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Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. (English) Zbl 1193.65114
Summary: Haar wavelet operational matrix has been widely applied in system analysis, system identification, optimal control and numerical solution of integral and differential equations. In the present paper we derive the Haar wavelet operational matrix of the fractional order integration, and use it to solve the fractional order differential equations including the Bagley-Torvik, Riccati and composite fractional oscillation equations. The results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply.

MSC:
65L05Initial value problems for ODE (numerical methods)
34A08Fractional differential equations
65T60Wavelets (numerical methods)
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References:
[1] Odibat, Z.; Momani, S.: Numerical methods for nonlinear partial differential equations of fractional order, Appl. math. Model. 32, 28-39 (2008) · Zbl 1133.65116 · doi:10.1016/j.apm.2006.10.025
[2] Momani, S.; Odibat, Z.: Numerical approach to differential equations of fractional order, J. comput. Appl. math. 207, 96-110 (2007) · Zbl 1119.65127 · doi:10.1016/j.cam.2006.07.015
[3] El-Wakil, S. A.; Elhanbaly, A.; Abdou, M. A.: Adomian decomposition method for solving fractional nonlinear differential equations, Appl. math. Comput. 182, 313-324 (2006) · Zbl 1106.65115 · doi:10.1016/j.amc.2006.02.055
[4] 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 · doi:10.1016/j.physleta.2007.06.016
[5] Das, S.: Analytical solution of a fractional diffusion equation by variational iteration method, Comput. math. Appl. 57, 483-487 (2009) · Zbl 1165.35398 · doi:10.1016/j.camwa.2008.09.045
[6] Arikoglu, A.; Ozkol, I.: Solution of fractional differential equations by using differential transform method, Chaos, solitons fract. 34, 1473-1481 (2007) · Zbl 1152.34306 · doi:10.1016/j.chaos.2006.09.004
[7] Arikoglu, A.; Ozkol, I.: Solution of fractional integro-differential equations by using fractional differential transform method, Chaos, solitons fract. 40, 521-529 (2009) · Zbl 1197.45001 · doi:10.1016/j.chaos.2007.08.001
[8] Darania, P.; Ebadian, A.: A method for the numerical solution of the integro-differential equations, Appl. math. Comput. 188, 657-668 (2007) · Zbl 1121.65127 · doi:10.1016/j.amc.2006.10.046
[9] Erturk, V. S.; Momani, S.: Solving systems of fractional differential equations using differential transform method, J. comput. Appl. math. 215, 142-151 (2008) · Zbl 1141.65088 · doi:10.1016/j.cam.2007.03.029
[10] Erturk, V. S.; Momani, S.; Odibat, Z.: Application of generalized differential transform method to multi-order fractional differential equations, Comm. nonlinear sci. Numer. simulat. 13, 1642-1654 (2008) · Zbl 1221.34022 · doi:10.1016/j.cnsns.2007.02.006
[11] Meerschaert, M.; Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations, Appl. numer. Math. 56, 80-90 (2006) · Zbl 1086.65087 · doi:10.1016/j.apnum.2005.02.008
[12] Odibat, Z.; Shawagfeh, N.: Generalized Taylor’s formula, Appl. math. Comput. 186, 286-293 (2007) · Zbl 1122.26006 · doi:10.1016/j.amc.2006.07.102
[13] I. Podlubny, The Laplace transform method for linear differential equations of the fractional order, 1997, eprint, <arxiv:funct-an/9710005>.
