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1. Introduction
In recent years, fractional calculus and differential equations have found enormous applications in mathematics, physics, chemistry, and engineering because of the fact that a realistic modeling of a physical phenomenon having dependence not only at the time instant but also on the previous time history can be successfully achieved by using fractional calculus. The applications of the fractional calculus have been demonstrated by many authors. For examples, it has been applied to model the nonlinear oscillation of earthquakes, fluid-dynamic traffic, frequency dependent damping behavior of many viscoelastic materials, continuum and statistical mechanics, colored noise, solid mechanics, economics, signal processing, and control theory [1–5]. However, during the last decade fractional calculus has attracted much more attention of physicists and mathematicians. Due to the increasing applications, some schemes have been proposed to solve fractional differential equations. The most frequently used methods are Adomian decomposition method (ADM) [6, 7], homotopy perturbation method [8], homotopy analysis method [9], variational iteration method (VIM) [10], fractional differential transform method (FDTM) [11, 12], fractional difference method (FDM) [13], power series method [14], generalized block pulse operational matrix method [15], and Laplace transform method [16]. Also, recently the Haar wavelets [17], Legendre wavelets [18, 19], and the Chebyshev wavelets of first kind [20–23] and second kind [24] have been developed to solve the fractional differential equations. It is worth noting that wavelets are localized functions, which are the basis for energy-bounded functions and in particular for
Approximation by orthogonal family of basis functions...