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.

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.

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 Torsional vibration of cracked carbon nanotubes with torsional restraints using Eringen’s nonlocal differential model

2018 ◽  
Vol 38 (1) ◽  
pp. 70-87 ◽  
Author(s):  
Mustafa Ö Yayli ◽  
Suheyla Y Kandemir ◽  
Ali E Çerçevik
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Torsional Vibration ◽  
Nonlocal Boundary ◽  
Coefficient Matrix ◽  
Vibration Frequencies ◽  
Sine Series ◽  
Differential Model ◽  
Fourier Sine ◽  
Fourier Sine Series

Free torsional vibration of cracked carbon nanotubes with elastic torsional boundary conditions is studied. Eringen’s nonlocal elasticity theory is used in the analysis. Two similar rotation functions are represented by two Fourier sine series. A coefficient matrix including torsional springs and crack parameter is derived by using Stokes’ transformation and nonlocal boundary conditions. This useful coefficient matrix can be used to obtain the torsional vibration frequencies of cracked nanotubes with restrained boundary conditions. Free torsional vibration frequencies are calculated by using Fourier sine series and compared with the finite element method and analytical solutions available in the literature. The effects of various parameters such as crack parameter, geometry of nanotubes, and deformable boundary conditions are discussed in detail.

P2438Deep learning model for atrial fibrillation detection based on single beat morphology

2019 ◽  
Vol 40 (Supplement_1) ◽  
Author(s):  
S W E Baalman ◽  
F E Schroevers ◽  
A Oakley ◽  
L A Ramos ◽  
R R Lopes ◽  
...  
Keyword(s):  
Atrial Fibrillation ◽  
Deep Learning ◽  
Sinus Rhythm ◽  
Learning Model ◽  
Heart Beat ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Specificity And Sensitivity ◽  
Fourier Cosine Series ◽  
Data Points

Abstract Background The electrocardiogram (ECG) is commonly used, but most recent rhythm discrimination algorithms still lack both specificity and sensitivity. Deep learning techniques have shown promising results in the classification of physiological signals like ECGs. Purpose To develop and test a deep learning (DL) model to discriminate between atrial fibrillation (AF) and sinus rhythm (SR). Methods For the development of the DL model we used 1499 ECGs sampled at 500 Hz of patients diagnosed with AF. All ECGs were labeled by two experienced investigators. Only ECGs labeled as SR or AF were included in the dataset. To simplify the learning process, solely the first ECG channel was used. The ECG waveforms were preprocessed using the Fourier cosine series to correct for baseline wander. Input data was generated by normalizing and scaling all different heartbeats by centralizing the R peak, leading to 15744 single heart beat samples of 80 data points (figure A). Multiple feedforward architectures were tested with different numbers of layers, filters and activation functions. The models were trained by equally splitting the data (50%SR, 50%AF) in a training (65%), validation (25%) and test set (15%). The best performing model was chosen based on the accuracy. Results A total of 1469 ECGs (1061 (72%)SR, 408 (28%)AF) were included. The model with the best performance was a feedforward model consisting three dense layers with ReLU activation and four dense layers with Linear activation. Training of the model was performed in 32 epochs. Validation of the model resulted in an accuracy of 96% (figure B), precision of 95% and recall of 96%. Conclusions The morphology based deep learning model developed in this study was able to discriminate atrial fibrillation from sinus rhythm with a fairly high accuracy using a limited size dataset and only one lead.

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