sweilam, N.H.: An efficient method for solving fractional Hodgkin–Huxley model. Phys. Lett. A 378(30), 1980–1984 (2014)
Sadatia, S.J., Ghaderi, R., Ranjbar, N.: Some fractional comparison results and stability theorem for fractional time delay systems. Rom. Rep. Phys. 65, 94–102 (2013)
Leo, R.A., Sicuro, G., Tempest, P.: A theorem on the existence of symmetries of fractional PDEs. C. R. Acad. Sci. Paris Ser. I 352, 219–222 (2014)
Gaur, M., Singh, K.: On group invariant solutions of fractional order Burgers–Poisson equation. Appl. Math. Comput. 244, 870–877 (2014)
Wang, G.W., Xu, T.Z.: The improved fractional sub-equation method and its applications to nonlinear fractional partial differential equations. Rom. Rep. Phys. 66(3), 595–602 (2014)
Liu, Z., Lu, P.: Stability analysis for HIV infection of CD4+ T-cells by a fractional differential time-delay model with cure rate. Adv. Differ. Equ. 2014, 298 (2014)
Huang, Q., Zhdanov, R.: Symmetries and exact solutions of the time fractional Harry–Dym equation with Riemann–Liouville derivative. Physica A 409, 110–118 (2014)
Takači, D., Takači, A., Takači, A.: On the operational solutions of fuzzy fractional differential equations. Frac. Calc. Appl. Anal. 17, 1100–1113 (2014)
Ghomashi, A., Salahshour, S., Hakimzadeh, A.: Approximating solutions of fully fuzzy linear systems: a financial case study. J. Intell. Fuzzy Syst. 26, 367–378 (2014)
Al-Khaled, K.: Numerical solution of time-fractional partial differential equations using sumudu decomposition method. Rom. J. Phys. 60, 99–110 (2015)
Bhrawy, A.H., Zaky, M.A.: A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comput. Phys. 281, 876–895 (2015)
Gao, F., Lee, X., Fei, F., Tong, H., Deng, Y., Zhao, H.: Identification time-delayed fractional order chaos with functional extrema model via differential evolution. Expert Syst. Appl. 41, 1601–1608 (2014)
Diethelm, K., Ford, N.J., Freed, A.D.: A predictor–corrector approach for the numerical solution of fractional differential equation. Nonlinear Dyn. 29, 3–22 (2002)
Guner, O., Cevikel, A.C.: A procedure to construct exact solutions of nonlinear fractional differential equations. Sci. World J. (2014). Article ID 489495, 10 pp
Xie, W., Xiao, J., Luo, Z.: Existence of extremal solutions for nonlinear fractional differential equation with nonlinear boundary conditions. Appl. Math. Lett. 41, 46–51 (2015)
Benchohraa, M., Lazreg, J.E.: Existence and uniqueness results for nonlinear implicit fractional differential equations with boundary conditions. J. Math. Comput. Sci. 4, 60–72 (2014)
Dehghan, M., Abbaszadeh, M., Mohebbi, A.: Error estimate for the numerical solution of fractional reaction–subdiffusion process based on a meshless method. J. Comput. Appl. Math. 280, 14–36 (2015)
Ye, H., Liu, F., Anh, V., Turner, I.: Maximum principle and numerical method for the multi-term time–space Riesz–Caputo fractional differential equations. Appl. Math. Comput. 277, 531–540 (2014)
Butera, S., Paola, M.D.: Mellin transform approach for the solution of coupled systems of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 20, 32–38 (2015)
Zhang, X.: Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions. Appl. Math. Lett. 39, 22–27 (2015)
Pedas, A., Tamme, E.: Numerical solution of nonlinear fractional differential equations by spline collocation methods. J. Comput. Appl. Math. 255, 216–230 (2014)
Deng, J., Deng, Z.: Existence of solutions of initial value problems for nonlinear fractional differential equations. Appl. Math. Lett. (2014). doi:10.1016/j.aml.2014.02.001
Dehghan, M., Abbaszadeh, M., Mohebbi, A.: An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations. Eng. Anal. Bound. Elem. 50, 412–434 (2015)
Gao, F., Lee, X., Tong, H., Fei, F., Zhao, H.: Identification of unknown parameters and orders via cuckoo search oriented statistically by differential evolution for noncommensurate fractional-order chaotic systems. Abstr. Appl. Anal. (2013). Article ID 382834
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, New York (2006)
Abdelkawy, M.A., Ahmed, E.A., Sanchez, P.: A method based on Legendre pseudo-spectral approximations for solving inverse problems of parabolic types equations. Math. Sci. Lett. 4, 81–90 (2015)
Bhrawy, A.H., Doha, E.H., Baleanu, D., Ezz-Eldien, S.S., Abdelkawy, M.A.: An accurate numerical technique for solving fractional optimal control problems. Proc. Rom. Acad. A 16, 47–54 (2015)
Xu, Q., Hesthaven, J.S.: Stable multi-domain spectral penalty methods for fractional partial differential equations. J. Comput. Phys. 257, 241–258 (2014)
