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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 nth 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

A brief introduction to the Hankel–Clifford transformations and its basic properties is given. The spaces [Formula: see text] and [Formula: see text] generalizing the spaces Hμ(I) and S(I), respectively are defined. It is given that the pseudo-differential operators h1,μ,a and h2,μ,a are automorphism of [Formula: see text] and [Formula: see text], respectively. Product and convolution on [Formula: see text] and [Formula: see text] are investigated. Some other spaces related to [Formula: see text] and [Formula: see text] are introduced and studied the continuity of the pseudo-differential operators h1,μ,a and h2,μ,a, respectively.

Pseudo-differential operator (p.d.o) associated with the symbol a(x, y) whose derivatives satisfy certain growth condition is defined and the Zemanian-type spaces Hμ(I) and S(I) are introduced. It is shown that the p.d.o is continuous linear map of the space Hμ(I) and S(I) into itself. An integral representation of p.d.o h1, μ, a is obtained. Using the Hankel convolution it is shown that p.d.o h1, μ, a satisfies a certain [Formula: see text]-norm inequality. Properties of Sobolev-type space [Formula: see text] are studied.

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