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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 Jun 01 2026
Journal Name
Journal Of Engineering
Temporal Shifts in Understanding of Privacy: Perspectives of House Users in Baghdad During Two Different Periods: Case Studies in Selected Neighborhoods
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The Dwelling is a Fundamental issue related to human beings; throughout time, man has always had to dwell in a place and build his house. Due to different styles and typologies appearing all over the world, all containing the character of privacy as the main element of human life in their dwellings. The present study aims to investigate the temporal shift in understanding privacy during the 20th and 21st centuries in Baghdadi houses, and what are the physical elements that are related to the meaning of privacy? The research methodology analyzed the concepts of privacy to extract the research indicators to be applied to (49) samples of individual houses in selected neighborhoods in two different periods. The data was collected from v

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Publication Date
Mon Jun 01 2026
Journal Name
Ecological Engineering & Environmental Technology
Towards low-energy desalination: Performance analysis and design of single and two-stage reverse osmosis systems to minimize specific energy consumption
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Lowering specific energy consumption (SEC) in reverse osmosis (RO) plants is one of the most important criteria in water treatment industry. The present work focused on best suggested RO pilot plant configurations for minimizing specific energy consumption, finding the best configurations experimentally to recover the water with minimizing salts in permeate side in RO module, study the factors affecting the performance of single and two stage systems, TFC membrane is used in spiral wound module. Brackish water is used as feed solution with different concentrations (2000, 3000, 4000 and 5000 ppm). The operating parameters studied were permeate flux, rejection and SEC. The experimental results showed the permeate concentration increased and w

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Scopus Crossref
Publication Date
Wed Jan 01 2020
Journal Name
Dental Research Journal
Impact of a mixture of nanofiller and intrinsic pigment on tear strength and hardness of two types of maxillofacial silicone elastomers
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Background: The ideal maxillofacial prosthesis should have fine and thin boundaries that bindwith the surrounding facial structures and possess high tear strength.This study aims to determinethe best percentages of nanofiller (TiO2) and intrinsic pigment (silicone functional intrinsic) thatcould be mixed in as additives to improve the tear strength of Cosmesil M511 andVST50F siliconeelastomers with the least effect on their hardness.Materials and Methods: In this in vitro experimental study, a total of 80 samples, 40 for eachelastomer, were fabricated. Each elastomer sample was split into two equal groups to test for tearstrength and Shore A hardness. Each group consisted of 20 samples, including 10 control sampleswithout additives and 10 e

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Scopus Crossref
Publication Date
Fri Sep 15 2017
Journal Name
Journal Of Baghdad College Of Dentistry
Comparison of Shear Bond Strength of Sapphire Bracket Bonded to Zirconium Surface after Using Different Surface Conditioning Methods (In Vitro Study)
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Background: The present study was carried out to compare shear bond strength of sapphire bracket bonded to zirconium surface after using different methods of surface conditioning and assessment of the adhesive remnant index. Materials and methods: The sample composed of 40 zirconium specimens divided into four groups; the first group was the control, the second group was conditioned by sandblast with aluminum oxide particle 50 μm, the third and fourth group was treated by (Nd: YAG) laser (1064nm)(0.888 Watt for 5 seconds) for the 1st laser group and (0.444 Watt for 10 seconds) for the 2nd laser group. All samples were coated by z-prime plus primer. A central incisor sapphire bracket was bonded to all samples with light cure adhesive res

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Publication Date
Wed Dec 18 2019
Journal Name
Baghdad Science Journal
The Calculation and Analysis of the Total Electron Content Over Different Latitudes and Seasons Using the Numerical Trapezoidal and Simpson Methods
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It has been shown in ionospheric research that calculation of the total electron content (TEC) is an important factor in global navigation system. In this study, TEC calculation was performed over Baghdad city, Iraq, using a combination of two numerical methods called composite Simpson and composite Trapezoidal methods. TEC was calculated using the line integral of the electron density derived from the International reference ionosphere IRI2012 and NeQuick2 models from 70 to 2000 km above the earth surface. The hour of the day and the day number of the year, R12, were chosen as inputs for the calculation techniques to take into account latitudinal, diurnal and seasonal variation of TEC. The results of latitudinal variation of TE

