The Gibbs' phenomenon

Fay, Temple H.; Kloppers, P. Hendrik

doi:10.1080/00207390150207077

Cited by 4 publications

(10 citation statements)

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“…Classroom notes Predictably, the other solutions do not do as well, 1 …P † º ¡0:0266 2 …P † º 0:000 600 3…”

Section: Numerical Solutionsmentioning

confidence: 91%

“…Thus we only obtain b n with odd subscripts that are signi®cant; we list the ®rst eight below. 3)Ðat least to 11 decimal places (more decimal places can be obtained by using higher precision calculations, see [1]). Of course this solution has period P: But moreover, from substituting f …t † back into the di erential equation, we obtain a residual everywhere less than 3:5 £ 10 ¡12 , thus suggesting 11 decimal place accuracy (see ®gure 13).…”

Section: Fourier Series Solutionsmentioning

confidence: 99%

“…This technique is simple and discussed in some depth in [1]. The basic idea is to use the numerical solution over the early portion, say 0 4 t 4 10; to estimate the period P of the solution and hence the frequency !ˆ2º=P: Once we know the frequency, we estimate the Fourier coe cients at that frequency from the numerical data over the time interval 0 4 t 4 P: This simple approach produces very good results as we will demonstrate.…”

Section: Fourier Series Solutionsmentioning

confidence: 99%

“…This work is a continuation of the ideas presented in [1] and is suitable for classroom discussion/presentation.…”

Section: Introductionmentioning

confidence: 96%

“…Thus we are led naturally to discuss numerical solutions to these equations. Due to the nonlinearity, we cannot expect numerical solutions to remain accurate over long time intervals, hence we use the approach from [1] that permits the calculation of (truncated) Fourier series solutions that have a bounded small error over the entire real line. Moreover, we can solve the large angle de¯ection pendulum problem as easily as we solve the small angle de¯ection problem.…”

Section: Introductionmentioning

confidence: 99%

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The pendulum equation

Fay¹

2002

International Journal of Mathematical Education in Science and

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We investigate the pendulum equation ‡ ¶ 2 sin ˆ0 and two approximations for it. On the one hand, we suggest that the third and ®fth-order Taylor series approximations for sin do not yield very good di erential equations to approximate the solution of the pendulum equation unless the initial conditions are appropriately chosen very small and the time interval is short. On the other, we suggest that computationally, there is no advantage taking these approximations. We further justify this by employing an approach to deriving Fourier series approximations to the pendulum equation accurate to at least eleven decimal places. Students can generate highly accurate Fourier series solutions to nonlinear equations and thus concentrate on the qualitative aspects of the model rather than the computational di culties.

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“…Classroom notes Predictably, the other solutions do not do as well, 1 …P † º ¡0:0266 2 …P † º 0:000 600 3…”

Section: Numerical Solutionsmentioning

confidence: 91%

Section: Fourier Series Solutionsmentioning

confidence: 99%

Section: Fourier Series Solutionsmentioning

confidence: 99%

“…This work is a continuation of the ideas presented in [1] and is suitable for classroom discussion/presentation.…”

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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The pendulum equation

Fay¹

2002

International Journal of Mathematical Education in Science and

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Reducing power line noise in EEG and MEG data via spectrum interpolation

2019

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Electroencephalographic (EEG) and magnetoencephalographic (MEG) signals can often be exposed to strong power line interference at 50 or 60 Hz. A widely used method to remove line noise is the notch filter, but it comes with the risk of potentially severe signal distortions. Among other approaches, the Discrete Fourier Transform (DFT) filter and CleanLine have been developed as alternatives, but they may fail to remove power line noise of highly fluctuating amplitude. Here we introduce spectrum interpolation as a new method to remove line noise in the EEG and MEG signal. This approach had been developed for electromyographic (EMG) signals, and combines the advantages of a notch filter, while synthetic test signals indicate that it introduces less distortion in the time domain. The effectiveness of this method is compared to CleanLine, the notch (Butterworth) and DFT filter. In order to quantify the performance of these three methods, we used synthetic test signals and simulated power line noise with fluctuating amplitude and abrupt on- and offsets that were added to an MEG dataset free of line noise. In addition, all methods were applied to EEG data with massive power line noise due to acquisition in unshielded settings. We show that spectrum interpolation outperforms the DFT filter and CleanLine, when power line noise is nonstationary. At the same time, spectrum interpolation performs equally well as the notch filter in removing line noise artifacts, but shows less distortions in the time domain in many common situations.

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Using the homotopy method to find periodic solutions of forced nonlinear differential equations

Fay¹,

Lott²

2002

International Journal of Mathematical Education in Science and

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This paper discusses a result of Li and Shen which proves the existence of a unique periodic solution for the di erential equationwhere k is a constant; g is continuous, continuously di erentiable with respect to x; and is periodic of period P in the variable t; " … t † is continuous and periodic of period P; and when @g=@x satis®es some additional boundedness conditions. This means that there exist initial values x … 0 †ˆ¬ ¤ and _ x x … 0 †ˆ ¤ so that the solution to the corresponding initial value problem is periodic of period P and is unique (up to a translation of the time variable) with this property. The proof of this result is constructive, so that starting with any initial conditions x … 0 †ˆ¬ and _ x x … 0 †ˆ ; a path in the phase plane can be produced, starting at … ¬; † and terminating at … ¬ ¤ ; ¤ † : Both the theoretical proof and a constructive proof are discussed and a Mathematica implementation developed which yields an algorithm in the form of a Mathematica notebook (which is posted on the webpage http://pax.st.usm.edu/downloads). The algorithm is robust and can be used on di erential equations whose terms do not satisfy Li and Shen's hypotheses. The ideas used reinforce concepts from beginning courses in ordinary di erential equations, linear algebra, and numerical analysis.

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The Gibbs' phenomenon

Cited by 4 publications

References 0 publications

The pendulum equation

The pendulum equation

Reducing power line noise in EEG and MEG data via spectrum interpolation

Using the homotopy method to find periodic solutions of forced nonlinear differential equations

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