FOURIER SERIES REPRESENTATION OF FRACTAL INTERPOLATION FUNCTION

Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050063
Author(s):  
XUEZAI PAN ◽  
MINGGANG WANG ◽  
XUDONG SHANG
Keyword(s):  
Fourier Series ◽  
Series Representation ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Sine Series ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fractal Interpolation Function ◽  
Fourier Cosine Series ◽  
Fourier Sine Series

The purpose of this research is to show how the complicated and irregular fractal interpolation function is represented by Fourier series. First, on the closed interval [0,1], even prolongation is operated to the fractal interpolation function generated by iterated function system constituted by affine transform and Fourier cosine series representation of fractal interpolation function is proved. Second, for fractal interpolation function, odd prolongation is done and Fourier sine series formula of fractal interpolation function is proved. Final, Fourier series expansion of fractal interpolation function on the closed interval [Formula: see text] is proved. The result shows that complex fractal interpolation function can be represented by Fourier sine series and Fourier cosine series, so relatively simple Fourier series can be used to represent relatively complicated fractal interpolation function.

Reducing the near boundary errors of nonhomogeneous heat equations by boundary consistent methods

2020 ◽  
Author(s):  
Chein-Shan Liu ◽  
Chih-Wen Chang
Keyword(s):  
Heat Equations ◽  
Series Method ◽  
Sine Series ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Sine ◽  
Fourier Cosine Series ◽  
Fourier Sine Series ◽  
Gibbs Phenomena ◽  
Nonhomogeneous Heat

Abstract In the paper, we point out a drawback of the Fourier sine series method to represent a given odd function, where the boundary Gibbs phenomena would occur when the boundary values of the function are non-zero. We modify the Fourier sine series method by considering the consistent conditions on the boundaries, which can improve the accuracy near the boundaries. The modifications are extended to the Fourier cosine series and the Fourier series. Then, novel boundary consistent methods are developed to solve the 1D and 2D heat equations. Numerical examples confirm the accuracy of the boundary consistent methods, accounting for the non-zeros of the source terms and considering the consistency of heat equations on the boundaries, which can not only overcome the near boundary errors but also improve the accuracy of solution about four orders in the entire domain, upon comparing to the conventional Fourier sine series method and Duhamel’s principle.

Free In-Plane Vibration Analysis of Rectangular Plates With Elastically Point-Supported Edges

2010 ◽  
Vol 132 (3) ◽  
Author(s):  
Jingtao Du ◽  
Zhigang Liu ◽  
Wen L. Li ◽  
Xuefeng Zhang ◽  
Wanyou Li
Keyword(s):  
Fourier Series ◽  
Vibration Analysis ◽  
Series Representation ◽  
Fourier Method ◽  
Rectangular Plates ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Plane Vibration ◽  
Auxiliary Functions ◽  
Time In Literature

In comparison with the transverse vibrations of rectangular plates, far less attention has been paid to the in-plane vibrations even though they may play an equally important role in affecting the vibrations and power flows in a built-up structure. In this paper, a generalized Fourier method is presented for the in-plane vibration analysis of rectangular plates with any number of elastic point supports along the edges. Displacement constraints or rigid point supports can be considered as the special case when the stiffnesses of the supporting springs tend to infinity. In the current solution, each of the in-plane displacement components is expressed as a 2D Fourier series plus four auxiliary functions in the form of the product of a polynomial times a Fourier cosine series. These auxiliary functions are introduced to ensure and improve the convergence of the Fourier series solution by eliminating all the discontinuities potentially associated with the original displacements and their partial derivatives along the edges when they are periodically extended onto the entire x-y plane as mathematically implied by the Fourier series representation. This analytical solution is exact in the sense that it explicitly satisfies, to any specified accuracy, both the governing equations and the boundary conditions. Numerical examples are given about the in-plane modes of rectangular plates with different edge supports. It appears that these modal data are presented for the first time in literature, and may be used as a benchmark to evaluate other solution methodologies. Some subtleties are discussed about corner support arrangements.

scholarly journals The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections

2021 ◽  
pp. 146134842110196
Author(s):  
Guofang Li ◽  
Gang Wang ◽  
Junfang Ni ◽  
Liang Li
Keyword(s):  
Fourier Series ◽  
Cross Section ◽  
Cross Sections ◽  
Timoshenko Beam ◽  
Functionally Graded ◽  
Variable Cross ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
High Convergence Rate

In this study, an investigation on the free vibration of the beam with material properties and cross section varying arbitrarily along the axis direction is studied based on the so-called Spectro-Geometric Method. The cross-section area and second moment of area of the beam are both expanded into Fourier cosine series, which are mathematically capable of representing any variable cross section. The Young’s modulus, the mass density and the shear modulus varying along the lengthwise direction of the beam, are also expanded into Fourier cosine series. The translational displacement and rotation of cross section are expressed into the Fourier series by adding some polynomial functions which are used to handle the elastic boundary conditions with more accuracy and high convergence rate. According to Hamilton’s principle, the eigenvalues and the coefficients of the Fourier series can be obtained. Some examples are presented to validate the accuracy of this method and study the influence of the parameters on the vibration of the beam. The results show that the first four natural frequencies gradually decrease as the coefficient of the radius [Formula: see text] increases, and decreases as the gradient parameter n increases under clamped–clamped end supports. The stiffness of the functionally Timoshenko beam with arbitrary cross sections is variable compared with the uniform beam, which makes the vibration amplitude of the beam have different changes.