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Publication Date
Wed Mar 30 2022
Journal Name
Journal Of Economics And Administrative Sciences
Planning the Production of the Electrical Distribution Converter (400KV/11) Using Time Series Methods and Goal Programming in the Fuzzy Environment
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This Paper aims to plan the production of the electrical distribution converter (400 KV/11) for one month at Diyala Public Company and with more than one goal for the decision-maker in a fuzzy environment. The fuzzy demand was forecasting using the fuzzy time series model. The fuzzy lead time for raw materials involved in the production of the electrical distribution converter (400 KV/11) was addressed using the fuzzy inference matrix through the application of the matrix in Matlab, and since the decision-maker has more than one goal, so a mathematical model of goal programming was create, which aims to achieve two goals, the first is to reduce the total production costs of the electrical distribution converter (400 KV/11) and th

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Crossref
Publication Date
Mon Mar 29 2021
Journal Name
Journal Of Engineering
Extracting Four Solar Model Electrical Parameters of Mono-Crystalline Silicon (mc-Si) and Thin Film (CIGS) Solar Modules using Different Methods
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Experimental measurements were done for characterizing current-voltage and power-voltage of two types of photovoltaic (PV) solar modules; monocrystalline silicon (mc-Si) and copper indium gallium di-selenide (CIGS). The conversion efficiency depends on many factors, such as irradiation and temperature. The assembling measures as a rule cause contrast in electrical boundaries, even in cells of a similar kind. Additionally, if the misfortunes because of cell associations in a module are considered, it is hard to track down two indistinguishable photovoltaic modules. This way, just the I-V, and P-V bends' trial estimation permit knowing the electrical boundaries of a photovoltaic gadget with accuracy. This measure

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Crossref
Publication Date
Fri Jun 05 2026
Journal Name
Indian Journal Of Information Sources And Services
Visual Perception of Population Maps of Baghdad Governorate (Comparative Study Between Cartographic Representation Using Graphical Methods within the ARC & GIS)
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This paper will juxtapose the effectiveness of conventional methods of graphical cartography and GIS-interpolation methods in presenting the population distribution of the Baghdad Governorate in 2023. The study touches upon the efficiency of two types of cartographic techniques to present demographic data and is concerned with the speed and clearness of the visual perception. The analysis involves the use of traditional techniques (square, triangle, column and divided circle) and the GIS techniques (Kernel Smoothing, Kriging and Inverse Distance weighting) to describe the male, female and aggregate population. Fifty respondents used their comprehension speed on the maps as the sample. The findings demonstrate that the traditional te

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Scopus Crossref
Publication Date
Sun Sep 13 2026
Journal Name
Journal Of Pharmaceutical Health Care And Sciences
Evaluating public acceptance of the official medicine label and mobile verification application: a mixed-methods study applying the technology acceptance model
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Abstract<sec> <title>Background

The Iraqi pharmaceutical sector faces significant challenges arising from the prevalence of substandard and falsified (S/F) medications, which pose a direct threat to public health safety. To strengthen regulatory oversight, health authorities have implemented national medicine track-and-trace system. However, the ultimate efficacy of these regulatory tools is heavily dependent on public awareness and the level of societal acceptance.

Objective

This study ai

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Crossref
Publication Date
Fri Nov 11 2022
Journal Name
Journal Of Accounting And Financial Studies ( Jafs )
Ambidexteritg leadership behaviors in organizational excellence Comparative : applied research in the two public companies, Al-Zawraa and the electrical and electronic industries
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Business organizations strive to reach organizational excellence, in instruction to maintain their position in the market, so the research started from the problem of the weakness of the senior leaders in the two companies investigated in the management and sustainability of work, as well as their inability to face challenges that could affect the work of the two companies in the future, especially the lack of organizational of the future and the prediction of those events, so the two companies seek to explore and invest opportunities to achieve organizational excellence. The research aims to know the role of Ambidexterity  leadership behaviors in organizational excellence in my company .and create ideas that will achieve organizati

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