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A new operational matrix for solving fractional-order differential equations. (English) Zbl 1189.65151
Summary: Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of a class of fractional differential equations. The fractional derivatives are described in the Caputo sense. Our main aim is to generalize the Legendre operational matrix to the fractional calculus. In this approach, a truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem. The method is applied to solve two types of fractional differential equations, linear and nonlinear. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.

##### MSC:
 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 26A33 Fractional derivatives and integrals 34A08 Fractional ordinary differential equations and fractional differential inclusions 45J05 Integro-ordinary differential equations
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##### References:
 [1] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004 [2] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002 [3] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010 [4] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier San Diego · Zbl 1092.45003 [5] Das, S., Functional fractional calculus for system identification and controls, (2008), Springer New York · Zbl 1154.26007 [6] J.H. He, Nonlinear oscillation with fractional derivative and its applications, in: International Conference on Vibrating Engineering’98, Dalian, China, 1998, pp. 288-291 [7] He, J.H., Some applications of nonlinear fractional differential equations and their approximations, Bull. sci. technol., 15, 2, 86-90, (1999) [8] Bagley, R.L.; Torvik, P.J., A theoretical basis for the application of fractional calculus to viscoelasticity, J. rheology, 27, 3, 201-210, (1983) · Zbl 0515.76012 [9] Bagley, R.L.; Torvik, P.J., Fractional calculus in the transient analysis of viscoelastically damped structures, Aiaa j., 23, 6, 918-925, (1985) · Zbl 0562.73071 [10] Mainardi, F., Fractional calculus: ‘some basic problems in continuum and statistical mechanics’, (), 291-348 · Zbl 0917.73004 [11] Mandelbrot, B., Some noises with 1/$$f$$ spectrum, a bridge between direct current and white noise, IEEE trans. inform. theory, 13, 2, 289-298, (1967) · Zbl 0148.40507 [12] Rossikhin, Y.A.; Shitikova, M.V., Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. mech. rev., 50, 15-67, (1997) · Zbl 0901.73030 [13] Baillie, R.T., Long memory processes and fractional integration in econometrics, J. econometrics, 73, 5-59, (1996) · Zbl 0854.62099 [14] Magin, R.L., Fractional calculus in bioengineering, Crit. rev. biomed. eng., 32, 1, 1-104, (2004) [15] Magin, R.L., Fractional calculus in bioengineering-part 2, Crit. rev. biomed. eng., 32, 2, 105-193, (2004) [16] Magin, R.L., Fractional calculus in bioengineering-part 3, Crit. rev. biomed. eng., 32, 3/4, 194-377, (2004) [17] Metzler, R.; Klafter, J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. phys. A, 37, 161-208, (2004) · Zbl 1075.82018 [18] Chow, T.S., Fractional dynamics of interfaces between soft-nanoparticles and rough substrates, Phys. lett. A, 342, 148-155, (2005) [19] Momani, S.; Shawagfeh, N.T., Decomposition method for solving fractional Riccati differential equations, Appl. math. comput., 182, 1083-1092, (2006) · Zbl 1107.65121 [20] Momani, S.; Noor, M.A., Numerical methods for fourth-order fractional integro-differential equations, Appl. math. comput., 182, 754-760, (2006) · Zbl 1107.65120 [21] Gejji, V.D.; Jafari, H., Solving a multi-order fractional differential equation, Appl. math. comput., 189, 541-548, (2007) · Zbl 1122.65411 [22] Ray, S.S.; Chaudhuri, K.S.; Bera, R.K., Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method, Appl. math. comput., 182, 544-552, (2006) · Zbl 1108.65129 [23] Wang, Q., Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method, Appl. math. comput., 182, 1048-1055, (2006) · Zbl 1107.65124 [24] Inc, M., The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method, J. math. anal. appl., 345, 476-484, (2008) · Zbl 1146.35304 [25] Momani, S.; Odibat, Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. lett. A, 355, 271-279, (2006) · Zbl 1378.76084 [26] Odibat, Z.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. nonlinear sci. numer. simul., 7, 271-279, (2006) · Zbl 1378.76084 [27] Momani, S.; Odibat, Z., Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. lett. A, 365, 345-350, (2007) · Zbl 1203.65212 [28] Sweilam, N.H.; Khader, M.M.; Al-Bar, R.F., Numerical studies for a multi-order fractional differential equation, Phys. lett. A, 371, 26-33, (2007) · Zbl 1209.65116 [29] Hashim, I.; Abdulaziz, O.; Momani, S., Homotopy analysis method for fractional ivps, Commun. nonlinear sci. numer. simul., 14, 674-684, (2009) · Zbl 1221.65277 [30] Rawashdeh, E.A., Numerical solution of fractional integro-differential equations by collocation method, Appl. math. comput., 176, 1-6, (2006) · Zbl 1100.65126 [31] Ervin, V.J.; Roop, J.P., Variational formulation for the stationary fractional advection dispersion equation, Numer. methods partial differential equations, 22, 558-576, (2005) · Zbl 1095.65118 [32] Kumar, P.; Agrawal, O.P., An approximate method for numerical solution of fractional differential equations, Signal processing, 86, 2602-2610, (2006) · Zbl 1172.94436 [33] Liua, F.; Anh, V.; Turner, I., Numerical solution of the space fractional Fokker-Planck equation, J. comput. appl. math., 166, 209-219, (2004) · Zbl 1036.82019 [34] Yuste, S.B., Weighted average finite difference methods for fractional diffusion equations, J. comput. phys., 216, 264-274, (2006) · Zbl 1094.65085 [35] Saadatmandi, A.; Razzaghi, M.; Dehghan, M., Hartley series approximations for the parabolic equations, Intern. J. comput. math., 82, 1149-1156, (2005) · Zbl 1075.65128 [36] Saadatmandi, A.; Dehghan, M., A tau method for the one-dimensional parabolic inverse problem subject to temperature overspecification, Comput. math. appl., 52, 933-940, (2006) · Zbl 1125.65340 [37] Saadatmandi, A.; Dehghan, M., Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the tau method, Commun. numer. methods. eng., 24, 1467-1474, (2008) · Zbl 1151.92017 [38] Saadatmandi, A.; Dehghan, M., Numerical solution of the one-dimensional wave equation with an integral condition, Numer. methods partial differential equations, 23, 282-292, (2007) · Zbl 1112.65097 [39] Caputo, M., Linear models of dissipation whose Q is almost frequency independent. part II, J. roy austral. soc., 13, 529-539, (1967) [40] Diethelm, K.; Ford, N.J.; Freed, A.D.; Luchko, Yu., Algorithms for the fractional calculus: A selection of numerical methods, Comput. methods appl. mech. eng., 194, 743-773, (2005) · Zbl 1119.65352 [41] Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. calculus appl. anal., 5, 367-386, (2002) · Zbl 1042.26003 [42] Diethelm, K.; Ford, N.J., Multi-order fractional differential equations and their numerical solution, Appl. math. comput., 154, 621-640, (2004) · Zbl 1060.65070 [43] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamic, (1988), Prentice-Hall Englewood Cliffs, NJ · Zbl 0658.76001 [44] Diethelm, K.; Ford, N.J., Numerical solution of the bagley – torvik equation, Bit, 42, 490-507, (2002) · Zbl 1035.65067 [45] 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) · Zbl 1009.65049
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