Chen, F., Xu, Q., Hesthaven, J.S.: A multi-domain spectral method for time-fractional differential equations. J. Comput. Phys. (2015). doi:10.1016/j.jcp.2014.10.016
Doha, E.H., Bhrawy, A.H., Hafez, R.M., Abdelkawy, M.A.: A Chebyshev-Gauss-Radau scheme for nonlinear hyperbolic system of first order. Appl. Math. Inf. Sci. 8(2), 535–544 (2014)
Zayernouri, M., Em Karniadakis, G.: Exponentially accurate spectral and spectral element methods for fractional ODEs. J. Comput. Phys. 257, 460–480 (2014)
Parand, K., Nikarya, M.: Application of Bessel functions for solving differential and integro-differential equations of the fractional order. Appl. Math. Model. 38, 4137–4147 (2014)
Doha, E.H., Abd-Elhameed, W.M., Bassuony, M.A.: On using third and fourth kinds Chebyshev operational matrices for solving Lane–Emden type equations. Rom. J. Phys. 60, 3–4 (2015)
Keshavarz, E., Ordokhani, Y., Razzaghi, M.: Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl. Math. Model. 38, 6038–6051 (2014)
Yang, L., Shen, C., Xie, D.: Multiple positive solutions for nonlinear boundary value problem of fractional order differential equation with the Riemann–Liouville derivative. Adv. Diff. Equ 2014, 284 (2014)
Doha, E.H., Bhrawy, A.H., Baleanu, D., Abdelkawy, M.A.: Numerical treatment of coupled nonlinear hyperbolic Klein–Gordon equations. Rom. J. Phys. 59, 247–264 (2014)
Ilati, M., Dehghan, M.: The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations. Eng. Anal. Bound. Elem. 52, 99–109 (2015)
Bhrawy, A.H., Doha, E.H., Ezz-Eldien, S.S., Abdelkawy, M.A.: A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations. Calcolo (2015). doi:10.1007/s10092-014-0132-x
Zhang, H., Yang, X., Hanc, X.: Discrete-time orthogonal spline collocation method with application to two-dimensional fractional cable equation. Comput. Math. Appl. 68, 1710–1722 (2014)
Yang, Y., Chen, Y., Huang, Y.: Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations. Acta Math. Sci. 34, 673–690 (2014)
Ma, X., Huang, C.: Spectral collocation method for linear fractional integro-differential equations. Appl. Math. Model. 38, 1434–1448 (2014)
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)
Saadatmandi, A., Dehghan, M.: A method based on the tau approach for the identification of a time-dependent coefficient in the heat equation subject to an extra measurement. J. Vib. Control 18, 1125–1132 (2012)
Bhrawy, A.H., Zaky, M.A., Machado, J.A.T.: Efficient Legendre spectral tau algorithm for solving two-sided space-time Caputo fractional advection-dispersion equation. J. Vib. Control (2015). doi:10.1177/1077546314566835
Lau, S.R., Price, H.: Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains. J. Comput. Phys. 231, 7695–7714 (2012)
Zayernouri, M., Ainsworth, M.: Em Karniadakis, G.: A unified Petrov-Galerkin spectral method for fractional PDEs. Comput. Methods Appl. Mech. Eng. (2014). doi:10.1016/j.cma.2014.10.051
Dehghan, M., Salehi, R.: A meshless local Petrov–Galerkin method for the time-dependent Maxwell equations. J. Comput. Appl. Math. 268, 93–110 (2014)
Yang, Y.: Jacobi spectral Galerkin methods for fractional integro-differential equations. Calcolo (2014). doi:10.1007/s10092-014-0128-6
Bhrawy, A.H.: An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system. Appl. Math. Comput. 247, 30–46 (2014)
Doha, E.H., Bhrawy, A.H., Abdelkawy, M.A., Hafez, R.M.: A Jacobi collocation approximation for nonlinear coupled viscous Burgers’ equation. Cent. Eur. J. Phys. 12, 111–122 (2014)
Doha, E.H., Bhrawy, A.H., Saker, M.A.: On the derivatives of Bernstein polynomials: an application for the solution of high even-order differential equations. Bound. Value Probl. (2011). doi:10.1155/2011/829543
Bhrawy, A.H., Zaky, M.A.: Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn. 80, 101–116 (2015)
Ozarslan, M.A.: On a singular integral equation including a set of multivariate polynomials suggested by Laguerre polynomials. Appl. Math. Comput. 229, 350–358 (2014)
Drivera, K., Muldoon, M.E.: Common and interlacing zeros of families of Laguerre polynomials. J. Approx. Theory (2014). doi:10.1016/j.jat.2013.11.013
Baleanu, D., Bhrawy, A.H., Taha, T.M.: Two efficient generalized Laguerre spectral algorithms for fractional initial value problems. Abstr. Appl. Anal. (2013). doi:10.1155/2013/546502
Bhrawy, A.H., Alghamdi, M.M., Taha, T.M.: A new modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line. Adv. Differ. Equ. 2012, 0:179 (2012)
Saadatmandi, A., Dehghan, M.: A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 59, 1326–1336 (2010)
Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: A new Jacobi operational matrix: an application for solving fractional differential equations. Appl. Math. Model. 36, 4931–4943 (2012)
