Criteria for Determining the Computational Error of Numerically Calculated Fourier Integral Transforms

1969 ◽  
Vol 8 (4) ◽  
pp. 745-748 ◽  
Author(s):  
C. J. Messa ◽  
W. L. Luyben ◽  
G. W. Poehlein
Keyword(s):  
Integral Transforms ◽  
Fourier Integral ◽  
Computational Error

Damping Effect on the Wave Propagation in Carbon Steel Pipelines Under Fluid Hammer Conditions

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Dimitrios G. Pavlou ◽  
Muk Chen Ong
Keyword(s):  
High Speed ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Generalized Function ◽  
Pressure Shock ◽  
Algebraic Form ◽  
Flow Parameters ◽  
Damping Characteristics ◽  
Research Gap ◽  
Hoop Stresses

A sudden reduction of the fluid flow yields a pressure shock, which travels along the pipeline with a high-speed. Due to this transient loading, dynamic hoop stresses are developed that may cause catastrophic damages in pipeline integrity. The vibration of the pipe wall is affected by the flow parameters as well as by the elastic and damping characteristics of the material. Most of the studies on dynamic response of pipelines: (a) neglect the effect of the material damping and (b) are usually limited to harmonic pressure oscillations. The present work is an attempt to fill the above research gap. To achieve this target, an analytic solution of the governing motion equation of pipelines under moving pressure shock is derived. The proposed methodology takes into account both elastic and damping characteristics of the steel. With the aid of Laplace and Fourier integral transforms and generalized function properties, the solution is based on the transformation of the dynamic partial differential equation into an algebraic form. Analytical inversion of the transformed dynamic radial deflection variable is achieved, yielding the final solution. The proposed methodology is implemented in an engineering example; and the results are shown and discussed.

The mellin integral transform in fractional calculus

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Yuri Luchko ◽  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Fourier Transforms ◽  
Integral Transform ◽  
Special Functions ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Monotone Functions ◽  
Completely Monotone ◽  
Mellin Convolution ◽  
New Applications

AbstractIn Fractional Calculus (FC), the Laplace and the Fourier integral transforms are traditionally employed for solving different problems. In this paper, we demonstrate the role of the Mellin integral transform in FC. We note that the Laplace integral transform, the sin- and cos-Fourier transforms, and the FC operators can all be represented as Mellin convolution type integral transforms. Moreover, the special functions of FC are all particular cases of the Fox H-function that is defined as an inverse Mellin transform of a quotient of some products of the Gamma functions.In this paper, several known and some new applications of the Mellin integral transform to different problems in FC are exemplarily presented. The Mellin integral transform is employed to derive the inversion formulas for the FC operators and to evaluate some FC related integrals and in particular, the Laplace transforms and the sin- and cos-Fourier transforms of some special functions of FC. We show how to use the Mellin integral transform to prove the Post-Widder formula (and to obtain its new modi-fication), to derive some new Leibniz type rules for the FC operators, and to get new completely monotone functions from the known ones.

ON q-EXTENSIONS OF MEHTA'S EIGENVECTORS OF THE FINITE FOURIER TRANSFORM

2006 ◽  
Vol 21 (23n24) ◽  
pp. 4993-5006 ◽  
Author(s):  
NATIG M. ATAKISHIYEV
Keyword(s):  
Fourier Transform ◽  
Hermite Polynomials ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Hermite Functions ◽  
Continuous Limit ◽  
Finite Fourier Transform ◽  
The Matrix ◽  
Matrix Formula

Mehta has shown that eigenvectors [Formula: see text] of the finite Fourier transform with the matrix [Formula: see text], 0 ≤ j, k ≤ N-1, can be defined in terms of the classical Hermite functions [Formula: see text] as [Formula: see text], where [Formula: see text]. We argue that the finite Fourier transform [Formula: see text] does actually govern also some q-extensions of Mehta's eigenvectors [Formula: see text], associated with certain well-known orthogonal q-polynomial families. For the pairs of the continuous q-Hermite and q-1-Hermite polynomials, the Rogers–Szegő and Stieltjes–Wigert polynomials, and the discrete q-Hermite polynomials of types I and II such links are explicitly derived. In the limit when the base q → 1 these q-extensions coincide with Mehta's eigenvectors [Formula: see text], whereas in the continuous limit (i.e. when the parameter N → ∞) they correspond to the classical Fourier integral transforms between the above-mentioned pairs of q-polynomial families.

scholarly journals Frictional Contact Problem of One Dimensional Hexagonal Piezoelectric Quasicrystals Layer

