Abstract
In this paper, the solvability of a class of convolution equations is discussed by using two-dimensional (2D) fractional Fourier transform (FRFT) in polar coordinates. Firstly, we generalize the 2D FRFT to the polar coordinates setting. The relationship between 2D FRFT and fractional Hankel transform (FRHT) is derived. Secondly, the spatial shift and multiplication theorems for 2D FRFT are proposed by using this relationship. Thirdly, in order to analyze the solvability of the convolution equations, a novel convolution operator for 2D FRFT is proposed, and the corresponding convolution theorem is investigated. Finally, based on the proposed theorems, the solvability of the convolution equations is studied.
Highlights
In recent years, convolution-type singular integral equations have received increasing attention from many mathematicians due to the wide applications in the field of engineering mechanics, fracture mechanics, and so on
The main objective of this paper is to propose a novel method to study the solvability of convolution Equation (3) by using two-dimensional (2D) fractional Fourier transform (FRFT) in polar coordinates
We study the spatial shift and multiplication theorem for 2D FRFT
Summary
Convolution-type singular integral equations have received increasing attention from many mathematicians due to the wide applications in the field of engineering mechanics, fracture mechanics, and so on. The main objective of this paper is to propose a novel method to study the solvability of convolution Equation (3) by using two-dimensional (2D) FRFT in polar coordinates. A novel convolution operator for 2D FRFT in polar coordinates is investigated, and the corresponding convolution theorem is proposed. The results of this paper studied some useful properties of FRFT in polar coordinates and provided a new way to study the solvability of integral equations. Applying this relationship, we study the spatial shift and multiplication theorem for 2D FRFT.
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