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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
Mon Sep 29 2025
Journal Name
Journal Of Physical Education
The effect of dynamic lactic training according to the target time fo developing special endurance, lactic acid concentration, and achievement for 800m runners under20 years old
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Publication Date
Wed Nov 01 2023
Journal Name
Ecological Engineering & Environmental Technology
Synthesis of Mn-Co-Ni Composite Electrode by Anodic and Cathodic Electrodeposition for Indirect Electro-oxidation of Phenol – Optimization of the Removal by Response Surface Methodology
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In the present work, Response Surface Methodology (RSM) was utilized to optimize process variables and find the best circumstances for indirect electrochemical oxidation of mimicked wastewater to remove phenol contaminants using prepared ternary composite electrode. The electrodeposition process is used for the synthesis of a ternary composite electrode of Mn, Co, and Ni oxides. The selected concentrations of metal salts of these elements were 0.05, 0.1, and 1.5 M, with constant molar ratio, current density, and electrolysis time of 1:1:1, 25 mA/cm2, and 2 h. Interestedly, the gathered Mn-Co-Ni oxides were deposited at both the anode and cathode. X-ray diffraction (XRD) and scanning electron microscopy (SEM) facilitated the qualitative char

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Scopus Crossref
Publication Date
Tue Jan 01 2019
Journal Name
Chemical Engineering Journal
Corrigendum to “Hierarchically porous zeolite X composites for manganese ion-exchange and solidification: Equilibrium isotherms, kinetic and thermodynamic studies” [Chem. Eng. J. 308 (2017) 476–491]
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Publication Date
Sun Dec 01 2013
Journal Name
Baghdad Science Journal
Flow injection analysis for the photometric determination of promethazine-HCl in pure and pharmaceutical preparation via oxidation by persulphate using Ayah 3SX3-3D solar micro photometer
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The first flow injection spectrophotometric method is characterized by its speed and sensitivity which have been developed for the determination of promethazine-HCl in pure and pharmaceutical preparation. It is based on the in situ detection of colored cationic radicals formed via oxidation of the drug with sodium persulphate to pinkish-red species and the same species was determined by using homemade Ayah 3SX3-3D solar flow injection photometer. Optimum conditions were obtained by using the high intensive green light emitted diode as a source. Linear dynamic range for the absorbance versus promethazine-HCl concentration was 0-7 mmol.L-1, with the correlation coefficient (r) was 0.9904 while the percentage linearity (r2%) was 98.09%. the L.

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Crossref
Publication Date
Sun Sep 26 2021
Journal Name
Modern Sport
The Effect Of Plank Exercises On The Lipid Component And The Rate Of Metabolism (Bmr) Among Students Of The Higher Institute For Security And Administrative Development
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The introduction of the research on the science of training and the physiology of sports was addressed from important sciences, where the physical effort drew the attention of scientists since the past centuries when they studied how the body performs its functions when performing physical exertion and observe the changes that occur in it and write down and study especially the positive effects of the practice of daily sports The aim of the study was to investigate the effect of plank exercises on the lipid component and the metabolic rate (bmr) of the female students of the Higher Institute for Security and Management Development. As for the third chapter, the two researchers used the experimental method on a sample of the female s

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Crossref
Publication Date
Tue Dec 01 2015
Journal Name
Journal Of Economics And Administrative Sciences
Integrated Framework for technologies to reduce costs and reengineering processes to develop strategies companyes An Empirical Study In some companies affiliated to the Ministry of Oil
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Began the process of re-engineering processes in the private sector as a way to assist organizations in re-thinking how to run the business in order to improve production processes and reduce operational cost, to get to compete on a global level. That was a major restructuring by further evolution in the use of technology to support innovative operations.

 Entered the technology in all areas of life and different regulations, This led to use as a change in all aspects The companies achieved success and progress today through the use of resources so as to ensure the wishes of the customers and their needs, and the requirements of the market primarily, Which is reflected on the basis of building strate

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Crossref
Publication Date
Fri May 15 2009
Journal Name
Journal Of Planner And Development
Study of Change in Regional Land Use Using Remote Sensing Technology and Geographic Information Systems / Study Area (Hawr Al Huweiza) for the Years 1973-1990-2004
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his project try to explain the using ability of spatial techniques for land cover change detection on regional level with the time parameter and did select for explain these abilities study case (Hewaizah marsh ) . this area apply to many big changes with the time. These changes made action on characters and behaviors of this area as well as all activities in it . This Project concerting to recognize the Using importance of remote sensing and GIS Methodology in data collecting for the changes of land use and the methodology for the analyses and getting the results for the next using as a base data for development and drawing the plans as well as in regional planning .This project focus on practical

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Publication Date
Sun Sep 15 2019
Journal Name
Al-academy
Role of the Internet in Spreading Rumors Social networking sites "Facebook" model For the duration of 1-7-2017 until 30-11-2017: يوسف محمد حسين
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This study is about the role of the Internet in spreading rumors, especially through social networking sites "Facebook" model as the effectiveness of social networks lies in the speed of transmission of events; these two characteristics are important to the public, making the Internet a strong contender for television and its relationship with the public. That's why we find that the Internet today has become a fertile environment for the growth and spread of rumors. The more limited the platforms and places of publication, the greater the responsibility in the search for the original source in   spreading this or that rumor, as the Internet is considered an easy means in the production, spreading  and re-spreading  of

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Crossref
Publication Date
Sun Mar 01 2020
Journal Name
Journal Of Accounting And Financial Studies ( Jafs )
The effect of economic factors on the appearance of unequal costs and rationalization of operational decision: An applied research in Wasit State Company for Textile Industries
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There is a set of economic factors that affect the rationalization of decisions on unexploited resources within the economic unit and here determines the problem of the search for the question of what economic factors cause the emergence of asymmetric costs, and aims to identify these factors in the costs of adjustment to resources, change in The size of the activity of the economic unit, the general trend of sales change in the previous period, and the economic level of the country. Rh measure the impact of these factors on economic unity, and taking into consideration the impact when formulating decisions.

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Publication Date
Wed May 05 2021
Journal Name
Mustansiriyah Journal Of Sports Science
The Effect of Parachute Exercises and Cloud Pockets in the Water Ocean on Some Special Physical Abilities and the Achievement of 50-Meter Swimming for Youth
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ولاء طارق حميد, Mustansiriyah Journal of Sports Science, 2021

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