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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
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
Tue May 26 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
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
Tue Jul 01 2025
Journal Name
Journal Of Studies In Humanities And Educational Sciences
The Effect of the Needham Constructivist Model on the Achievement of Third Year Biology Department Students in the Subject of Teaching Methods
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The current research aims to know the effect of Needham's constructivist model on the achievement of third-year students in the Life Sciences Department in the teaching methods subject. To achieve the research objectives, the experimental method was followed for the experimental and control groups with dimensional measurement of the achievement variable of the teaching methods subject. The research sample included (62) students in the third year of the Life Sciences Department, distributed into two equal groups in the variables (self-assessment of learning methods - chronological age in years - intelligence level - previous information). To measure the level of students' achievement, an achievement test was constructed consisting of (40) te

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Publication Date
Sat Nov 30 2019
Journal Name
Journal Of Engineering And Applied Sciences
The use of Resource Consumption Accounting and Balanced Scorecard Methods as an Integrated Framework in Correcting the Economic Unit Performance: Applied Study
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Publication Date
Sat Oct 28 2023
Journal Name
Baghdad Science Journal
Qualitative and Quantitative Determination of Dapagliflozin Propanediol Monohydrate and Its Related Substances and Degradation Products Using LC-MS and Preparative Chromatography Methods
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Dapagliflozin is a novel sodium-glucose cotransporter type 2 inhibitor. This work aims to develop a new
validated sensitive RP-HPLC coupled with a mass detector method for the determination of dapagliflozin, its
alpha isomer, and starting material in the presence of dapagliflozin major degradation products and an internal
standard (empagliflozin). The separation was achieved on BDS Hypersil column (length of 250mm, internal
diameter of 4.6 mm and 5-μm particle size) at a temperature of 35℃. Water and acetonitrile were used as
mobile phase A and B by gradient mode at a flow rate of 1 mL/min. A wavelength of 224nm was selected to
perform detection using a photo diode array detector. The method met the

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Publication Date
Tue Jun 01 2021
Journal Name
Journal Of The College Of Languages (jcl)
The methods used to translate medical terms between Arabic and Spanish: Los métodos utilizados para traducir términos médicos entre árabe y español
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         The present paper deals with medical terms translation and its relationship with the medical text of Arabic and Spanish. Medical translation is the process of transferring texts related to the field of health and medicine to achieve an accurate effective translation from the source language text to the equivalent target language text. The most prominent medical translations are from English to Arabic as most of the syllabuses in Arab countries are taught in English.

       Translation is an innovative work intended to render the original text in the source language into the target language with the highest level of linguistic and intellec

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Publication Date
Wed Oct 17 2018
Journal Name
Journal Of Economics And Administrative Sciences
Effect Of Using Quantitative Methods Of Demand Forecasting In Improving Of Supply Chain Performance:" Case Study In One Of An Industerial Organization"
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Objecte The study aims to test the effect of using the appropriate quantitative method of demand forecasting in improving the performance of supply chain of the aviation fuel product ( The study sample), One of the products of the Doura refinery (The study site), By testing a set of quantitative methods of demand forecasting using forecasting error measurements, and choosing the least faulty, most accurate and reliable method and adept it in the building  chain.

Is the study of problem through a starting with the fol

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Publication Date
Wed Aug 01 2018
Journal Name
Journal Of Economics And Administrative Sciences
A Comparative Study of Some Methods of Estimating Robust Variance Covariance Matrix of the Parameters Estimated by (OLS) in Cross-Sectional Data
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Abstract

The Classical Normal Linear Regression Model Based on Several hypotheses, one of them is Heteroscedasticity as it is known that the wing of least squares method (OLS), under the existence of these two problems make the estimators, lose their desirable properties, in addition the statistical inference becomes unaccepted table. According that we put tow alternative,  the first one is  (Generalized Least Square) Which is denoted by (GLS), and the second alternative is to (Robust covariance matrix estimation) the estimated parameters method(OLS), and that the way (GLS) method neat and certified, if the capabilities (Efficient) and the statistical inference Thread on the basis of an acceptable

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Publication Date
Wed Jan 01 2025
Journal Name
Smart Innovation, Systems And Technologies
The Effect of Two Training Approaches with (Hanging–Heavy) Ropes in Developing Some Physical Abilities and Accuracy of Shooting—In Front of Handball
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