2021 ◽  
Author(s):  
RuKai Huang ◽  
Sheng hu Ding ◽  
Xin Zhang ◽  
Xing Li
Keyword(s):  
Contact Problem ◽  
Frictional Contact ◽  
Three Dimensional ◽  
Integral Transforms ◽  
Fourier Integral ◽  
General Solutions ◽  
One Dimensional ◽  
Frictional Contact Problem ◽  
Material Development ◽  
Influence Coefficients

Abstract Based on three-dimensional (3D) general solutions for one-dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs), this paper studied the frictional contact problem of 1D-hexagonal PEQCs layer. The frequency response functions (FRFs) for 1D-hexagonal PEQCs layer are analytically derived by applying double Fourier integral transforms to the general solutions and boundary conditions, which are consequently converted to the corresponding influence coefficients (ICs). The conjugate gradient method (CGM) is used to obtain the unknown pressure distribution, while the discrete convolution-fast Fourier transform technique (DC-FFT) is applied to calculate the displacements and stresses of phonon and phason, electric potentials and electric displacements. Numerical results are given to reveal the influences of material parameters and loading conditions on the contact behavior. The obtained 3D contact solutions are not only helpful further analysis and understanding of the coupling characteristics of phonon, phason and electric field, but also provide a reference basis for experimental analysis and material development.

Thermoelasticity of Cylindrically Anisotropic Generally Laminated Cylinders

1976 ◽  
Vol 43 (1) ◽  
pp. 124-130 ◽  
Author(s):  
J. Padovan
Keyword(s):  
Numerical Experiments ◽  
Differential Operators ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Material Anisotropy ◽  
Complex Power ◽  
Different Types ◽  
Power Series Expansions ◽  
Cylindrically Anisotropic ◽  
Stated Problem

The effects of cylindrically curvilinear mechanical and thermal material anisotropy on the stationary thermoelastic fields of generally laminated cylinders are studied. To model the stated problem, each individual ply of the cylinder is considered to be composed of both mechanically and thermally cylindrically anisotropic media whose governing fields satisfy the 3-D elasticity and conduction equations. Based on finite and infinite Fourier integral transforms together with the use of complex adjoint differential operators and complex power series expansions, a nonhomogeneous pseudo-stiffness procedure is used to develop the general solution form for the stated problem. Through the use of the model and its solution, several numerical experiments are presented which emphasize the significant effects of cylindrical material anisotropy on the governing fields of several different types of laminate configuration.

scholarly journals Stationary Dynamic Displacement Solutions for a Rectangular Load Applied within a 3D Viscoelastic Isotropic Full Space—Part II: Implementation, Validation, and Numerical Results

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Josue Labaki ◽  
Edivaldo Romanini ◽  
Euclides Mesquita
Keyword(s):  
Present Article ◽  
Integral Transform ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Constant Load ◽  
Numerical Results ◽  
Dynamic Displacement ◽  
Loading Case ◽  
Fourier Integral Transform ◽  
Stationary Loading

In part I of the present article the formulation for a dynamic stationary semianalytical solution for a spatially constant load applied over a rectangular surface within a viscoelastic isotropic full-space has been presented. The solution is obtained within the frame of a double Fourier integral transform. These inverse integral transforms must be evaluated numerically. In the present paper, the technique to evaluate numerically the inverse double Fourier integrals is described. The procedure is validated, and a number of original displacement results for the stationary loading case are reported.

scholarly journals Stationary Dynamic Stress Solutions for a Rectangular Load Applied within a 3D Viscoelastic Isotropic Full-Space

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
E. Romanini ◽  
J. Labaki ◽  
E. Mesquita ◽  
R. C. Silva
Keyword(s):  
Fourier Transforms ◽  
Damping Ratio ◽  
Three Dimensional ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Wave Number ◽  
Influence Functions ◽  
Time Harmonic ◽  
Final Stress ◽  
Stress Influence

This paper presents stress influence functions for uniformly distributed, time-harmonic rectangular loads within a three-dimensional, viscoelastic, isotropic full-space. The coupled differential equations relating displacements and stresses in the full-space are solved through double Fourier integral transforms in the wave number domain, in which they can be solved algebraically. The final stress fields are expressed in terms of double indefinite integrals arising from the Fourier transforms. The paper presents numerical schemes with which to integrate these functions accurately. The article presents numerical validation of the synthesized stress kernels and their behavior for high frequencies and large distances from the excitation source. The influence of damping ratio on the dynamic results is also investigated. This article is complementary to previous results of the authors in which the corresponding displacement solutions were derived. Stress influence functions, together with their displacement counterparts, are a fundamental part of many numerical methods of discretization such the boundary element method.

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