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Solving fractional integral equations by the Haar wavelet method. (English) Zbl 1170.65106
Summary: Haar wavelets for the solution of fractional integral equations are applied. Fractional Volterra and Fredholm integral equations are considered. The proposed method also is used for analysing fractional harmonic vibrations. The efficiency of the method is demonstrated by three numerical examples.

65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
26A33 Fractional derivatives and integrals
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
65T60 Numerical methods for wavelets
Full Text: DOI
[1] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004
[2] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, North-holland mathematic studies, vol. 204, (2006), Elsevier
[3] Achar, B.N.N.; Hanneken, J.W.; Enck, T.; Clarke, T., Dynamics of the fractional oscillator, Physica A, 297, 361-367, (2001) · Zbl 0969.70511
[4] Achar, B.N.N.; Hanneken, J.W.; Clarke, T., Response characteristics of a fractional oscillator, Physica A, 309, 275-288, (2002) · Zbl 0995.70017
[5] Eab, C.H.; Lim, S.C., Path integral representation of fractional harmonic oscillator, Physica A, 371, 303-316, (2006)
[6] Miyakoda, T., On a almost free damping vibration equation using N-fractional calculus, J. comp. appl. math., 144, 233-240, (2002) · Zbl 1007.34006
[7] Rabei, E.M.; Nawafleh, K.I.; Hijjawi, R.S.; Muslin, S.I.; Baleanu, D., The Hamilton formalism with fractional derivatives, J. math. anal. appl., 327, 891-897, (2007) · Zbl 1104.70012
[8] Daftardar-Gejji, V.; Jafari, H., Solving a multi-order fractional differential equation using Adomian decomposition, Appl. math. comp., 189, 541-548, (2007) · Zbl 1122.65411
[9] Sheu, L.-J.; Chen, H.-K.; Chen, J.-H.; Tam, L.-M., Chaotic dynamics of the fractionally damped Duffing equation, Chaos soliton fract., 32, 1459-1468, (2007) · Zbl 1129.37015
[10] Ahmad, W.M.; Sprott, J.C., Chaos in fractional-order autonomous nonlinear systems, Chaos soliton fract., 16, 339-351, (2003) · Zbl 1033.37019
[11] Al-Assaf, Y.; El-Khazali, R.; Ahmad, W., Identification of fractional chaotic system parameters, Chaos soliton fract., 22, 897-905, (2004) · Zbl 1129.93492
[12] Ü. Lepik, Application of the Haar wavelets for solution of linear integral equations, in: Dynamical Systems and Applications, Proceedings, with E. Tamme, Antalaya, 2004, pp. 494-507. · Zbl 1339.65248
[13] Lepik, Ü., Numerical solutions of differential equations using Haar wavelets, Math. comp. simul., 68, 127-143, (2003) · Zbl 1072.65102
[14] Lepik, Ü., Haar wavelet method for nonlinear integro – differential equations, Appl. math. comp., 176, 324-333, (2006) · Zbl 1093.65123
[15] Lepik, Ü., Numerical solution of evolution equations by the Haar wavelet method, Appl. math. comp., 185, 695-704, (2007) · Zbl 1110.65097
[16] Chen, C.F.; Hsiao, C.H., Haar wavelet method for solving lumped and distributed parameter systems, IEE-proc.: control theory appl., 144, 87-94, (1997) · Zbl 0880.93014
[17] Schäfer, I.; Kempfle, S., Impulse responses of fractional damped systems, Nonlinear dynam., 38, 61-68, (2004) · Zbl 1097.70016
[18] Diethelm, K.; Ford, N.J.; Freed, A.D., A predictor – corrector approach for the numerical solution of fractional differential equations, Nonlinear dynam., 29, 3-22, (2002) · Zbl 1009.65049
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