scholarly journals Matrix Fourier Transforms for Consistent Mathematical Models

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Oleg Yaremko ◽  
Natalia Yaremko
Keyword(s):  
Mathematical Models ◽  
Fourier Transforms ◽  
Integral Form ◽  
Integral Transforms ◽  
Mathematical Physics ◽  
Discontinuous Coefficients ◽  
Piecewise Homogeneous Media ◽  
Fourier Matrix ◽  
Matrix Transforms ◽  
Homogeneous Media

We create a matrix integral transforms method; it allows us to describe analytically the consistent mathematical models. An explicit constructions for direct and inverse Fourier matrix transforms with discontinuous coefficients are established. We introduce special types of Fourier matrix transforms: matrix cosine transforms, matrix sine transforms, and matrix transforms with piecewise trigonometric kernels. The integral transforms of such kinds are used for problems solving of mathematical physics in homogeneous and piecewise homogeneous media. Analytical solution of iterated heat conduction equation is obtained. Stress produced in the elastic semi-infinite solid by pressure is obtained in the integral form.

Extending the unified transform: curvilinear polygons and variable coefficient PDEs

2018 ◽  
Vol 40 (2) ◽  
pp. 976-1004 ◽  
Author(s):  
Matthew J Colbrook
Keyword(s):  
Fourier Transforms ◽  
Elliptic Boundary ◽  
Neumann Boundary ◽  
Integral Transforms ◽  
Variable Coefficients ◽  
Central Component ◽  
Unknown Boundary ◽  
Boundary Values ◽  
Global Relation ◽  
Chebyshev Interpolation

Abstract We provide the first significant extension of the unified transform (also known as the Fokas method) applied to elliptic boundary value problems, namely, we extend the method to curvilinear polygons and partial differential equations (PDEs) with variable coefficients. This is used to solve the generalized Dirichlet-to-Neumann map. The central component of the unified transform is the coupling of certain integral transforms of the given boundary data and of the unknown boundary values. This has become known as the global relation and, in the case of constant coefficient PDEs, simply links the Fourier transforms of the Dirichlet and Neumann boundary values. We extend the global relation to PDEs with variable coefficients and to domains with curved boundaries. Furthermore, we provide a natural choice of global relations for separable PDEs. These generalizations are numerically implemented using a method based on Chebyshev interpolation for efficient and accurate computation of the integral transforms that appear in the global relation. Extensive numerical examples are provided, demonstrating that the method presented in this paper is both accurate and fast, yielding exponential convergence for sufficiently smooth solutions. Furthermore, the method is straightforward to use, involving just the construction of a simple linear system from the integral transforms, and does not require knowledge of Green’s functions of the PDE. Details on the implementation are discussed at length.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

Derivation by a Transform Method of Integral Equations of Unsteady Lifting Surface Theory in Subsonic and Supersonic Flow

1979 ◽  
Vol 30 (4) ◽  
pp. 529-543
Author(s):  
Shigenori Ando ◽  
Akio Ichikawa
Keyword(s):  
Fourier Transforms ◽  
Integral Transforms ◽  
Physical Models ◽  
Streamwise Direction ◽  
Fourier Inversion ◽  
Transform Method ◽  
Inviscid Flows ◽  
Surface Theory ◽  
Method Of Integral Equations ◽  
Lifting Surface

SummaryApplications of “integral transforms of in-plane coordinate variables” in order to formulate unsteady planar lifting surface theories are demonstrated for both sub- and supersonic inviscid flows. It is concise and pithy. Fourier transforms are exclusively used, except for only Laplace transform in the supersonic streamwise direction. It is found that the streamwise Fourier inversion in the subsonic case requires some caution. Concepts based on the theory of distributions seem to be essential, in order to solve the convergence difficulties of integrals. Apart from this caution, the method of integral transforms of in-plane coordinate variables makes it be pure-mathematical to formulate the lifting surface problems, and makes aerodynamicist’s experiences and physical models such as vortices or doublets be useless.

scholarly journals Response Due To Impulsive Force In Generalized Thermomicrostretch Elastic Solid

