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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
Sat Dec 15 2018
Journal Name
Journal Of The College Of Education For Women
The Reasons for the Behavior of Bullying school Students in the first grade average from the point of view of teachers and teachers and methods of modification
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 هدف البحث التعرف الى اسباب سلوك التنمر لدى طلاب الصف الاول المتوسط من وجهة نظر المدرسين والمدرسات واساليب تعديله، واستعمل الباحثان المنهج الوصفي واختيار عينة عشوائية من المدرسين والمدرسات في متوسطة أرض الرافدين ومتوسطة الرحمن للبنين وكان عددهم (46) مدرساً ومدرسة بواقع (32) مدرساً و(14) مدرسة، واعتمد الباحثان الاستبانة أداة للتعرف الى اسباب سلوك التنمر واساليب تعديله، واشارت نتائج البحث الى تنوع اسباب التن

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Publication Date
Mon Jan 01 2024
Journal Name
Aip Conference Proceedings
The dynamical behavior of AIDS and HCV infection model with two modes of transmission
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The aim of this paper is to describe an epidemic model when two SI-Type of diseases are transmitted vertically as well as horizontally through one population. The population contains two subclasses: susceptible and infectious, while the infectious are divided into three subgroups: Those infected by AIDS disease, HCV disease, and by both diseases. A nonlinear mathematical model for AIDS and HCV diseases is Suggested and analyzed. Both local and global stability for each feasible equilibrium point are determined theoretically by using the stability theory of differential equations, Routh-Hurwitz and Gershgorin theorem. Moreover, the numerical simulation was carried out on the model parameters in order to determine their impact on the disease

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Publication Date
Tue Jun 20 2023
Journal Name
Bulletin Of The Iraq Natural History Museum
A COMPARATIVE-MORPHOLOGICAL STUDY OF SKULLS IN TWO SPECIES OF CARNIVOROUS AND HERBIVOROUS MAMMALS
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The skull is one of the largest bones in the body. It is classified into flat bones that maintain the important organic structures; which are the brain, eyes, and tongue. The skull is a strong support for preserving these organs but they are various according to the type of animals and the environments in which they live and the nature of their nutrition. There are many differences among living organisms in terms of the bones in the skull, their difference or disappearance and their length in the shape of the head. The samples were taken from the scientific storage in the Iraq Natural History Research Center and Museum; Cape hare Lepus capensis (Linnaeus, 1758) and Red fox Vulpes vulpes (Linnaeus, 1758) and the study was conducted o

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Publication Date
Fri Jan 08 2010
Journal Name
Al-kindy College Medical Journal
Maternal and fetal outcomes in emergency versus elective two or more previous cesarean deliveries
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Publication Date
Tue May 26 2026
Journal Name
Journal Of Baghdad College Of Dentistry
Effect of canal dryness and flaring on the accuracy of two electronic apex locators
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Background: This in vitro study evaluated the effect of canal dryness and flaring on the accuracy of two electronic apex locators for working length (WL) determination. Materials and methods: Sixty extracted teeth were used, after access opening was done, the occlusal surface was flattened to create stable reference point. The teeth were randomly divided into two equal main groups of flared and unflared group each one of 30 teeth. The flaring was done with Gates Glidden drills. The two main groups were further subdivided into two subgroups: dry canal and wet canal using 5.25% sodium hypochlorite groups, Electronic WL of each sample was determined using both Root ZX and i-Root apex locator. Consequently, histologic WL was determined by shav

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Publication Date
Sat Apr 01 2017
Journal Name
International Journal Of Science And Research
Evaluating the Behavior of Ring Footing on Two-Layered Soil Subjected to Inclined Load
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Evaluating the behavior of a ring foundation resting on multi-layered soil is one of the important issues facing civil engineers. Many researchers have studied the behavior of ring foundation rests on multi-layered soil with vertical loads acting on the foundation. In real life ring foundation can be subjected to both vertical and horizontal loads at the same time due to wind or the presence of soil. In this research, the behavior of ring footing subjected to inclined load has been studied using PLAXIS software. Furthermore, the effect of multi-layered soil has been simulated in the model. The results showed that both vertical and horizontal stresses are mainly affected when the inclination angle of the load exceeded 45 degrees with a reduc

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Publication Date
Fri Nov 09 2018
Journal Name
Iraqi National Journal Of Nursing Specialties
Evaluation of Anemia among Lactating Mothers and their Children Under Two Years of Age
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Objective: The aim of this study is to evaluate anemia among lactating women and their children less than 2 years of
age.
Methodology: The study was done on (148) lactating mothers and their children under 2 years of age in the primary
health care center at AL-Salam Quarter/Baghdad from l/10/2009 to 15/10/2010.
Result: This study recommends that there is a significant relationship between anemic mothers and their children. The
study also revealed that there is (77.8%) from nursing mothers in the age groups (25-29) who suffered from anemia,
while (23.1%) for the age group (20-24) did not have anemia
Recommendation: We encourage the use of breast milk or iron-fortified infant formula only for any milk-based part
of

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Publication Date
Sun Mar 03 2013
Journal Name
Baghdad Science Journal
The Predication of the Type of Jupiter Radio Storm from Two Different Iraqi Locations
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A program in Visual Basic language was designed to predict the type of radio storm that emitted from Jupiter at specific Local Time (LT) from two different Iraqi locations (Baghdad and Basra), such storms result from the Central Meridian Longitude (CML) of system ??? for Jupiter and phase of Io’s satellite (?Io). Some of these storms are related to position of Io (Io- A,B,C,D) and others are unrelated (non-Io-A,B,C,D) to its position. The input parameters for this program were user specified by determining the observer’s location (longitude), year, month and day. The output program results in form of tables provides the observer with information about the date and the LT of beginning and end of each type of emitted storm. Two Io-storm r

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Publication Date
Mon Sep 23 2019
Journal Name
Civil Engineering Journal
Energy Absorption Evaluation of CFRP-Strengthened Two-Spans Reinforced Concrete Beams under Pure Torsion
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For more than a decade, externally bonded carbon fiber reinforced polymer (CFRP) composites successfully utilized in retrofitting reinforced concrete structural elements. The function of CFRP reinforcement in increasing the ductility of reinforced concrete (RC) beam is essential in such members. Flexural and shear behaviors, ductility, and confinement were the main studied properties that used the CFRP as a strengthening material. However, limited attention has been paid to investigate the energy absorption of torsion strengthening of concrete members, especially two-span concrete beams. Hence, the target of this work is to investigate the effectiveness of CFRP-strengthening technique with regard to energy absorption of two-span RC

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Publication Date
Sat Dec 24 2022
Journal Name
Iraqi Journal Of Pharmaceutical Sciences ( P-issn 1683 - 3597 E-issn 2521 - 3512)
Possible protective effects of two different doses of cyanocobalamin against methotrexate nephrotoxicity in rats
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Abstract

   Nephrotoxicity is defined as rapid deterioration in kidney functions. It arises from direct exposure to drugs or their metabolites. Methotrexate is a famous chemotherapeutic drug with anti-inflammatory and immunosuppressive properties. A high-dose methotrexate-induced renal dysfunction can be life threatening. Cyanocobalamin, one of the forms of vitamin B12, acts as a coenzyme in the conversion of homocysteine to methionine in the cytosol, and the conversion of methylmalonyl-CoA to succinyl-CoA in the mitochondrion. This study is designed to examine the effect of cyanocobalamin in two different doses each co-administered with methotrexate at 20 mg/kg induced nephrotoxicity in rat

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