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Two efficient methods for solving Schlömilch’s integral equation
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Purpose

In this paper, the exact solutions of the Schlömilch’s integral equation and its linear and non-linear generalized formulas with application are solved by using two efficient iterative methods. The Schlömilch’s integral equations have many applications in atmospheric, terrestrial physics and ionospheric problems. They describe the density profile of electrons from the ionospheric for awry occurrence of the quasi-transverse approximations. The paper aims to discuss these issues.

Design/methodology/approach

First, the authors apply a regularization method combined with the standard homotopy analysis method to find the exact solutions for all forms of the Schlömilch’s integral equation. Second, the authors implement the regularization method with the variational iteration method for the same purpose. The effectiveness of the regularization-Homotopy method and the regularization-variational method is shown by using them for several illustrative examples, which have been solved by other authors using the so-called regularization-Adomian method.

Findings

The implementation of the two methods demonstrates the usefulness in finding exact solutions.

Practical implications

The authors have applied the developed methodology to the solution of the Rayleigh equation, which is an important equation in fluid dynamics and has a variety of applications in different fields of science and engineering. These include the analysis of batch distillation in chemistry, scattering of electromagnetic waves in physics, isotopic data in contaminant hydrogeology and others.

Originality/value

In this paper, two reliable methods have been implemented to solve several examples, where those examples represent the main types of the Schlömilch’s integral models. Each method has been accompanied with the use of the regularization method. This process constructs an efficient dealing to get the exact solutions of the linear and non-linear Schlömilch’s integral equation which is easy to implement. In addition to that, the accompanied regularization method with each of the two used methods proved its efficiency in handling many problems especially ill-posed problems, such as the Fredholm integral equation of the first kind.

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Publication Date
Sat Nov 01 2025
Journal Name
Chemical Engineering Research And Design
Efficient degradation of methyl red in wastewater via induced electro-Fenton process
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Publication Date
Wed Jan 01 2025
Journal Name
Journal Of Computer Sciences And Informatics
Edge Detection Methods: A Review
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This article studies a comprehensive methods of edge detection and algorithms in digital images which is reflected a basic process in the field of image processing and analysis. The purpose of edge detection technique is discovering the borders that distinct diverse areas of an image, which donates to refining the understanding of the image contents and extracting structural information. The article starts by clarifying the idea of an edge and its importance in image analysis and studying the most noticeable edge detection methods utilized in this field, (e.g. Sobel, Prewitt, and Canny filters), besides other schemes based on distinguishing unexpected modifications in light intensity and color gradation. The research as well discuss

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Publication Date
Wed Dec 01 2010
Journal Name
Iraqi Journal Of Physics
Study of electron energy distribution function and transport parameters for CF4 and Ar gases discharge by using the solution of Boltzmann equation-Part I
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The Boltzmann transport equation is solved by using two- terms approximation for pure gases . This method of solution is used to calculate the electron energy distribution function and electric transport parameters were evaluated in the range of E/N varying from . 172152110./510.VcmENVcm
From the results we can conclude that the electron energy distribution function of CF4 gas is nearly Maxwellian at (1,2)Td, and when E/N increase the distribution function is non Maxwellian. Behavior of electrons transport parameters is nearly from the experimental results in references. The drift velocity of electron in carbon tetraflouride is large compared with other gases

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Publication Date
Wed Dec 01 2010
Journal Name
Iraqi Journal Of Physics
Study of electron energy distribution function and transport parameters for CF4, Ar gases mixture discharge by using the solution of Boltzmann equation-Part II
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The Boltzmann transport equation is solved by using two- terms approximation for pure gases and mixtures. This method of solution is used to calculate the electron energy distribution function and electric transport parameters were evaluated in the range of E/N varying from . 172152110./510.VcmENVcm
The electron energy distribution function of CF4 gas is nearly Maxwellian at (1,2)Td, and when E/N increase the distribution function is non Maxwellian. Also, the mixtures are have different energy values depending on transport energy between electron and molecule through the collisions. Behavior of electrons transport parameters is nearly from the experimental results in references. The drift velocity of electron in carbon tetraflouride i

