zbMATH — the first resource for mathematics

The construction of operational matrix of fractional derivatives using B-spline functions. (English) Zbl 1276.65015
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. Here we construct the operational matrix of a fractional derivative of order \(\alpha \) in the Caputo sense using the linear B-spline functions. 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 we can solve directly 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 new technique presented in the current paper.

65D25 Numerical differentiation
65D07 Numerical computation using splines
35R11 Fractional partial differential equations
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
26A33 Fractional derivatives and integrals
34K37 Functional-differential equations with fractional derivatives
34A08 Fractional ordinary differential equations and fractional differential inclusions
Full Text: DOI
[1] de Boor, C., A practical guide to spline, (1978), Springer-Verlag New York · Zbl 0406.41003
[2] Caputo, M., Linear models of dissipation whose Q is almost frequency independent. part II, J R austral soc, 13, 529-539, (1967)
[3] Chui, C.K., An introduction to wavelets, (1992), Academic Press San Diego, Calif · Zbl 0925.42016
[4] Das, S., Functional fractional calculus for system identification and controls, (2008), Springer New York · Zbl 1154.26007
[5] Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math comput simul, 71, 16-30, (2006) · Zbl 1089.65085
[6] Dehghan, M.; Manafian, J.; Saadatmandi, A., Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer methods partial differ eqn, 26, 448-479, (2010) · Zbl 1185.65187
[7] Dehghan, M.; Manafian, J.; Saadatmandi, A., The solution of the linear fractional partial differential equations using the homotopy analysis method, Z naturforsch, 65a, 935-949, (2010)
[8] Dehghan, M.; Salehi, R., A semi-numeric approach for solution of the eikonal partial differential equation and its applications, Numer methods partial differ eqn, 26, 702-722, (2010) · Zbl 1189.65237
[9] Dehghan, M.; Shakourifar, M.; Hamidi, A., The solution of linear and nonlinear systems of Volterra functional equations using Adomian-pade technique, Chaos soliton fract, 39, 2509-2521, (2009) · Zbl 1197.65223
[10] Dehghan, M.; Yousefi, S.A.; Lotfi, A., The use of he’s variational iteration method for solving the telegraph and fractional telegraph equations, Int J numer methods biomed eng, 27, 219-231, (2011) · Zbl 1210.65173
[11] 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
[12] Ervin, V.J.; Roop, J.P., Variational formulation for the stationary fractional advection dispersion equation, Numer methods partial differ eqn, 22, 558-576, (2005) · Zbl 1095.65118
[13] 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
[14] Gejji, V.D.; Jafari, H., Solving a multi-order fractional differential equation, Appl math comput, 189, 541-548, (2007) · Zbl 1122.65411
[15] Goswami, J.C.; Chan, A.K., Fundamentals of wavelets: theory, algorithms, and applications, (1999), John Wiley & Sons, Inc. · Zbl 1209.65156
[16] Hashim, I.; Abdulaziz, O.; Momani, S., Homotopy analysis method for fractional ivps, Commun nonlinear sci numer simul, 14, 674-684, (2009) · Zbl 1221.65277
[17] 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
[18] Jafari, H.; Daftardar-Gejji, V., Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl math comput, 180, 700-706, (2006) · Zbl 1102.65136
[19] Jafari, H.; Momani, S., Solving fractional diffusion and wave equations by modified homotopy perturbation method, Phys lett A, 370, 388-396, (2007) · Zbl 1209.65111
[20] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier San Diego · Zbl 1092.45003
[21] Kumar, P.; Agrawal, O.P., An approximate method for numerical solution of fractional differential equations, Signal process, 86, 2602-2610, (2006) · Zbl 1172.94436
[22] Lakestani, M.; Razzaghi, M.; Dehghan, M., Solution of nonlinear fredholm – hammerstein integral equations by using semiorthogonal spline wavelets, Math prob eng, 1, 113-121, (2005) · Zbl 1073.65568
[23] Lakestani, M.; Razzaghi, M.; Dehghan, M., Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations, Math prob eng, 1-12, (2006) · Zbl 1200.65112
[24] 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
[25] Lotfi A, Dehghan M, Yousefi SA. A numerical technique for solving fractional optimal control problems. Comput Math Appl in press. doi:10.1016/j.camwa.2011.03.044. · Zbl 1228.65109
[26] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[27] Momani, S.; Noor, M.A., Numerical methods for fourth-order fractional integro-differential equations, Appl math comput, 182, 754-760, (2006) · Zbl 1107.65120
[28] 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
[29] 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
[30] Momani, S.; Shawagfeh, N.T., Decomposition method for solving fractional Riccati differential equations, Appl math comput, 182, 1083-1092, (2006) · Zbl 1107.65121
[31] 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
[32] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004
[33] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[34] Rawashdeh, E.A., Numerical solution of fractional integro-differential equations by collocation method, Appl math comput, 176, 1-6, (2006) · Zbl 1100.65126
[35] 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
[36] Saadatmandi, A.; Dehghan, M., A new operational matrix for solving fractional-order differential equations, Comput math appl, 59, 1326-1336, (2010) · Zbl 1189.65151
[37] Saadatmandi, A.; Dehghan, M., A tau approach for solution of the space fractional diffusion, Comput math appl, 62, 1135-1142, (2011) · Zbl 1228.65203
[38] Saadatmandi A, Dehghan M, Legendre A. collocation method for fractional integro – differential equations. J Vib Control. in press. doi:10.1177/1077546310395977.
[39] Shakeri, F.; Dehghan, M., Solution of delay differential equations via a homotopy perturbation method, Math comput model, 48, 486-498, (2008) · Zbl 1145.34353
[40] 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
[41] Tatari, M.; Dehghan, M., On the convergence of he’s variational iteration method, J comput appl math, 207, 121-128, (2007) · Zbl 1120.65112
[42] Tenreiro Machado, J.; Kiryakova, V.; Mainardi, F., Recent history of fractional calculus, Commun nonlinear sci numer simul, 16, 1140-1153, (2011) · Zbl 1221.26002
[43] Wang, Q., Numerical solutions for fractional kdv – burgers equation by Adomian decomposition method, Appl math comput, 182, 1048-1055, (2006) · Zbl 1107.65124
[44] Wu, J.L., A wavelet operational method for solving fractional partial differential equations numerically, Appl math comput, 214, 31-40, (2009) · Zbl 1169.65127
[45] Yousefi SA, Lotfi A, Dehghan M. The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems. J Vib Control in press. doi:10.1177/1077546311399950.
[46] Yuste, S.B., Weighted average finite difference methods for fractional diffusion equations, J comput phys, 216, 264-274, (2006) · Zbl 1094.65085
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.