An analytical technique for solving general linear integral equations of the second kind and its application in analysis of flash lamp control circuit

Hetmaniok, Edyta�; Słota, Damian; Trawiński, T.; Wituła, Roman

doi:10.2478/bpasts-2014-0043

Cited by 3 publications

(6 citation statements)

References 38 publications

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“…It concerns the case of nonlinear equations as well. Whereas for h = −1 the method is identical with the homotopy perturbation method (see [23][24][25]). …”

Section: Theorem 8 If Inequality (25) Is Fulfilled and N ∈ N Then We mentioning

confidence: 99%

“…Respective theorems can be then formulated in the following way. (23) and (24). Then, if series in (7) is convergent, the sum of this series is the solution of (14).…”

Section: Linear Integral Equationmentioning

confidence: 99%

“…Obtained results indicate that the method is very rapidly convergent and calculation of only few first terms of the series ensures very good approximation of the exact solution. Example 2 In the next example we deal with equation which may be practically applied for calculating the charge in supply circuit of flash lamps used in cameras [15,24]. Supply circuit of flash lamps may be pictured as a simple electrical circuit consisting of source and series connected ideal switch, resistor and capacitor.…”

Section: Examplesmentioning

confidence: 99%

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Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind

et al. 2013

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The paper presents an application of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind. In this method a series is created, sum of which (if the series is convergent) gives the solution of discussed equation. Conditions ensuring convergence of this series are presented in the paper. Error of approximate solution, obtained by considering only partial sum of the series, is also estimated. Examples illustrating usage of the investigated method are presented as well, including the example having practical application for calculating the charge in supply circuit of flash lamps used in cameras.

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“…It concerns the case of nonlinear equations as well. Whereas for h = −1 the method is identical with the homotopy perturbation method (see [23][24][25]). …”

Section: Theorem 8 If Inequality (25) Is Fulfilled and N ∈ N Then We mentioning

confidence: 99%

“…Respective theorems can be then formulated in the following way. (23) and (24). Then, if series in (7) is convergent, the sum of this series is the solution of (14).…”

Section: Linear Integral Equationmentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

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Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind

et al. 2013

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“…The most common methods used to solve this equation include harmonic balance [1,3,24] and multiscale methods [25], averaging method [26], discrete singular convolution [27], the homotopy perturbation method [28,29] and the perturbation method (the small parameter method or the Poincaré method) [30,31]. …”

Section: Introductionmentioning

confidence: 99%

Dynamic stability of moderately thick beams and frames with the use of harmonic balance and perturbation methods

Obara

Gilewski

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. The objective of this paper is to determine dynamic instability areas of moderately thick beams and frames. The effect of moderate thickness on resonance frequencies is considered, with transverse shear deformation and rotatory inertia taken into account. These relationships are investigated using the Timoshenko beam theory. Two methods, the harmonic balance method (HBM) and the perturbation method (PM) are used for analysis. This study also examines the influence of linear dumping on induced parametric vibration. Symbolic calculations are performed in the Mathematica programme environment. The majority of studies reported in the literature use numerical analysis for determining resonance areas. Only a few researchers have adopted an analytical approach.The purpose of this article is to propose two methods for determining parametric resonance areas for moderately thick beams and frames under various support conditions. These methods, never before used for this purpose, are the harmonic balance method (HBM) and the perturbation method (PM). The latter of these two methods represents a surprisingly simple tool for achieving the goal, as its results are very close to those obtained from the standard harmonic balance method which is far more troublesome in application.An analytical approach is used to analyze simply supported moderately thick beams, and a numerical technique, based on the finite element method with the use of physical shape functions [32] is applied to other beams and frames. The analysis covers the effect of moderate thickness on resonance frequencies and the influence of type of fixing used in the beams and frames on the instability areas. For this purpose, shear deformation and rotatory inertia are taken into account. The Timoshenko beam theory is applied to examine how these factors affect resonance frequency values. The results are compared with those obtained with the Bernoulli-Euler beam theory.In addition, the effect of linear dumping of induced parametric vibration is considered. Analysis of a simply supported beamThe following assumptions are adopted in physical and numerical modeling:• The beam is made of an isotropic homogenous linear elastic material with Young's modulus E, shear modulus G and Poisson's ratio v.

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“…This method has found an application for solving many problems formulated with the aid of ordinary and partial differential equations [24][25][26][27], including the heat conduction problems [28][29][30][31], fractional differential equations [32,33] (for some other applications of the fractional calculus see for example [34][35][36]), integral equations [37][38][39], integro-differential equations [40,41] and others. A particular case of the homotopy analysis method is the homotopy perturbation method [16,17,42].…”

Section: Introductionmentioning

confidence: 99%

An analytical method for solving the two-phase inverse Stefan problem

Hetmaniok¹,

Słota²,

Wituła³

et al. 2015

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. In the paper we present an application of the homotopy analysis method for solving the two-phase inverse Stefan problem. In the proposed approach a series is created, having elements which satisfy some differential equation following from the investigated problem. We reveal, in the paper, that if this series is convergent then its sum determines the solution of the original equation. A sufficient condition for this convergence is formulated. Moreover, the estimation of the error of the approximate solution, obtained by taking the partial sum of the considered series, is given. Additionally, we present an example illustrating an application of the described method.

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An analytical technique for solving general linear integral equations of the second kind and its application in analysis of flash lamp control circuit

Cited by 3 publications

References 38 publications

Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind

Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind

Dynamic stability of moderately thick beams and frames with the use of harmonic balance and perturbation methods

An analytical method for solving the two-phase inverse Stefan problem

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