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Publication Date
Sat Apr 09 2022
Journal Name
Engineering, Technology & Applied Science Research
A Semi-Empirical Equation based on the Strut-and-Tie Model for the Shear Strength Prediction of Deep Beams with Multiple Large Web Openings
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The behavior and shear strength of full-scale (T-section) reinforced concrete deep beams, designed according to the strut-and-tie approach of ACI Code-19 specifications, with various large web openings were investigated in this paper. A total of 7 deep beam specimens with identical shear span-to-depth ratios have been tested under mid-span concentrated load applied monotonically until beam failure. The main variables studied were the effects of width and depth of the web openings on deep beam performance. Experimental data results were calibrated with the strut-and-tie approach, adopted by ACI 318-19 code for the design of deep beams. The provided strut-and-tie design model in ACI 318-19 code provision was assessed and found to be u

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Publication Date
Wed Jan 30 2019
Journal Name
Journal Of The College Of Education For Women
Detecting the effectiveness of two tools for detecting the talented from their teachers' points of view
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Education specialists have differed about determining the best ways to detect the
talented. Since the appearance of the mental and psychological measurement movement, some
scholars adopted intelligence ratios as a criterion to identify the talented and others went to
rely on the degree of academic achievement. Each of these two methods has its own flaws and
mistakes and a large number of talented children were victims of these two methods.
Therefore the need to use other scales for the purpose of detection of talented children
appeared because they provide valuable information which may not be obtained easily
through objective tests and these scales are derived from consecutive studies of gifted andtalented children

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Publication Date
Sat Dec 01 2012
Journal Name
Iraqi Journal Of Physics
The transition rates for 232Th using the two component particle-hole state density with different corrections
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The particle-hole state densities have been calculated for 232Th in
the case of incident neutron with  ,  1 Z Z T T T T and   2 Z T T .
The finite well depth, surface effect, isospin and Pauli correction are
considered in the calculation of the state densities and then the
transition rates. The isospin correction function ( ) iso f has been
examined for different exciton configurations and at different
excitation energies up to 100 MeV. The present results are indicated
that the included corrections have more affected on transition rates
behavior for        , , and    above 30MeV excitation energy

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Publication Date
Sat Jul 01 2023
Journal Name
Electric Power Systems Research
Analytical and measurement-based wideband two-port modeling of DC-DC converters for electromagnetic transient studies
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Power-electronic converters are essential elements for the effective interconnection of renewable energy sources to the power grid, as well as to include energy storage units, vehicle charging stations, microgrids, etc. Converter models that provide an accurate representation of their wideband operation and interconnection with other active and passive grid components and systems are necessary for reliable steady state and transient analyses during normal or abnormal grid operating conditions. This paper introduces two Laplace domain-based approaches to model buck and boost DC-DC converters for electromagnetic transient studies. The first approach is an analytical one, where the converter is represented by a two-port admittance model via mo

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Publication Date
Sun Sep 25 2022
Journal Name
American Scientific Research Journal For Engineering, Technology, And Sciences
Design and Fabrication of an Electromechanical Tester to Perform Two-dimensional Tensile Testing for Flexible Materials
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There are many diseases that affect the arteries, especially those related to their elasticity and stiffness, and they can be guessed by estimating and calculating the modulus of elasticity. Hence, the accurate calculation of the elastic modulus leads to an accurate assessment of these diseases, especially in their early stages, which can contribute to the treatment of these diseases early. Most of the calculations used the one-dimensional (1D) modulus of elasticity. From a mechanical point of view, the stresses to which the artery is subjected are not one-dimensional, but three-dimensional. Therefore, estimating at least a two-dimensional (2D) modulus of elasticity will necessarily be more accurate. To the knowledge of researchers, there i

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Publication Date
Mon Dec 31 2012
Journal Name
Al-khwarizmi Engineering Journal
Genetic Algorithm Based PID Controller Design for a Precise Tracking of Two-Axis Piezoelectric Micropositioning Stage
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 In this paper, an intelligent tracking control system of both single- and double-axis Piezoelectric Micropositioner stage is designed using Genetic Algorithms (GAs) method for the optimal Proportional-Integral-Derivative (PID) controller tuning parameters. The (GA)-based PID control design approach is a methodology to tune a (PID) controller in an optimal control sense with respect to specified objective function. By using the (GA)-based PID control approach, the high-performance trajectory tracking responses of the Piezoelectric Micropositioner stage can be obtained. The (GA) code was built and the simulation results were obtained using MATLAB environment. The Piezoelectric Micropositioner simulation model with th

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