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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
Fri Dec 01 2017
Journal Name
Journal Of Economics And Administrative Sciences
Comparing the Sequential Nonlinear least squared Method and Sequential robust M method to estimate the parameters of Two Dimensional sinusoidal signal model:
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Estimation of the unknown parameters in 2-D sinusoidal signal model can be considered as important and difficult problem. Due to the difficulty to find estimate of all the parameters of this type of models at the same time, we propose sequential non-liner least squares method and sequential robust  M method after their development through the use of sequential  approach in the estimate suggested by Prasad et al to estimate unknown frequencies and amplitudes for the 2-D sinusoidal compounds but depending on Downhill Simplex Algorithm in solving non-linear equations for the purpose of obtaining non-linear parameters estimation which represents frequencies and then use of least squares formula to estimate

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Publication Date
Wed Mar 11 2026
Journal Name
Research On Crops
Effect of organic acid, amino acids and nano-fertilizer on growth, yield and nitrate concentration of lettuce plant under two farming systems
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Publication Date
Sat Feb 03 2018
Journal Name
Chinese Journal Of Physics
A true random number generator based on the photon arrival time registered in a coincidence window between two single-photon counting modules
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True random number generators are essential components for communications to be conconfidentially secured. In this paper a new method is proposed to generate random sequences of numbers based on the difference of the arrival times of photons detected in a coincidence window between two single-photon counting modules

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Publication Date
Thu Jun 15 2017
Journal Name
Journal Of Baghdad College Of Dentistry
Effects of Two Different Colorant Solutions on the Color Stability of Bleached Enamel in Association with CPP-ACPF: An In Vitro Study
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Background: One of the drawbacks of vital teeth bleaching is color stability. The aim of the present study was to evaluate the effects of tea and tomato sauce on the color stability of bleached enamel in association with the application of MI Paste Plus (CPP-ACPF). Materials and Methods: Sixty enamel samples were bleached with 10% carbamide peroxide for two weeks then divided into three groups (A, B and C) of 20 samples each. After bleaching, the samples of each group were subdivided into two subgroups (n=10). While subgroups A1, B1 and C1 were kept in distilled water, A2, B2, and C2 were treated with MI Paste Plus. Then, the samples were immersed in different solutions as follow: A1 and A2 in distilled water (control); B1 and B2 in black

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Publication Date
Fri Nov 11 2022
Journal Name
Journal Of Accounting And Financial Studies ( Jafs )
Ambidexteritg leadership behaviors in organizational excellence Comparative : applied research in the two public companies, Al-Zawraa and the electrical and electronic industries
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Business organizations strive to reach organizational excellence, in instruction to maintain their position in the market, so the research started from the problem of the weakness of the senior leaders in the two companies investigated in the management and sustainability of work, as well as their inability to face challenges that could affect the work of the two companies in the future, especially the lack of organizational of the future and the prediction of those events, so the two companies seek to explore and invest opportunities to achieve organizational excellence. The research aims to know the role of Ambidexterity  leadership behaviors in organizational excellence in my company .and create ideas that will achieve organizati

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Publication Date
Fri Oct 02 2020
Journal Name
Biochem. Cell. Arch.
MORPHOLOGICAL DESCRIPTION AND COMPARATIVE HISTOLOGICAL STUDY OF THE LIVER IN TWO IRAQI MAMMALS : WEASEL (HERPESTES JAVANICUS) AND EASTERN GRAY SQUIRREL (SCIURUS CAROLINENSIS
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The results of the current study showed that the liver of H. javanicus appeared as large lobulated organ divided into six distinct lobes, that filled the cranial region and little extended to the middle region of abdominal cavity. On the other hand, liver of S. carolinensis laid against the diaphragm, occupied the cranial region of the abdominal cavity and consisted of five lobes. The liver is surrounded with a thin capsule of dense regular collagenous connective tissue and few numbers of smooth muscles fibers can be seen in the capsule that covered the squirrel liver. The liver parenchyma divided into a large number of interconnected hepatic lobules marked only by the abundant amount of connective tissue bordered the triads, and within the

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Publication Date
Wed Mar 11 2026
Journal Name
Journal Of Baghdad College Of Dentistry
Comparison of regional bond strength of post space of fiber-reinforced post luted with two types of cements at different testing times
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Back ground: This in vitro study was carried out to investigate the effect of post space regions (coronal, middle and apical), Time and the mode of polymerization (dual, self-cured) of the cements used on the bond strength between translucent fiber post and root dentin by using push-out test. Materials and Methods: Forty eight extracted mandibular first premolars (single root) were instrumented with ProTaper Universal system files (for hand use) and obturated with gutta percha for ProTaper and AH26® root canal sealer following the manufacturer instructions, after 24 hours post space was prepared using FRC postec® plus drills no.3 creating 8 mm depth post space. The prepared samples were randomly divided into two main groups (24 samples ea

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Publication Date
Thu Feb 01 2018
Journal Name
Journal Of Economics And Administrative Sciences
Estimation of parameters of two-dimensional sinusoidal signal model by employing Deferential Evaluation algorithm and the use of Sequential approach in estimation
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Estimation the unknown parameters of a two-dimensional sinusoidal signal model is an important and a difficult problem , The importance of this model  in modeling Symmetric gray- scale texture image . In this paper, we propose employment Deferential Evaluation algorithm and the use of Sequential approach to estimate the unknown frequencies and amplitudes of the 2-D sinusoidal components when the signal is affected by noise. Numerical simulation are performed for different sample size, and various level of standard deviation to observe the performance of this method in estimate the parameters of 2-D sinusoidal signal model , This model was used for modeling  the Symmetric gray scale texture image and estimating by using

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Publication Date
Sun Jun 30 2019
Journal Name
Iraqi National Journal Of Nursing Specialties
Effect of Non-Pharmacological Pain Management Methods on Reduction the Duration of Labor Stages in Primigravida Women at AL-Elwyia Maternity Teaching Hospital
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Objectives: To evaluate the effect of non-pharmacological pain relief methods on duration of labor stage.Methodology: A quasi-experimental study design was conducted during the period of (4th July 2018 through 24th October 2018) on non-probability of (60) women (30) of them were a control group and (30) were the study group whom admitted to Al-Elwyia Maternity Teaching Hospital suffering from labor pain. A questionnaire was used as a tool of data collection Descriptive& Inferential statistical analyses were used to analyze the data.Result: The highest percentages of study and control groups were in age group (< 20) years old, primary schools graduates, housewife, from "urban area", within low category of socioeconomic scale,

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Publication Date
Thu Dec 15 2022
Journal Name
Al-academy
Strategies of brain-based learning theory and its impact on the achievement of students of the Department of Art Education in Teaching Methods
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The aim of the current research is to reveal the effect of using brain-based learning theory strategies on the achievement of Art Education students in the subject of Teaching Methods. The experimental design with two equal experimental and control groups was used. The experimental design with two independent and equal groups was used, and the total of the research sample was (60) male and female students, (30) male and female students represented the experimental group, and (30) male and female students represented the control group. The researcher prepared the research tool represented by the cognitive achievement test consisting of (20) questions, and it was characterized by honesty and reliability, and the experiment lasted (6) weeks

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