[14] Wu, J. L.: A wavelet operational method for solving fractional partial differential equations numerically, Appl. math. Comput. 214, 31-40 (2009) · Zbl 1169.65127 · doi:10.1016/j.amc.2009.03.066
[15] Lepik, Ü.: Solving fractional integral equations by the Haar wavelet method, Appl. math. Comput. 214, 468-478 (2009) · Zbl 1170.65106 · doi:10.1016/j.amc.2009.04.015
[16] Chen, C.; Hsiao, C.: Haar wavelet method for solving lumped and distributed-parameter systems, IEE P.-contr. Theor. ap. 144, 87-94 (1997) · Zbl 0880.93014 · doi:10.1049/ip-cta:19970702
[17] Bujurke, N.; Salimath, C.; Shiralashetti, S.: Numerical solution of stiff systems from nonlinear dynamics using single-term Haar wavelet series, Nonlinear dynam. 51, 595-605 (2008) · Zbl 1171.65407 · doi:10.1007/s11071-007-9248-8
[18] Karimi, H.; Lohmann, B.; Maralani, P.; Moshiri, B.: A computational method for solving optimal control and parameter estimation of linear systems using Haar wavelets, Int. J. Comput. math. 81, 1121-1132 (2004) · Zbl 1068.65088 · doi:10.1080/03057920412331272225
[19] Pawlak, M.; Hasiewicz, Z.: Nonlinear system identification by the Haar multiresolution analysis, IEEE T. Circuits-I 45, 945-961 (1998) · Zbl 0952.93021 · doi:10.1109/81.721260
[20] Hsiao, C.; Wang, W.: Optimal control of linear time-varying systems via Haar wavelets, J. optimiz. Theory app. 103 (1999) · Zbl 0941.49018 · doi:10.1023/A:1021740209084
[21] Karimi, H.; Moshiri, B.; Lohmann, B.; Maralani, P.: Haar wavelet-based approach for optimal control of second-order linear systems in time domain, J. dyn. Control syst. 11, 237-252 (2005) · Zbl 1063.49002 · doi:10.1007/s10883-005-4172-z
[22] Babolian, E.; Shahsavaran, A.: Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets, J. comput. Appl. math. 225, 87-95 (2009) · Zbl 1159.65102 · doi:10.1016/j.cam.2008.07.003
[23] Bujurke, N. M.; Shiralashetti, S. C.; Salimath, C. S.: An application of single-term Haar wavelet series in the solution of nonlinear oscillator equations, J. comput. Appl. math. 227, 234-244 (2009) · Zbl 1162.65040 · doi:10.1016/j.cam.2008.03.012
[24] Guf, J.; Jiang, W.: The Haar wavelets operational matrix of integration, Int. J. Syst. sci. 27, 623-628 (1996) · Zbl 0875.93116 · doi:10.1080/00207729608929258
[25] Maleknejad, K.; Mirzaee, F.: Using rationalized Haar wavelet for solving linear integral equations, Appl. math. Comput. 160, 579-587 (2005) · Zbl 1067.65150 · doi:10.1016/j.amc.2003.11.036
[26] Maleknejad, K.; Mirzaee, F.; Abbasbandy, S.: Solving linear integro-differential equations system by using rationalized Haar functions method, Appl. math. Comput. 155, 317-328 (2004) · Zbl 1056.65144 · doi:10.1016/S0096-3003(03)00778-1
[27] Ordokhani, Y.: Solution of nonlinear Volterra -- Fredholm -- Hammerstein integral equations via rationalized Haar functions, Appl. math. Comput. 180, 436-443 (2006) · Zbl 1102.65141 · doi:10.1016/j.amc.2005.12.034
[28] Razzaghi, M.; Ordokhani, Y.: An application of rationalized Haar functions for variational problems, Appl. math. Comput. 122, 353-364 (2001) · Zbl 1020.49026 · doi:10.1016/S0096-3003(00)00050-3
[29] Razzaghi, M.; Yousefi, S.: The Legendre wavelets operational matrix of integration, Int. J. Syst. sci. 32, 495-502 (2001) · Zbl 1006.65151 · doi:10.1080/002077201300080910
[30] Reihani, M. H.; Abadi, Z.: Rationalized Haar functions method for solving Fredholm and Volterra integral equations, J. comput. Appl. math. 200, 12-20 (2007) · Zbl 1107.65122 · doi:10.1016/j.cam.2005.12.026
[31] Oldham, K.; Spanier, J.: The fractional calculus, (1974) · Zbl 0292.26011
[32] Kilicman, A.; Zhour, Z. A. A. Al: Kronecker operational matrices for fractional calculus and some applications, Appl. math. Comput. 187, 250-265 (2007) · Zbl 1123.65063 · doi:10.1016/j.amc.2006.08.122
[33] Diethelm, K.; Ford, J.: Numerical solution of the bagley -- torvik equation, Bit. numer. Math. 42, 490-507 (2002) · Zbl 1035.65067
[34] El-Mesiry, A.; El-Sayed, A.; El-Saka, H.: Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. math. Comput. 160, 683-699 (2005) · Zbl 1062.65073 · doi:10.1016/j.amc.2003.11.026