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The Legendre wavelet method for solving fractional differential equations. (English) Zbl 1222.65063
Summary: Fractional differential equations are solved using the Legendre wavelets. An operational matrix of fractional order integration is derived and is utilized to reduce the fractional differential equations to system of algebraic equations. The illustrative examples are provided to demonstrate the applicability, simplicity of the numerical scheme based on the Legendre wavelets.

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34B15 Nonlinear boundary value problems for ordinary differential equations
34A08 Fractional ordinary differential equations
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[2] Kulish, V.V.; Lage, José L., Application of fractional calculus to fluid mechanics, J fluids eng, 124, (2002)
[3] Oldham, K.B., Fractional differential equations in electrochemistry, Adv eng soft, 41, 9-12, (2010) · Zbl 1177.78041
[4] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[5] Diethelm, K.; Ford, N.J.; Freed, A.D., A predictor corrector approach for the numerical solution of fractional differential equation, Nonlin dyn, 29, 3-22, (2002) · Zbl 1009.65049
[6] Hashim, I.; Abdulaziz, O.; Momani, S., Homotopy analysis method for fractional ivps, Commun nonlin sci num simul, 14, 674-684, (2009) · Zbl 1221.65277
[7] Lepik, Ü., Solving fractional integral equations by the Haar wavelet method, Appl math comp, 214, 468-478, (2009) · Zbl 1170.65106
[8] Chen, C.; Hsiao, C., Haar wavelet method for solving lumped and distributed-parameter systems, IEE P-contr theor ap, 144, 8794, (1997) · Zbl 0880.93014
[9] Kilicman, A.; Al Zhour, Z.A.A., Kronecker operational matrices for fractional calculus and some applications, Appl math comp, 187, 250-265, (2007) · Zbl 1123.65063
[10] Saadatmandi, A.; Dehghan, M., A new operational matrix for solving fractional-order differential equations, Comp math appl, 59, 1326-1336, (2010) · Zbl 1189.65151
[11] Li, Yuanlu; Zhao, Weiwei, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl math comput, 216, 2276-2285, (2010) · Zbl 1193.65114
[12] Li, Yuanlu, Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun nonl sci num sim, 15, 2284-2292, (2010) · Zbl 1222.65087
[13] Saeedi, H.; Mohseni Moghadam, M.; Mollahasani, N.; Chuev, G.N., A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order, Commun nonl sci num sim, 16, 1154-1163, (2011) · Zbl 1221.65354
[14] Saeedi, H.; Mohseni Moghadam, M., Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets, Appl math comput, 16, 1216-1226, (2011) · Zbl 1221.65140
[15] Liu, N.; Lin, En-Bing, Legendre wavelet method for numerical solutions of partial differential equations, SIAM J math anal, 29, 1040-1065, (1998)
[16] Ali Yousefi, S., Legendre wavelets method for solving differential equations of lane – emden type, Appl math comp, 181, 417-1422, (2006) · Zbl 1105.65080
[17] Zheng, X.; Yang, X., Techniques for solving integral and differential equations by Legendre wavelets, Int J syst sci, 40, 1127-1137, (2009) · Zbl 1292.65146
[18] Maleknejad, K.; Sohrabi, S., Numerical solution of Fredholm integral equations of the first kind by using Legendre wavelets, Appl math comput, 186, 836-843, (2007) · Zbl 1119.65126
[19] Razzaghi, M.; Yousefi, S., The Legendre wavelets operational matrix of integration, Int J syst sci, 32, 495-502, (2001) · Zbl 1006.65151
[20] Khellat, F.; Yousefi, S.A., The linear Legendre mother wavelets operational matrix of integration and its application, J Frank inst, 343, 181-190, (2006) · Zbl 1127.65105
[21] Caputo, M., Linear models of dissipation whose Q is almost frequency independent. part II, J roy austral soc, 13, 529-539, (1967)
[22] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, North-holland mathematics studies, vol. 204, (2006), Elsevier
[23] Bai, Z.B.; Lü, H.S., Positive solutions for boundary value problem of nonlinear fractional differential equation, J math anal appl, 311, 495-505, (2005) · Zbl 1079.34048
[24] Su, X., Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl math lett, 22, 64-69, (2009) · Zbl 1163.34321
[25] Bagley, R.L.; Torvik, P.J., On the appearance of the fractional derivative in the behavior of real materials, ASME J appl mech, 51, 294-298, (1984) · Zbl 1203.74022
[26] Saha Ray, S.; Bera, R.K., Analytical solution of the bagley – torvik equation by Adomian decomposition method, Appl math comp, 168, 398-410, (2005) · Zbl 1109.65072
[27] Diethelm, K.; Ford, N.J., Numerical solution of the bagley – torvik equation, Bit, 42, 490-507, (2002) · Zbl 1035.65067
[28] Wang, Z.H.; Wang, X., General solution of the bagley – torvik equation with fractional – order derivative, Commun nonlin sci num simul, 15, 1279-1285, (2010) · Zbl 1221.34020
[29] Kumar, P.; Agrawal, O.P., An approximate method for numerical solution of fractional differential equations, Signal process, 86, 2602-2610, (2006) · Zbl 1172.94436
[30] Erturk, V.S.; Momani, S.; Odibat, Z., Application of generalized differential transform method to multi-order fractional differential equations, Commun nonlin sci num simul, 13, 1642-1654, (2008) · Zbl 1221.34022
[31] Narahari Achar, B.N.; Hanneken, J.W.; Enck, T.; Clarke, T., Dynamics of the fractional oscillator, Physica A, 297, 361-367, (2001) · Zbl 0969.70511
[32] Momani, S.; Odibat, Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos solitons fract, 31, 1248-1255, (2007) · Zbl 1137.65450
[33] Arikoglu, A.; Ozkol, I., Solution of fractional differential equations by using differential transform method, Chaos solitons fract, 34, 1473-1481, (2007) · Zbl 1152.34306
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