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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 Oct 31 2025
Journal Name
Revista Iberoamericana De Psicología Del Ejercicio Y El Deporte, Issn 1886-8576, Vol. 18, Nº. 4, 2023, Págs. 361-363
The Effect of a Training Curriculum in Developing the Speed Endurance and Strength Characterized By Speed for the Muscles of the Legs for Female Basketball Students
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Autorías: Mariam Liwa Abdel Fattah, Liqaa Abdullah Ali. Localización: Revista iberoamericana de psicología del ejercicio y el deporte. Nº. 4, 2023. Artículo de Revista en Dialnet.

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Scopus (1)
Scopus
Publication Date
Sun Dec 05 2004
Journal Name
Journal Of Physical Education
The evaluate of usage educational method for jump shooting and it effect on speed strength in basketball for specialized students in College Sport of Dayla University
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The research aim to the usage educational method for jump shooting and it effect on speed strength in basketball for the specialist students in College Sport of Dayla University, which used the following statistic treatment (The T.test for compatible specimens), so after statistic treatment which appears theres a tow moral differences in speed strength and jump shooting tests results to (legs & arms) for the before and after tests, and after that the conclusions we positive and the second the special drills effect immaterial speed strength to legs and arms, so the tow researches recommended to looking after the best for educational methods that used in our sport colleges in Iraq.

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Publication Date
Sun Feb 01 2015
Journal Name
Journal Of Economics And Administrative Sciences
Use of model sales man fuzzy multi-objective linear for speed up and rationing in the transport mechanism used in the General Company for grain Processing
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     Applications of quantitative methods, which had been explicit attention during previous period (the last two centuries) is the method of application sales man or traveling salesman method. According to this interest by the actual need for a lot of the production sectors and companies that distribute their products, whether locally made or the imported for customers or other industry sectors where most of the productive sectors and companies distributed always aspired to (increase profits, imports, the production quantity, quantity of exports. etc. ...) this is the part of the other hand, want to behave during the process of distribution routes that achieve the best or the least or most appropriate.

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Crossref
Publication Date
Sun Oct 04 2015
Journal Name
Iraqi Journal Of Market Research And Consumer Protection
Study the optimum conditions for production of inulunase from isolate Kluyveromyces marxianus AY2.: Study the optimum conditions for production of inulunase from isolate Kluyveromyces marxianus AY2.
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Results showed that the optimum conditions for production of inulunase from isolate Kluyveromyces marxianus AY2 by submerged culture could be achieved by using inulin as carbon source at a concentration of 2% with mixture of yeast extract and ammonium sulphate in a ratio of 1:1 in a concentration of 1% at initial pH 5.5 after incubation for 42 hours at 30ºC.

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Publication Date
Wed Jun 01 2016
Journal Name
Journal Of Economics And Administrative Sciences
High Commitment Management and their Impact in Organizational Excellence Afield Research for opinions a sample of managers in the company general Alfurat for chemical industries \Babylon
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In its theoretical framework, this study dealt with the subjects of high commitment management and organizational excellence, as the study came in response to the growing developments and changes in the fields of management. It includes an analysis of correlation and effect between high commitment management, which has been attracting a lot of attention recently due to the intensifying rivalry between organizations because of certain external factors like globalization and world markets liberation, and its effect in achieving organizational excellence.

The practical framework, on the other hand, dealt with the analysis of correlation and effect between the study's variables. The problem

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Crossref (3)
Crossref
Publication Date
Mon Feb 01 2016
Journal Name
Al-academy
Impact organization of configuration for still life in paintings of first grade students for department of Art Education in subject of colors: احمد مطشر لطيف السبتي
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Problem of research as defined, including the following:what is impact organizing of configuration for still life in paintings of students in the Department of Art Education in subject of colors ?Study of well training for organized in painting of still life is necessary for a teacher of art education to contribute effectively to the performance of his letter, so is this research in response to the public goals for a subject of art education. The current research aims to:1. Impact organization of configuration for still life in paintings of students in the Department of Art Education for the first grade in subject of colors2. Role organization of configuration for still life in the painting to serve the educational process in subject of

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Crossref
Publication Date
Sat Jan 01 2022
Journal Name
Journal Of Algebraic Statistics
The effect of compound exercises for the arm least used in developing the accuracy of the passing skill of the handball for ages (15-17) years
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Publication Date
Fri Oct 31 2025
Journal Name
Journal Of Physical Education
Exercises With Different Ranges Of Motion With Significance Of Electrical Activity for Muscle in Strength With Speed Of Lower Limbs For Weight Lifters Of Physical Strength
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Publication Date
Fri Feb 01 2019
Journal Name
Journal Of Economics And Administrative Sciences
Organizational spirituality and its implications for organizational performance: A survey of the opinions of employees of the General Company for Food Industries in Abu Ghraib (Baghdad)
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The research discusses with organizational spirituality and its implications on the organizational performance in the General Company for Food Industries in Abu Ghraib (Baghdad). The aim of the research was to determine the contribution of organizational spirituality in the organizational performance of the surveyed company. In order to achieve the objectives of the research، two main hypotheses were adopted. Several sub-hypotheses centered on the relationship between organizational spirituality and organizational performance in terms of its dimensions (Meaning work، self-esteem، community affiliation،

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Crossref
Publication Date
Thu Apr 30 2015
Journal Name
Iraqi Journal Of Market Research And Consumer Protection
Detection of Commercial Cheating for Some Kinds of Local Markets retailed Medicinal Oils: Detection of Commercial Cheating for Some Kinds of Local Markets retailed Medicinal Oils
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The aims of this study are to explore the commercial artifacts in the following three kinds of vegetables oils, Nigella Sativa, Trigonella foenum-graecum Linn,and Zingiber officinale. These oils have been very popular medicinal plants which are commonly used in traditional medicine .These commercial oils have been compared with the extracts of these plants.
The physical properties of extracts and commercial oils of these plants have been stuied. We observed that the refractive index of the plants matches and non-significant, while specific gravity of Nigella Sativa has similar specific gravity in both extracts and commercial oil in contrast with Trigonella foenum Linn,and Zingiber officinale and we found significant difference (P&lt

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