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Solving a nonlinear fractional differential equation using Chebyshev wavelets. (English) Zbl 1222.65087
Summary: Chebyshev wavelet operational matrix of the fractional integration is derived and used to solve a nonlinear fractional differential equations. Some examples are included to demonstrate the validity and applicability of the technique.

MSC:
65L99 Numerical methods for ordinary differential equations
34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
45J05 Integro-ordinary differential equations
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[1] Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, (1999), Academic Press New York · Zbl 0924.34008
[2] Gaul, L.; Klein, P.; Kemple, S., Damping description involving fractional operators, Mech syst signal pr, 5, 81-88, (1991)
[3] Suarez, L.; Shokooh, A., An eigenvector expansion method for the solution of motion containing fractional derivatives, J appl mech, 64, 629-635, (1997) · Zbl 0905.73022
[4] Momani, S., An algorithm for solving the fractional convection – diffusion equation with nonlinear source term, Commun nonlinear sci numer simul, 12, 7, 1283-1290, (2007) · Zbl 1118.35301
[5] Jafari, H.; Seifi, S., Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun nonlinear sci numer simul, 14, 5, 1962-1969, (2009) · Zbl 1221.35439
[6] Sweilam, N.H.; Khader, M.M.; Al-Bar, R.F., Numerical studies for a multi-order fractional differential equation, Phys lett A, 371, 1-2, 26-33, (2007) · Zbl 1209.65116
[7] Das, S., Analytical solution of a fractional diffusion equation by variational iteration method, Comput math appl, 57, 3, 483-487, (2009) · Zbl 1165.35398
[8] Arikoglu, A.; Ozkol, I., Solution of fractional integro-differential equations by using fractional differential transform method, Chaos solitons fract, 40, 2, 521-529, (2009) · Zbl 1197.45001
[9] Erturk, V.S.; Momani, S.; Odibat, Z., Application of generalized differential transform method to multi-order fractional differential equations, Commun nonlinear sci numer simul, 13, 8, 1642-1654, (2008) · Zbl 1221.34022
[10] Meerschaert, M.; Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl numer math, 56, 1, 80-90, (2006) · Zbl 1086.65087
[11] Odibat, Z.; Shawagfeh, N., Generalized taylor’s formula, Appl math comput, 186, 1, 286-293, (2007) · Zbl 1122.26006
[12] Wu, J.L., A wavelet operational method for solving fractional partial differential equations numerically, Appl math comput, 214, 1, 31-40, (2009) · Zbl 1169.65127
[13] Lepik, Solving fractional integral equations by the Haar wavelet method, Appl math comput, 214, 2, 468-478, (2009) · Zbl 1170.65106
[14] Chen, C.; Hsiao, C., Haar wavelet method for solving lumped and distributed-parameter systems, IEE P-contr theor appl, 144, 1, 87-94, (1997) · Zbl 0880.93014
[15] Bujurke, N.; Salimath, C.; Shiralashetti, S., Numerical solution of stiff systems from nonlinear dynamics using single-term Haar wavelet series, Nonlinear dyn, 51, 4, 595-605, (2008) · Zbl 1171.65407
[16] 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, 9, 1121-1132, (2004) · Zbl 1068.65088
[17] Pawlak, M.; Hasiewicz, Z., Nonlinear system identification by the Haar multiresolution analysis, IEEE trans circuits I, 45, 9, 945-961, (1998) · Zbl 0952.93021
[18] Hsiao, C.; Wang, W., Optimal control of linear time-varying systems via Haar wavelets, J optim theory appl, 103, 3, 641-655, (1999) · Zbl 0941.49018
[19] 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, 2, 237-252, (2005) · Zbl 1063.49002
[20] Sadek, I.; Abualrub, T.; Abukhaled, M., A computational method for solving optimal control of a system of parallel beams using Legendre wavelets, Math comput model, 45, 9-10, 1253-1264, (2007) · Zbl 1117.49026
[21] 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, 2, 234-244, (2009) · Zbl 1162.65040
[22] Babolian, E.; Masouri, Z.; Hatamzadeh-Varmazyar, S., Numerical solution of nonlinear volterra – fredholm integro-differential equations via direct method using triangular functions, Comput math appl, 58, 2, 239-247, (2009) · Zbl 1189.65306
[23] Kajani, M.; Vencheh, A., The Chebyshev wavelets operational matrix of integration and product operation matrix, Int J comput math, 86, 7, 1118-1125, (2008) · Zbl 1169.65072
[24] Reihani, M.H.; Abadi, Z., Rationalized Haar functions method for solving Fredholm and Volterra integral equations, J comput appl math, 200, 1, 12-20, (2007) · Zbl 1107.65122
[25] Khellat, F.; Yousefi, S., The linear Legendre mother wavelets operational matrix of integration and its application, J franklin inst, 343, 2, 181-190, (2006) · Zbl 1127.65105
[26] Razzaghi, M.; Yousefi, S., The Legendre wavelets operational matrix of integration, Int J syst sci, 32, 4, 495-502, (2001) · Zbl 1006.65151
[27] Tenreiro Machado, J.A., Fractional derivatives: probability interpretation and frequency response of rational approximations, Commun nonlinear sci numer simul, 14, 9-10, 3492-3497, (2009)
[28] Kilicman, A.; Al Zhour, Z.A.A., Kronecker operational matrices for fractional calculus and some applications, Appl math comput, 187, 1, 250-265, (2007) · Zbl 1123.65063
[29] Odibat, Z.; Momani, S., Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos solitons fract, 36, 1, 167-174, (2008) · Zbl 1152.34311
[30] El-Mesiry, A.; El-Sayed, A.; El-Saka, H., Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl math comput, 160, 3, 683-699, (2005) · Zbl 1062.65073
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