scholarly journals A Concretization of an Approximation Method for Non-Affine Fractal Interpolation Functions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 767
Author(s):  
Alexandra Băicoianu ◽  
Cristina Maria Păcurar ◽  
Marius Păun
Keyword(s):  
Approximation Method ◽  
Finite System ◽  
Countable System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Finite Systems ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Finite Set ◽  
Interpolation Functions

The present paper concretizes the models proposed by S. Ri and N. Secelean. S. Ri proposed the construction of the fractal interpolation function(FIF) considering finite systems consisting of Rakotch contractions, but produced no concretization of the model. N. Secelean considered countable systems of Banach contractions to produce the fractal interpolation function. Based on the abovementioned results, in this paper, we propose two different algorithms to produce the fractal interpolation functions both in the affine and non-affine cases. The theoretical context we were working in suppose a countable set of starting points and a countable system of Rakotch contractions. Due to the computational restrictions, the algorithms constructed in the applications have the weakness that they use a finite set of starting points and a finite system of Rakotch contractions. In this respect, the attractor obtained is a two-step approximation. The large number of points used in the computations and the graphical results lead us to the conclusion that the attractor obtained is a good approximation of the fractal interpolation function in both cases, affine and non-affine FIFs. In this way, we also provide a concretization of the scheme presented by C.M. Păcurar .

scholarly journals A Mode-Matching Solution for TE-Backscattering from an Arbitrary 2D Rectangular Groove in a PEC

2020 ◽  
Vol 20 (3) ◽  
pp. 159-163 ◽  
Author(s):  
Mehdi Bozorgi
Keyword(s):  
Singular Integrals ◽  
Logarithmic Singularity ◽  
Linear Equations ◽  
Mode Matching ◽  
Electric Conductor ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Highly Oscillatory Integrals ◽  
Rectangular Groove

In this paper, the simple yet effective mode-matching technique is utilized to compute TE-backscattering from a 2D filled rectangular groove in an infinite perfect electric conductor (PEC). The tangential magnetic fields inside and outside of the groove are represented as the sums of infinite series of cosine harmonics (half-range Fourier cosine series). By applying the continuity of the tangential magnetic field, these modes are matched on the groove to obtain the series coefficients by solving a system of linear equations. For this purpose, some oscillatory logarithmic singular integrals involving Hankel and trigonometric functions are solved numerically, starting by removing the logarithmic singularity via integration by parts. In the following, the new well-behaved highly oscillatory integrals are computed using efficient methods, and several comparisons are made to demonstrate the validity and ability of the presented procedure.

scholarly journals An Exploration of Fractal Based Prognostic Model and Comparative Analysis for Second Wave of COVID-19 Diffusion

2021 ◽  
Author(s):  
Easwaramoorthy D. ◽  
Gowrisankar A. ◽  
Manimaran A. ◽  
Nandhini S. ◽  
Santo Banerjee ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Death Rate ◽  
Transmission Rate ◽  
The United States ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Statistical Tool ◽  
Dimension Estimate ◽  
Fractal Interpolation Function ◽  
Second Wave

Abstract The coronavirus disease 2019 (COVID-19) pandemic has fatalized 216 countries across the world and has claimed the lives of millions of people globally. Researches are being carried out worldwide by scientists to understand the nature of this catastrophic virus and find a potential vaccine for it. The most possible efforts have been taken to present this paper as a form of contribution to the understanding of this lethal virus in the first and second wave. This paper presents a unique technique for the methodical comparison of disastrous virus dissemination in two waves amid five most infested countries and the death rate of the virus in order to attain a clear view on the behaviour of the spread of the disease. For this study, the dataset of the number of deaths per day and the number of infected cases per day of the most affected countries, The United States of America, Brazil, Russia, India, and The United Kingdom have been considered in first and second wave. The correlation fractal dimension has been estimated for the prescribed datasets of COVID-19 and the rate of death has been compared based on the correlation fractal dimension estimate curve. The statistical tool, analysis of variance has also been used to support the performance of the proposed method. Further, the prediction of the daily death rate has been demonstrated through the autoregressive moving average model. In addition, this study also emphasis a feasible reconstruction of the death rate based on the fractal interpolation function. Subsequently, the normal probability plot is portrayed for the original data and the predicted data, derived through the fractal interpolation function to estimate the accuracy of the prediction. Finally, this paper neatly summarized with the comparison and prediction of epidemic curve of the first and second waves of COVID-19 pandemic to picturize the transmission rate in the both times.

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