2015 ◽  
Vol 20 (3) ◽  
pp. 487-502
Author(s):  
V. Kumar ◽  
R. Singh
Keyword(s):  
Fourier Transforms ◽  
Normal Force ◽  
Elastic Solid ◽  
Integral Transforms ◽  
Normal Displacement ◽  
Impulsive Force ◽  
Field Equations ◽  
Laplace And Fourier Transforms ◽  
Eigen Value ◽  
Plain Strain

Abstract A two dimensional Cartesian model of a generalized thermo-microstretch elastic solid subjected to impulsive force has been studied. The eigen value approach is employed after applying the Laplace and Fourier transforms on the field equations for L-S and G-L model of the plain strain problem. The integral transforms have been inverted into physical domain numerically and components of normal displacement, normal force stress, couple stress and microstress have been illustrated graphically.

General Definitions of Integral Transforms for Mathematical Physics

2020 ◽  
Vol 70 (9) ◽  
pp. 759-765
Author(s):  
Dongseung KANG ◽  
Hoewoon KIM ◽  
Bongwoo LEE*
Keyword(s):  
Integral Transforms ◽  
Mathematical Physics

scholarly journals Some Schemata for Applications of the Integral Transforms of Mathematical Physics

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 254 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Integral Transform ◽  
First Integral ◽  
Integral Transforms ◽  
Mathematical Physics ◽  
Basic Properties ◽  
Survey Article ◽  
Laplace Integral ◽  
Generating Operators

In this survey article, some schemata for applications of the integral transforms of mathematical physics are presented. First, integral transforms of mathematical physics are defined by using the notions of the inverse transforms and generating operators. The convolutions and generating operators of the integral transforms of mathematical physics are closely connected with the integral, differential, and integro-differential equations that can be solved by means of the corresponding integral transforms. Another important technique for applications of the integral transforms is the Mikusinski-type operational calculi that are also discussed in the article. The general schemata for applications of the integral transforms of mathematical physics are illustrated on an example of the Laplace integral transform. Finally, the Mellin integral transform and its basic properties and applications are briefly discussed.

scholarly journals Airworthiness Support Measures Analogy to the Prospective Roundabouts Alternatives: Theoretical Aspects

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Andriy Viktorovich Goncharenko
Keyword(s):  
Mathematical Models ◽  
Traffic Flow ◽  
Lagrange Equation ◽  
Integral Form ◽  
Environmental Safety ◽  
Transport Infrastructure ◽  
Transportation Services ◽  
Modeling And Optimization ◽  
Support Measures ◽  
Capacity Model

The goal of this paper is to investigate theoretically the possible directions of some specified methods for the alternative roundabouts effectiveness modeling and optimization. The out-coming criteria have an economical interpretation. Those are the objective functionals of the alternative roundabouts effectiveness as the profit gained in the course of the traffic flow changes in the view of the integral form. This is modeled taking into consideration the transport infrastructure functioning elements such as the traffic flow of a capacity model. It takes into account two major components of the transportation services which are the alternative roundabouts business’ incomes and expenses relating to the roundabouts transportation worthiness support. The prototypic approach is that one from the aircraft airworthiness support measures models. Corresponding managerial influences with respect to environmental, safety, utility, and other issues, as well as probable impacts, are modeled with the construction of the relevant under-integral expressions, equations, and appropriate coefficients and parameters of the mathematical models. The achieved theoretical results, on the basis of the Euler-Lagrange equation and accepted assumptions, have been checked for the sufficiency of the objective functional maximum presence at the “point” with the use of the conducted computer simulation. The necessary diagrams are plotted in order to illustrate the theoretical contemplations and speculations, as well as to check the correctness of the applied mathematical derivations and visualize the models’ preciseness and abilities. The theoretically constructed mathematical models have a significance of the prognostic values applicability required at the alternative roundabouts effectiveness modeling and optimization ensuring their design progress and evolutions.

Solving spectral problems in piecewise homogeneous media

1995 ◽  
Vol 75 (4) ◽  
pp. 1866-1871
Author(s):  
V. G. Prikazchikov ◽  
A. S. Medvedyev
Keyword(s):  
Spectral Problems ◽  
Piecewise Homogeneous Media ◽  
Homogeneous Media
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