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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 Feb 28 2026
Journal Name
Al–bahith Al–a'alami
International Public Relations methods in building the state's image through Cyberspace
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The researcher has studied in his research (International Public Relations methods in building the state's image through Cyberspace)
, analytical study of the Facebook and twitter pages for British foreign office , the role was played by the International Public Relations in building the mental image of British , especially after the new media and internet have became influential role in political life . and became an important tools used by political institutions as ministries of foreign affairs in the twenty: one century .
The researcher identified the problem of this study with the following question:
(what is the role of the International Public Relations in building the mental image of state through Cyberspace)
To answer

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Publication Date
Sun Jul 02 2017
Journal Name
Journal Of Educational And Psychological Researches
Down Syndrome causes, symptoms, and most important methods of functional treatment
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This study is descriptive and theory of Dawn syndrome as the problem of research lies in the need to identify the identification of the causes of Dawn syndrome and its symptoms and methods of dealing with it, which has become a problem that needs treatment, especially after the numbers have become high in Iraq, which has not yet taken the necessary importance for treatment and care.

The objectives of the research were summarized in the identification of the most important causes of Dawn syndrome and its symptoms and diagnosis and ways or methods of dealing with people with Dawn syndrome in order to develop therapeutic plans for him.

       

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Publication Date
Mon Aug 17 2020
Journal Name
International Journal Of Applied Mechanics And Engineering
Analysis of Structural Concrete Bar Members Based on Secant Stiffness Methods
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Abstract<p>In this paper, the behavior of structural concrete linear bar members was studied using numerical model implemented in a computer program written in MATLAB. The numerical model is based on the modified version of the procedure developed by Oukaili. The model is based on real stress-strain diagrams of concrete and steel and their secant modulus of elasticity at different loading stages. The behavior presented by normal force-axial strain and bending moment-curvature relationships is studied by calculating the secant sectional stiffness of the member. Based on secant methods, this methodology can be easily implemented using an iterative procedure to solve non-linear equations. A compari</p> ... Show More
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Publication Date
Sat Feb 28 2026
Journal Name
Journal Mustansiriyah Of Sports Science
The effect of using Daniel's model for people with two types of brain control (left and right) to learn the skill of the Cartwheel in artistic gymnastics for second-stage students
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The research problem focused through the researcher's experience in the gymnastics game and the lack of use of educational models that give the student an important role in the educational process, so it became necessary to identify the type of prevailing style for students, and the need for diversity in the use of educational models based on scientific theories, including the Daniel Document model. Based on three theories of learning, which are structural, behavioral, and meaningful learning. The research aimed to identify the effect of using the Daniel model for people with two types of brain control (left and right) to learn the skill of the Cartwheel in artistic gymnastics for students of the second stage. The researcher used the experi

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Publication Date
Sun Jun 01 2014
Journal Name
International Journal Of Physics And Research (ijpr)
CALCULATIONS OF MOLTIPOLE MIXING RATIOS FOR GAMMA TRANSITIONS OF Yb POPULATED FROM Yb REACTION USING -RATIO, CONSTANT STATISTICAL TENSOR AND LEAST SQUARES FITTING METHODS
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The - mixing ratios of -transitions from levels in populated in the reactions are calculated in present work using - ratio, constant statisticalTensor and least squares fitting methods The results obtained are in general, in good agreement or consistent, within the associated uncertainties, with these reported in Ref.[9],the discrepancies that occurs are due to inaccuracy existing in the experimental data The results obtained in the present work confirm the –method for mixed transitions better than that for pure transition because this method depends only on the experimental data where the second method depends on the pure or those considered to be pure -transitions, the same results occur in – method

Publication Date
Wed Aug 01 2018
Journal Name
Collegian
Behaviour change interventions to improve medication adherence in patients with cardiac disease: Protocol for a mixed methods study including a pilot randomised controlled trial
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Publication Date
Wed Mar 16 2022
Journal Name
Journal Of Educational And Psychological Researches
The Effectiveness of an Educational Program in Light Of Behavioral Cognitive Theory to Develop Efficient Response to Students Affected by Crises
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The aim of this research is to construct an educational program in light of the theory of behavioral cognitive and its impact on the development of the efficient response to students affected by crises (centers of your right to education). To achieve the objectives of the research, two scales were developed by the researcher in addition to two equivalent hypotheses were formulated. The scale contains (26) items divided into five fields; for its validity and reliability were derived based on the measure of efficient response, an educational program based on the theory of behavioral cognition. The test and the educational program were applied to a sample of (60) students from the centers of your right to education, divided into experimenta

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Publication Date
Thu Apr 01 2021
Journal Name
Basra Journal Of Science
Organic Field Effect Transistor Based on P3HT with Two Different Gate Dielectrics
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The electrical performance of bottom-gate/top source-drain contact for p-channel organic field-effect transistors (OFETs) using poly(3-hexylthiophene) (P3HT) as an active semiconductor layer with two different gate dielectric materials, Polyvinylpyrrolidone (PVP) and Hafnium oxide (HfO2), is investigated in this work. The output and transfer characteristics were studied for HfO2, PVP and HfO2/PVP as organic gate insulator layer. Both characteristics show a high drain current at the gate dielectric HfO2/PVP equal to -0.0031A and -0.0015A for output and transfer characteristics respectively, this can be attributed to the increasing of the dielectric capacitance. Transcondactance characteristics also studied for the three organic mater

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Publication Date
Sat Feb 02 2019
Journal Name
Journal Of The College Of Education For Women
Design of an Electron Gun Using two Electrod Low Aberration Immersion Lens
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Abstract
A two electrode immersion electrostatic lens used in the design
of an electron gun, with small aberration, has been designed using
the finite element method (FEM). By choosing the appropriate
geometrical shape of there electrodes the potential V(r,z) and the
axial potential distribution have been computed using the FEM to
solve Laplace's equation.
The trajectory of the electron beam and the optical properties of
this lens combination of electrodes have been computed under
different magnification conditions (Zero and infinite magnification
conditions) from studying the properties of the designed electron
gun can be supplied with Abeam current of 5.7*10-6 A , electron
gun with half acceptance

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Publication Date
Sat Feb 28 2026
Journal Name
Iraqi Journal Of Science
UNSTEADY PRESSURE DROP AND HEAT TRANSFER OFMAGNETOHYDRODYNAMIC ANNULAR TWO-PHASE INRECTANGULAR CHANNEL
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An annular two-phase, steady and unsteady, flow model in which a conductingfluid flow under the action of magnetic field is concavely. Two models arepresented, in the model one; the magnetic field is perpendicular to the long side ofthe channel, while in the model two is perpendicular to the short side. Also, westudy, to some extent the single-phase liquid flow.It is found that the motion and heat transfer equations are controlled by differentdimensionless parameters namely, Reynolds, Hartmann, Prandtl, and Poiseuilleparameters. The Laplace transform technique is used to solve each of the motion andheat transfer equations. The effects of each of dimensionless parameters upon thevelocity and heat transfer is analyzed.A comprehensive study fo

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