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Application of Integral Transforms in Flood Studies
Published in Saeid Eslamian, Faezeh Eslamian, Flood Handbook, 2022
Vahid Nourani, Mehran Dadashzadeh, Saeid Eslamian
There are many other integral transforms, including the Mellin transform, the Hankel transform, the Hilbert transform, and the wavelet transform, which are widely used to solve a broad range of problems in mathematics, science, and engineering. Although Mellin (1854–1933) presented an elaborate discussion of his transform and its inversion formula, it was G. Bernhard Riemann (1826–1866) who first recognized the Mellin transform and its inversion formula. Hermann Hankel (1839–1873) introduced the Hankel transform with the Bessel function as its kernel. Although the Hilbert transform was named after one of the greatest mathematicians of the 20th century, David Hilbert (1862–1943), this transform and its properties are basically studied by G. H. Hardy (1877–1947) and E. C. Titchmarsh (1899–1963). The original idea of wavelet transform belongs to Fourier with his theories of frequency analysis. In 1980, Grossman and Morlet provided a way of thinking for wavelets based on physical intuition. Then, Stephane Mallat gave wavelets an additional jump-start through his work in digital signal processing in 1985. Ingrid Daubechies used Mallat's work to construct a set of wavelet orthonormal basis functions that are perhaps the most elegant and have become the cornerstone of wavelet applications today.
Sturm-Liouville Expansions, Discrete Polynomial Transforms and Wavelets
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
In analogy with the way the concept of the Fourier series of functions on the circle is extended to the Fourier transform of functions on the line, a transform based on Bessel functions can be defined. It is called the mth order Hankel transform8f^m(p)=∫0∞f(x)Jm(px)xdx
Introduction
Published in Richard C. Dorf, Circuits, Signals, and Speech and Image Processing, 2018
Of the two branches of calculus, integral calculus is richer in its applications, as well as in its theoretical content. Though the theory is not emphasized here, important applications such as finding areas, lengths, volumes, centroids, and the work done by a nonconstant force are included. Both cylindrical and spherical polar coordinates are discussed, and a table of integrals is included. Vector analysis is summarized in a separate section and includes a summary of the algebraic formulas involving dot and cross multiplication, frequently needed in the study of fields, as well as the important theorems of Stokes and Gauss. The part on special functions includes the gamma function, hyperbolic functions, Fourier series, orthogonal functions, and both Laplace and z-transforms. The Laplace transform provides a basis for the solution of differential equations and is fundamental to all concepts and definitions underlying analytical tools for describing feedback control systems. The z-transform, not discussed in most applied mathematics books, is most useful in the analysis of discrete signals as, for example, when a computer receives data sampled at some prespecified time interval. The Bessel functions, also called cylindrical functions, arise in many physical applications, such as the heat transfer in a “long” cylinder, whereas the other orthogonal functions discussed—Legendre, Hermite, and Laguerre polynomials—are needed in quantum mechanics and many other subjects (e.g., solid-state electronics) that use concepts of modern physics.
The Fourier–Bessel method for the inverse scattering problem of cavities
Published in Applicable Analysis, 2023
Xiaoying Yang, Xinwei Du, Yinglin Wang
The total field u satisfies where denotes the Dirac delta distribution. The main idea is to approximate the scattered field to the problem (1)–(2) by the linear combination of the Fourier–Bessel functions where , are the Fourier coefficients, is the Bessel function of the first kind of order n under the polar coordinates , , is the solution of the Helmholtz equation Let where is the measured scattered data on the circle S with radius R. Then where stands for the inner product on .
Analytical modeling of sound transmission through FGM sandwich cylindrical shell immersed in convected fluids
Published in Mechanics of Advanced Materials and Structures, 2023
Substituting Eq. (12) into Eq. (13) and collecting the coefficients of and the equations of cylindrical shell impinged by a plane sound wave are obtained as: where and are the incident, reflected, and transmitted wave, respectively. It must be mentioned that since the Helmholtz equation is quietly well-known and one may refer to [34, 35] for more details of the cylindrical coordinate description of the harmonic plane wave, and its details are not repeated herein, wherein the incident wave can be expressed as follows: where where is the amplitude of three incident plane wave, is the incident angle, and represents the Bessel function of the first kind of integer order and can be and
The existence and evolution of fast-decaying Bessel modes in cylindrical hollow waveguides and in free space
Published in Journal of Modern Optics, 2019
G. Nyitray, A. Nagyváradi, M. Koniorczyk
According to our working hypothesis if the initial field of the cylindrical waveguide is a symmetrical mode (see Figure 2) the symmetry transformation generates the concentric rings of . The guided field can be considered as an interference structure of waves launched from a virtual light source. If one looks into the waveguide from the exit towards the illuminated entrance this virtual ring structure becomes observable (see Figure 3) and can be referred to as a FWF of the cylindrical hollow waveguide [14]. As Figure 2 shows each root of the Bessel function corresponds to a particular (transverse magnetic) waveguide mode. Having an intuitive picture of the waveguides in argument let us turn now our attention to the details of our system.