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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 Jan 01 2022
Journal Name
Specialusis Ugdymas
The Effect of the Idea Filtering Strategy in Second Grade Middle School Female Students for the Subject of Science in their productive habits of the mind
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RHS Nasser, NHY Al-Afoun, SPECIALUSIS UGDYMAS, 2022

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Publication Date
Mon Apr 01 2024
Journal Name
Materials Science In Semiconductor Processing
Rational design of novel 0D/0D Bi2Sn2O7/CeO2 in the core-shell nanostructure for boosting the photocatalytic decomposition of antibiotics in wastewater: S-type-based mechanism
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Publication Date
Mon Sep 08 2025
Journal Name
Retos
The effect of mental training for sensory perceptions of skill performance in some indicators of electrical activity and the special physical abilities of young pole vaulters
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Objective: The aim of the research is to prepare skilled mental exercises according to spatial and temporal perceptions and the awareness of the strength of young pole-vaulters, and to recognize the impact of these exercises on improving the indicators of electrical activity of working muscles and some special physical abilities and accomplishing this effectiveness Research methodology: The researcher used the experimental curriculum (one experimental group), and included the sample of research on (5) two joint jumpers in the Iraqi club championship, all from the center of talent in athletics, the sample is trained on the same curriculum prepared by the coach himself but accompanied by a mental training approach that Prepared by the

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Crossref
Publication Date
Tue Apr 16 2024
Journal Name
International Development Planning Review
RESISTANCE TRAINING IN THE WEIGHT OF DIFFERENT BODY PARTS AND THEIR EFFECT ON SOME MECHANICAL BIO VARIABLES AND THE ACHIEVEMENT OF THE HIGH JUMP FOR YOUNG PEOPLE
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The study aims to identify the effect of training with weight added to different parts of the body in improving the biomechanical variables of the last step and upgrading the high jump for young people, the research was applied to one of the elite high jumpers for young people, and video imaging and kinetic analysis were conducted to extract the variables under study, and then the jumper underwent For the proposed exercises by 3 training units per week for a period of (8) weeks, in which emphasis was placed on improving the rotational forces (moments) of these parts, and then the post-tests were conducted, and the researcher concluded that the exercises applied and the added weights for all parts of the body (leg, thigh, trunk, and arms) le

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Publication Date
Sun Sep 28 2025
Journal Name
Revista Iberoamericana De Psicología Del Ejercicio Y El Deporte, Issn 1886-8576, Vol. 17, Nº. 6, 2022, Págs. 402-404
The Effect Exercises Of Stretching Featured Strength On Special Flexibility, Insulin Hormone, And The Achievement Of The Snatch, Clean And Jerk Lifts For Advanced Lifters, Baghdad Clubs
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Autorías: Ghassan Adeeb Abdulhasan, Rasha Raed Hamid Hameed, Hussein Jabber Abood. Localización: Revista iberoamericana de psicología del ejercicio y el deporte. Nº. 6, 2022. Artículo de Revista en Dialnet.

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Scopus
Publication Date
Tue Nov 09 2021
Journal Name
Journal Of Accounting And Financial Studies ( Jafs )
Strategic leadership and its role in achieving competitiveness in industrial establishments": "A field study in Bashir Al-Siksik & Co. for the manufacture of sanitary and plastic tools
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This study aims to know the role of strategic leadership to achieving competitiveness in industrial establishments by identifying the respondents’ perceptions about the level of availability of dimensions of leadership strategies (creativity and innovation, risk tolerance, available opportunities) in Bashir Al-Siksek & Partners Company for the manufacture of sanitary and plastic ware in Gaza strip

To achieve this, a questionnaire was developed and distributed to a sample of managers, auditors, accountants, and administrative employees in the study sample company. The questionnaire tool was distributed to 60 employees and employees, of which (52) were retrieved, or 86.6%, and (8) were excluded for la

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Publication Date
Sun Sep 28 2025
Journal Name
Revista Iberoamericana De Psicología Del Ejercicio Y El Deporte, Issn 1886-8576, Vol. 16, Nº. 3, 2021
The effect of a rehabilitation program using a tens device for the treatment of pain and impaired mobility of the wrist and limbs joint in basketball players
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Autorías: Hadeer Idan Ghanim, Ishraq Mahmood. Localización: Revista iberoamericana de psicología del ejercicio y el deporte. Nº. 3, 2021. Artículo de Revista en Dialnet.

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Scopus
Publication Date
Fri Jul 21 2023
Journal Name
Journal Of Engineering
The designed housing projects as an alternative for the informal building and their impact in addressing the heterogeneity of the urban escape in the built residential areas
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The phenomenon of informal building Spread recently in Iraqi residential areas, in general, and in Baghdad, in particular, due to the urgent housing need, on the one hand, and lack of commitment to building controls, on the other hand, to highlight the phenomenon of uncommitted building to controls and housing governing legislation in Iraq, leading to heterogeneity in both building densities and plot areas, and disorder in the urban fabric and urban escape of those areas. Research problem identified as the absence of a clear vision about the General aspects of the phenomenon of informal building in residential street scene, and the role of designed housing projects as a substitute for informal building in built residential areas. The des

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Crossref
Publication Date
Fri Oct 21 2022
Journal Name
International Journal Of Professional Business Review
The Impact of Dumping Policy on the Food Gap of Chicken Meat in Iraq For the Period (2004-2019) - Turkish Imports Of Chicken Meat a Case Study
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Purpose: As a result of the sudden and ill-considered trade openness of Iraq after 2003 to the countries of the world in general and the neighboring countries in particular, and in the absence of the necessary support for the national productive forces and the lack of effective standardization and quality control devices, this led to the exposure of most local products, especially agricultural ones, to decline and inability On the competition and thus dumping the Iraqi market, especially the agricultural products, with imported products, this study came to find out the effect that dumping has on the local production of chicken meat and the impact of that impact on the size of the food gap, and whether the results of the practica

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Scopus (2)
Crossref (1)
Scopus Crossref
Publication Date
Thu Feb 09 2023
Journal Name
Journal Of Educational And Psychological Researches
An Evaluation Study of the Classroom Activities within the Developed English Course (Flying High) for the High School First- Grade Students in View of Creative Thinking Skills
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The study aims to examine the classroom activities of the developed English course (Flying High) for the high school first-grade students, identify creative thinking skills appropriate for this grade, and show the extent the classroom activities involve these skills from the female- teachers ‘point of view. The study adopted the descriptive survey method. The study community consists of all (50) English female-teachers who teach high school first grade in Arar city during the academic year (1440 -1441 A.H, the first semester). The study was applied to all respondents. The researcher used a questionnaire as a study tool. The study revealed that the female-teachers reported their disagreement and refusal of the classroom activities in th

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