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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
Thu Mar 31 2022
Journal Name
Journal Of The College Of Education For Women
News of Zubarah Family- Unionists in Nishapur and their Scientific Contributions in the two Centuries (4-6 AH/10-12 AD)
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This research is concerned with qualitatively studying the history of an ancient family, one of the most prominent noble, Alawite families, that assumed the position of the Student Union for the Alawite Supervision in the city of Nishapur in Khorasan during the centries (4-6 H./ 10-12 A.H.). Thus, the research aims to investgate the historical news, and the outstanding Zubarah figures who experience the supereme the religious, historical and scientific position in Khorasan generally and Nishapur particularily. It further aims at arriving at the true meaning of the name Zubarah, and to whom it was called, and the extent of this family’s affiliation to the noble Alawi House, and how they moved to live and settle in the cities of the East

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Publication Date
Fri Oct 09 2026
Journal Name
Sciences Journal Of Physical Education
Comparison of maximum strength and peak electrical activity (EMG) of the muscles of the two legs among bodybuilding players and Weightlifting
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Publication Date
Sat Oct 01 2022
Journal Name
Al–bahith Al–a'alami
Language communication, and the impact of journalism after 2003 AD (a study in the two newspapers, Al-Takahi and Al-Sabah)
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The methods used by the print media (press) is varied in conveying its messages and its orientations to the readers. But language takes the leading method among these methods, or a real competitor - so far - due to the characteristics and advantages of languages, including: ease, abbreviation, and development, etc.

In the midst of such events, political tensions, and what they carry of the global developments, and among which was the most important the fall of the former Iraqi regime in the spring of 2003, the press had the greatest share, and a large portion on the level of monitoring, analyzing, and interpreting the various consequences of the event. The interpreting of the news varies according to the evolving political trends

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Publication Date
Thu Jun 01 2017
Journal Name
Journal Of The College Of Languages (jcl)
Plain Language Movement and Legal Translation : An Analytical Study of Two Translations of a Contract of Lease from Arabic into English
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The present paper addresses one of the most challenging topics in translation; namely legal translation in the framework of two different approaches; the classical (formal) and the more recent (functional). The latter approach is the outcome of the process of simplifying legal language known technically as Plain Language Movement. The advent of this movement dates back to the 1950s, in response to the widely-held complain about the awkwardness of the legal register. Within this framework, the salient features of legal language, at the various linguistic and textual levels, underwent reconsideration in favor of more publicly digested expressions. The paper then subjects two translations of a lease contract to analysis in the ligh

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Publication Date
Sun Feb 25 2024
Journal Name
Baghdad Science Journal
Prioritized Text Detergent: Comparing Two Judgment Scales of Analytic Hierarchy Process on Prioritizing Pre-Processing Techniques on Social Media Sentiment Analysis
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Most companies use social media data for business. Sentiment analysis automatically gathers analyses and summarizes this type of data. Managing unstructured social media data is difficult. Noisy data is a challenge to sentiment analysis. Since over 50% of the sentiment analysis process is data pre-processing, processing big social media data is challenging too. If pre-processing is carried out correctly, data accuracy may improve. Also, sentiment analysis workflow is highly dependent. Because no pre-processing technique works well in all situations or with all data sources, choosing the most important ones is crucial. Prioritization is an excellent technique for choosing the most important ones. As one of many Multi-Criteria Decision Mak

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Publication Date
Mon Apr 03 2023
Journal Name
Journal Of Genetic And Environmental Resources Conservation, 2023, 11(1):53-59.
Comparative histomorphological and immunohistochemical study of parathyroid gland in two Iraqi mammals (Weasel, Herpestes javanicus and long-ear hedgehog, Hemiechinus auritus)
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The present study aimed to investigate the anatomy, histology, and immunohistochemistry of parathyroid gland in two Iraqi mammals (Weasel, Herpestes javanicus and Long-ear hedgehog, Hemiechinus auritus) as a comparative study. A total of (20) animal for each species were used in the present study. Animals collected were immediately anesthesia and dissected to get the parathyroid gland. Methods of Humason and Bancroft and Stevens were employed for histological techniques. Different stains were used (Hematoxylin- Eosin stain-(H & E), Periodic Acid Schiff stain-(PAS), Azan stain, and Methyl Blue stain-(MB)) for staining the histological sections. Anti-calcitonin, code140778 marker used for immune-histochemical study. Results of the present stu

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Publication Date
Mon Nov 05 2018
Journal Name
Iraqi National Journal Of Nursing Specialties
Effect of Mechanical Surface Treatment on the Tensile Bond Strength of Repaired Acrylic Denture Base Resin Cured by Two Different Techniques
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Objective: To evaluate and compare the effect of mechanical surface treatment (groove, aluminum oxide particles)
with 45 degree bevel type of joint on tensile bond strength of acrylic specimens repaired by two curing methods
(microwave and water both).
Methodology: Eighty specimens (80) were prepared from pink heat cure acrylic resin. They were divided into two
main groups (40 specimen repaired by microwave energy and 40 specimens repaired by water bath method).Each
group can be divided into four subgroups of ten according to the surface treatment. The control group A was left
intact, group B received no surface treatment, group C and D received surface treatment by (groove, 50 m aluminum
oxide particles). Specimens

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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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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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