Kazem, S., Abbasbandy, S., Kumar, S.: Fractional-order Legendre functions for solving fractional-order differential equations. Appl. Math. Model. 37, 5498–5510 (2013)
Yin, F., Song, J., Wu, Y., Zhang, L.: Numerical solution of the fractional partial differential equations by the two-dimensional fractional-order Legendre functions. Abstr. Appl. Anal. (2013). Article ID 562140, 13 pages
Ahmadian, A., Suleiman, M., Salahshour, S.: An operational matrix based on Legendre polynomials for solving fuzzy fractional-order differential equations. Abstr. Appl. Anal. (2013). Article ID 505903 29
Ishteva, M., Boyadjiev, L.: On the C-Laguerre functions. C.R. Acad. Bulg. Sci. 58(9), 1019–1024 (2005)
Ishteva, M., Boyadjiev, L., Scherer, R.: On the Caputo operator of fractional calculus and C-Laguerre functions. Math. Sci. Res. 9(6), 161–170 (2005)
Yin, F., Song, J., Leng, H., Lu, F.: Couple of the variational iteration method and fractional-order Legendre functions method for fractional differential equations. Sci. World J. (2014). Article ID 928765, 9 pp
Dehghan, M., Yousefi, S.A., Lotfi, A.: The use of Hes variational iteration method for solving the telegraph and fractional telegraph equations. Int. J. Numer. Methods Biomed. Eng. 27, 219–231 (2011)
Ford, N.J., Connolly, J.A.: Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations. Comput. Appl. Math. 229, 382–391 (2009)
Lakestani, M., Dehghan, M., Irandoust-pakchin, S.: The construction of operational matrix of fractional derivatives using B-spline functions. Commun. Nonlinear Sci. Numer. Simul. 17, 1149–1162 (2012)
Bhrawy, A.H., Tharwat, M.M., Alghamdi, M.A.: A new operational matrix of fractional integration for shifted Jacobi polynomials. Bull. Malays. Math. Sci. Soc. (2) 37(4), 983–995 (2014)
Bhrawy, A.H., Abdelkawy, M.A.: A fully spectral collocation approximation for multi-dimensional fractional Schrodinger equations. J. Comput. Phys. 294, 462–483 (2015)
Bhrawy, A.H., Alofi, A.S.: The operational matrix of fractional integration for shifted Chebyshev polynomials. Apl. Math. Lett. 26, 25–31 (2013)
Bhrawy, A.H., Zaky, M.A., Baleanu, D.: New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method. Rom. Rep. Phys. 67(2) (2015)
Mokhtary, P.: Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations. J. Comput. Appl. Math. 279, 145–158 (2015)
Abdelkawy, M.A., Taha, T.M.: An operational matrix of fractional derivatives of Laguerre polynomials. Walailak J. Sci. Technol. 11, 1041–1055 (2014)
Heydari, M.H., Hooshmandasl, M.R., Mohammadi, F.: Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions. Appl. Math. Comput. 234, 267–276 (2014)
Prakash, P., Harikrishnan, S., Benchohra, M.: Oscillation of certain nonlinear fractional partial differential equation with damping term. Appl. Math. Lett. 43, 72–79 (2015)
El-Wakil, S.A., Abulwafa, E.M.: Formulation and solution of space–time fractional Boussinesq equation. Nonlinear Dyn. 80, 167–175 (2015)
Stokes, P.W., Philippa, B., Read, W., White, R.D.: Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart. J. Comput. Phys. 282, 334–344 (2015)
Chen, Y., Ke, X., Wei, Y.: Numerical algorithm to solve system of nonlinear fractional differential equations based on wavelets method and the error analysis. Appl. Math. Comput. 251, 475–488 (2015)
Szegö, G.: Orthogonal Polynomials. In: American Mathematical Society Colloquium Publications, vol. 23, 4th edn. American Mathematical Society, Providence, RI (1975)
Funaro, D.: Polynomial Approximations of Differential Equations. Springer, Berlin (1992)
Doha, E.H.: On the coefficients of differentiated expansions and derivatives of Jacobi polynomials. J. Phys. A Math. Gen. 35, 3467–3478 (2002)
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)
Luke, Y.: The Special Functions and Their Approximations, vol. 2. Academic Press, New York (1969)
Saadatmandi, A.: Bernstein operational matrix of fractional derivatives and its applications. Appl. Math. Model. 38, 1365–1372 (2014)
Diethelm, K., Ford, N.J.: Multi-order fractional differential equations and their numerical solutions. Appl. Math. Comput. 154, 621–640 (2004)
Bhrawy, A.H., Alhamed, Y.A., Baleanu, D., Al-Zahrani, A.A.: New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions. Frac. Calc. Appl. Anal. 17, 1137–1157 (2014)
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Bhrawy, A.H., Taha, T.M. & Machado, J.A.T. A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dyn 81, 1023–1052 (2015). https://doi.org/10.1007/s11071-015-2087-0
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DOI: https://doi.org/10.1007/s11071-015-2087-0