×

A new Jacobi operational matrix: an application for solving fractional differential equations. (English) Zbl 1252.34019

Summary: We derive the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.

MSC:

34A45 Theoretical approximation of solutions to ordinary differential equations
34A08 Fractional ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Miller, K.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), John Wiley, Sons Inc New York · Zbl 0789.26002
[2] Oldham, K.B.; Spanier, J., Fractional calculus: theory and applications, differentiation and integration to arbitrary order, (1974), Academic Press, Inc New York, London · Zbl 0292.26011
[3] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[4] Amairi, M.; Aoun, M.; Najar, S.; Abdelkrim, M.N., A constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation, Appl. math. comput., 217, 2162-2168, (2010) · Zbl 1250.34006
[5] 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
[6] Girejko, E.; Mozyrska, D.; Wyrwas, M., A sufficient condition of viability for fractional differential equations with the Caputo derivative, J. math. anal. appl., 382, 1, 146-154, (2011) · Zbl 1222.34007
[7] Ray, S.S.; Bera, R.K., Solution of an extraordinary differential equation by Adomian decomposition method, J. appl. math., 4, 331-338, (2004) · Zbl 1080.65069
[8] Dehghan, M.; Manafian, J.; Saadatmandi, A., Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. methods partial diff. eq., 26, 448-479, (2010) · Zbl 1185.65187
[9] Hashim, I.; Abdulaziz, O.; Momani, S., Homotopy analysis method for fractional ivps, Commun. nonlinear sci. numer. simul., 14, 674-684, (2009) · Zbl 1221.65277
[10] Odibat, Z.; Momani, S.; Xu, H., A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations, Appl. math. model., 34, 593-600, (2010) · Zbl 1185.65139
[11] Yang, 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
[12] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1989), Springer-Verlag New York
[13] Saadatmandi, A.; Dehghan, M., A new operational matrix for solving fractional-order differential equations, Comput. math. appl., 59, 1326-1336, (2010) · Zbl 1189.65151
[14] Doha, E.H.; Bhrawy, A.H.; Ezz-Eldien, S.S., Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. math. model., 35, 5662-5672, (2011) · Zbl 1228.65126
[15] 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
[16] 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
[17] Ghoreishi, F.; Yazdani, S., An extension of the spectral tau method for numerical solution of multi-order fractional differential equations with convergence analysis, Comput. math. appl., 61, 30-43, (2011) · Zbl 1207.65108
[18] Vanani, S.K.; Aminataei, A., Tau approximate solution of fractional partial differential equations, Comput. math. appl., 62, 1075-1083, (2011) · Zbl 1228.65205
[19] Pedas, A.; Tamme, E., On the convergence of spline collocation methods for solving fractional differential equations, J. comput. appl. math., 235, 3502-3514, (2011) · Zbl 1217.65154
[20] Esmaeili, S.; Shamsi, M., A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations, Commun. nonlinear sci. numer. simul., 16, 3646-3654, (2011) · Zbl 1226.65062
[21] Esmaeili, S.; Shamsi, M.; Luchko, Y., Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials, Comput. math. appl., 62, 918-929, (2011) · Zbl 1228.65132
[22] Szegö, G., Orthogonal polynomials, Am. math. soc. colloq. pub., 23, (1985) · JFM 65.0278.03
[23] Doha, E.H.; Bhrawy, A.H.; Hafez, R.M., Jacobi – jacobi dual-petrov – galerkin method for third- and fifth- order differential equations, Math. comput. model., 53, 1820-1832, (2011) · Zbl 1219.65077
[24] 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
[25] Luke, Y., The special functions and their approximations, vol. 2, (1969), Academic Press New York
[26] Diethelm, K.; Ford, N.J., Multi-order fractional differential equations and their numerical solutions, Appl. math. comput., 154, 621-640, (2004) · Zbl 1060.65070
[27] Mdallal, Q.M.; Syam, M.I.; Anwar, M.N., A collocation-shooting method for solving fractional boundary value problems, Commun. nonlinear sci. numer. simul., 15, 3814-3822, (2010) · Zbl 1222.65078
[28] 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
[29] Jafari, H.; Das, S.; Tajadodi, H., Solving a multi-order fractional differential equation using homotopy analysis method, J. King saud university sci., 23, 151-155, (2011)
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.