Abstract
Abstract The main objective of this paper is to study the fractional Hankel transformation and the continuous fractional Bessel wavelet transformation and some of their basic properties. Applications of the fractional Hankel transformation (FrHT) in solving generalized n th order linear nonhomogeneous ordinary differential equations are given. The continuous fractional Bessel wavelet transformation, its inversion formula and Parseval’s relation for the continuous fractional Bessel wavelet transformation are also studied. MSC:46F12, 26A33.
Highlights
Pathak and Dixit [ ] introduced continuous and discrete Bessel wavelet transformations and studied their properties by exploiting the Hankel convolution of Haimo [ ] and Hirschman [ ]
Upadhyay et al [ ] studied the continuous Bessel wavelet transformation associated with the Hankel-Hausdorff operator
Using the fractional Hankel transformation (FrHT), we investigate the solution of the generalized differential equation
Summary
Pathak and Dixit [ ] introduced continuous and discrete Bessel wavelet transformations and studied their properties by exploiting the Hankel convolution of Haimo [ ] and Hirschman [ ]. We define a one-dimensional fractional Hankel transformation (FrHT) with parameter θ of φ(x) for μ ≥ – / and < θ < π as follows: φμθ (y) = hθμφ (y) = Kμθ (x, y)φ(x) dx, where the kernel. = φ(y) τxθ ψ (y) dy, where the fractional Hankel translation of the function φ ∈ L (R+) is defined by τxθ φ φθ φ(z)Dθμ(x, y, z) dz, and. We define the continuous fractional Bessel wavelet transformation and study some of its properties using the theory of fractional Hankel convolution ( ) corresponding to [ ]. The fractional Bessel wavelets ψbθ,a are generated from one single function ψ ∈ L (R+) by dilation and translation with parameters a > and b ≥ respectively by ψbθ,a(x)
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