In the present paper, a fractional wavelet transform of real order α is introduced, and various useful properties and results are derived for it. These include (for example) Perseval's formula and inversion formula for the fractional wavelet transform. Multiresolution analysis and orthonormal fractional wavelets associated with the fractional wavelet transform are studied systematically. Fractional Fourier transforms of the Mexican hat wavelet for different values of the order α are compared with the classical Fourier transform graphically, and various remarkable observations are presented. A comparative study of the various results, which we have presented in this paper, is also represented graphically.

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We define a window function ψ as an element of L 2 (R n ) satisfying certain boundedness properties with respect to the L 2 (R n ) norm and prove that it satisfies the admissibility condition if and only if the integral ofalong the real line is zero. We also prove that each of the window functions is an element of L 1 (R n ). A function ψ ∈ L 2 (R n ) satisfying the admissibility condition is a wavelet. We define the wavelet transform of f ∈ L 2 (R n ) (which is a window function) with respect to the wavelet ψ ∈ L 2 (R n ) and prove an inversion formula interpreting convergence in L 2 (R n ). It is also proved that at a point of continuity of f the convergence of our wavelet inversion formula is in a pointwise sense.

In this article, we study a two point boundary value problem of non linear differential equation on a semi infinite domain that describes the unsteady flow of gas through a porous medium. Under special transform, we convert this problem to boundary value problem in compactly supported domain [0,1]. An algorithm provided for obtaining solution by Legendre wavelet collocation method. This method is effectively used to determine y (t) and its initial slope at the origin. The convergence and stability analysis is provided. The results thus obtained are compared with the those obtained from modified decomposition method [5], Variational iterational method [6], rational Chebyshev functions method (RCM) [7] and radial basis function (RBF) collocation method [10]. It has been observed that the proposed method provide better results with lesser computational complexity.

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