Solving fractional integral equations by the Haar wavelet method

Haar wavelets for the solution of fractional integral equations are applied. Fractional Vol-terra 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. Although the conception of the fractional derivatives was introduced already in the middle of the 19th century by Rie-mann and Liouville, the first work, devoted exclusively to the subject of fractional calculus, is the book by Oldham and Spa-nier [1] published in 1974. After that the number of publications about the fractional calculus has rapidly increased. The reason for this is that some physical processes as anomalous diffusion, complex viscoelasticity, behaviour of mechatronic and biological systems, rheology etc. cannot be described adequately by the classical models. At the present time we possess several excellent monographs about fractional calculus for example the book [2] by Kilbas et al., to which is also included a rather large and up-to-date Bibliography (928 items). Because of the enormous number of the papers about this topic we shall cite here only some papers which are more close to subject of this paper. In a number of papers fractional differential equations are discussed; mostly these equations are transformed to fractional Volterra integral equations. For solution different techniques, as Fourier and Laplace transforms, power spectral density, Adomian decomposition method, path integration etc., are applied. One-dimensional fractional harmonic oscillator is analysed in [3–6]. In [3,4] the solution is obtained in terms of Mittag– Leffler functions using Laplace transforms; several cases of the forcing function equation are considered. In [5] the fractional equation of motion is solved by the path integral method. In [6] the case, where the fractional derivatives only slightly differ from the ordinary derivatives, is analysed. Fractional Hamilton's equations are discussed in [7]. In [8] multiorder fractional differential equations are solved by using the Adomian decomposition. In several papers fractional chaotic systems are discussed. In [8] a three-dimensional fractional chaotic oscillator model is proposed. Chaotic dynamics of the fractionally damped Duffing equation is investigated in [9]. Two chaotic models for third-order chaotic nonlinear systems are analysed in [10]. It is somewhat surprising that among different solution techniques the wavelet method has not attained much attention. We found only one paper [11] in which the wavelet method is applied for solving fractional differential equations; for this purpose the Daubechies wavelet functions are used. Among the different …