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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
Wed Jun 30 2021
Journal Name
Gsc Biological And Pharmaceutical Sciences
Differences in some cranial bones between two Cyprinidae species, Common carp Cyprinus carpio (Linnaeus, 1758) and Crucian Carp Carassius carassius (Linnaeus, 1758) Collected from Tigris River, Iraq
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The present study attempts to identify some of the differences between the skull bones of two species Cyprinus carpio and Carassius carassius, which belong to the Cyprinidae family. The study is a taxonomic diagnostic study between the two species which are considered local fish abundant in the Iraqi aquatic environment

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Publication Date
Wed Mar 10 2021
Journal Name
Baghdad Science Journal
Effect of spraying fertilizer paper and aqueous extract of licorice in the content of the fruits of two varieties of sweet peppers total Azaúbh solids and vitamin
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Studied the effect of foliar fertilizers Alaongrin results showed that spraying fertilizer Alaongrin and Fertilizers and Ministry of Agriculture and rack licorice extract every three weeks after thirty days from planting seedlings

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Publication Date
Wed Oct 08 2025
Journal Name
Journal Of Baghdad College Of Dentistry
Evaluation of the Efficiency of Three Different Obturation Techniques to Obturate the Isthmus Area of Roots Canals Prepared by Two Different Instrumentation Techniques (An In Vitro Study)
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Background: The isthmus is a difficult area in the root canal complex to manage. The research aimed to evaluate the efficiency of three different obturation techniques (lateral condensation, EandQ (thermoplasticized gutta percha system) and Soft Core (thermoplasticized core carrier gutta percha system)) to obturate the isthmus area of roots prepared by two different instrumentation techniques (rotary ProTaper universal and ProTaper Next systems). Material and method: Sixty freshly extracted teeth were randomly divided into two main groups (A and B) of 30 teeth each. Group A was prepared by rotary ProTaper Universal whereas group B was prepared by ProTaper Next system. Each main group was then randomly subdivided into three subgroups of 10 t

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Publication Date
Tue May 01 2018
Journal Name
Journal Of Physics: Conference Series
Synthesis, Characterization and the Corrosion Inhibition Study of Two Schiff Base Ligands Derived From Urea and Thiourea and Their Complexes with Cu(II) and Hg(II) Ions
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Publication Date
Wed Jun 29 2022
Journal Name
College Of Islamic Sciences
term of the first difference between the Imam of the Two Holy Mosques, Imam Taj Al-Din al-Subki and its impact on the doctrine of Al-Shafeiyah
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Praise be to Allah, the Lord of the Worlds, the best prayer and the best prayer, on our master Muhammad, and on his pure God, and his companions and the faithful, and who followed them by charity to the day of religion. This relationship between emphasis, is a sincere, and this harmony, such as the relationship between water, and the green, but it is good, but, with greenery, it is better, and as well as alone, is a beautiful view, but the most beautiful, with the most beautiful. From here he was starting on myself in writing the fundamentalist research of jurisprudence, to show the depth of this interconnection. The doctrine as a new term, then the taj al-Din al-Suobki came after three centuries, and a

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Publication Date
Sun Jun 02 2013
Journal Name
Baghdad Science Journal
Comparison of The Effect of Aqueous Extracts of two Plants, Origanum Vulgare L. and Fenugreek Seeds with Anticancer Drug Cis-Platin on the Growth of Cancer Cell Lines
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This study involved the effect of the aqueous extracts of two plants, Origanum vulgare L.(1), Trigonella Foenum Graecum L. (Fenugreek) seeds(2) on the growth of cancer cell lines. Rhabdomyo sarcomas (RD) of human cell line and female intestine cells of Albino mice (L20B) in vitro System. These extracts were compared with the known anticancer drug Cis-platinum(Cis-Pt) as a positive control. The phytochemical tests were used for screening the active compounds in plants. The inhibition activity assay was used as a parameter of the cytotoxic effect of these extracts. Cancer cell lines were treated with four concentrations of Cis-platin, 31.25, 62.5, 125 and 250 ?g/ml for 72 hour exposure time. The same concentrations were used for the other ext

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Publication Date
Sun Jan 01 2017
Journal Name
Al–bahith Al–a'alami
The Coverage of Health Topics in Iraqi Press An analytical Study of Two Newspapers entitled 'al-Sabah and al-Mada' from January 1, 2007 to October 31, 2001
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The current study aims to cover  the health topics  in the Iraqi  journalism ,  to know  the concern 's level  of the Iraqi journalism  when  covering  the health subjects and to know the size of its performance  under the diversity of  visual and audio medias. The researcher has discussed his research by wording it with a set of inquiries   that should be answered via the research and study.  He could have determined that by the following:

What is the concern's level of Iraqi journalism over the health subjects?  What are the forms and press arts that are functionalized by the Iraqi journalism as to get the health message to the republics?  What is t

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Publication Date
Tue Jan 01 2019
Journal Name
Mustansiriyah Journal Of Sports Science
The use of training methods in accordance with special exercises to develop flexibility and perform the skills of shooting from above the chest (abduction) to wrestlers ages (14-15)"
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Evolution has become a feature of this era because of the speed that makes it open multiple horizons and many to identify everything that is new in different areas and also characterized by the competitive position of emotional attitudes changing depending on the positions of winning and defeat, and the use of training methods are the most important pillars of the game of wrestling, The methods contribute to raising the level of the wrestler and refining his physical and skill potential. The problem of the research is that the shooting exercises from above the chest are very important in Roman wrestling and can be terminated by the player. Through very personal interviews for coaches and concluded that there is a weakness in the level of fl

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Publication Date
Sun Mar 06 2011
Journal Name
Baghdad Science Journal
Isolation and Identification of Vibrio cholerae causes Epidemic cholera 2007 in Iraq by Rapid Method (Immunochromatographic one step rapid visual test) and comparing it with the traditional Bacterlogical Methods
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This study was for searching for Cholera Bacteria serotype which causes epidemiology Cholera in the 2007 in a fast method which contains (Rapid Visual Test) (Crystal V.C.) which was used for the first time in Iraq to diagnosis of Cholera Bacteria & compared with the traditional bacteriology method. The Cholera disease is one of the most dangerous epidemiological diseases which lead to death with a percentage of (50 – 70) % in the severe cases for untreated patients . For this purpose, 100 samples of stool from the patients from a (13) hospitals in Baghdad Governorate in the period from August to the end of December. The Cholera was diagnosis in two methods, 1st method was the fast method using the nitrocellulose which is coated with anti-

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Publication Date
Tue Jan 24 2023
Journal Name
Archives Of Razi Institute
PCR-RFLP Analysis of Insulin-Like Growth Factor 2 Gene Polymorphisms in Two Commercial Broiler Chicken Strains (Cobb 500 and Hubbard F-15) and Their Associations with Performance Traits
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The present research aimed to study the polymorphisms of the chicken insulin-like growth factor 2 (IGF2) in two commercial broiler breeds (Cobb 500 and Hubbard F-15). In total, 300 avian blood samples were obtained. The genomic DNA was isolated using a fast salt-extraction technique. Moreover, polymerase chain reaction (PCR) was used to amplify 1146 bp fragments of the gene. The amplified fragments were subjected to restriction enzyme digestion using the HinfI endonuclease enzyme, and the digested products were separated on a 2% agarose gel. The findings indicated that there were two alleles, T and C, for the target locus, with frequencies of 73.3% and 26.7%, respectively. Three distinct genotype variations, TT, TC, and CC, were found, with

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