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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
Sun Jan 01 2023
Journal Name
Journal Of Oral Medicine And Oral Surgery
Reliability of two difficulty indexes in predicting the surgical extraction difficulty of impacted mandibular third molars
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Introduction: The aim of this study was to compare the reliability and the agreement of the Pederson and Pernambuco difficulty indexes in predicting the surgical extraction difficulty of the impacted mandibular third molars. Materials and methods: A prospective observational cohort study was conducted on 83 patients who had undergone surgical removal of impacted mandibular third molar. The difficulty of extraction was determined preoperatively according to the total scores obtained from the Pederson and Pernambuco difficulty indexes, the operative difficulty was determined by the surgical technique and the duration of extraction. The accuracy of prediction of the surgical difficulty and

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Publication Date
Tue Feb 12 2019
Journal Name
Iraqi Journal Of Physics
Calculated the diffuse and direct parts of global solar radiation in Baghdad city for the period (1983-2005) depending on clearness index by applying the two world models of Liu -Jordan
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In this paper solar radiation was studied over a region of Baghdad (Latitude 33.3o and longitude 44.4o). The two parts of global solar radiation: diffuse and direct solar radiation were estimated depending on the clearance index of measured data (Average Monthly mean global solar radiation). Metrological data of measured (average monthly mean diffuse and direct solar radiation) were used to comparison the results and show the agreement between them. Results are determined by applying Liu and Jordan two models (1960). Excel 2007program is used in calculation, graphics and comparison the results.

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Publication Date
Sun Mar 08 2026
Journal Name
Journal Of Baghdad College Of Dentistry
The influence of cavity design for cusp coverage on fracture strength of weakened maxillary first premolars using two esthetic restorative systems (CAD/CAM hybrid ceramic and nanohybrid composite) (An in vitro study)
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Background: Maxillary first premolar with wide MOD cavity more susceptible to fracture. The aim of this study was to assess the influence of cavity design for cusp coverage on the fracture resistance of weakened maxillary first premolar restored with CAD/CAM hybrid ceramic versus nanohybide composite. Materials and Methods: Fifty six intact maxillary first premolars of approximately comparable sizes were divided into seven groups eight for each: Group A: Intact teeth (control group); Group B: teeth prepared for MOD inlay; Group C: teeth prepared for MOD onlay covering the lingual cusp; Group D: teeth prepared for MOD covering buccal and lingual cusps ,the previous three groups indirectly restored with nanohybrid composite (3M ESPE Z 250 X

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Publication Date
Sun Mar 08 2026
Journal Name
Sciences Journal Of Physical Education
Cognitive bias and its relationship to the performance of the two skills of standing on (hands and the human wheel) on the floor movement carpet device in the artistic gymnastics for women
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Publication Date
Sun Dec 20 2020
Journal Name
Journal Of Accounting And Financial Studies ( Jafs )
The tax examination by using some statistical methods: An applied research in the General Commission of taxes
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this research aims at a number of objectives including Developing the tax examination process and raise its efficiency without relying on comprehensive examination method using some statistical methods in the tax examination and Discussing the most important concepts related to the statistical methods used in the tax examination and showing its importance and how they are applied. the research represents an applied study in the General Commission of taxes. In order to achieve its objectives the research has used in the theoretical side the descriptive approach (analytical), and in the practical side Some statistical methods applied to the sample of the final accounts for the contracting company (limited) and the pharmaceutical industry (

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Publication Date
Tue Dec 15 2020
Journal Name
Al-academy
Methods of Acting Performance and Watching Industry in Street Theatre- Play (Sold) - A Model: بهاء زهير كاظم
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  The audience is one of the important practical elements in the theatrical show and its importance is not confined to its static activity as a receiver element only, rather it went beyond that issue as an effective and influential element in the proceedings of the show and the process of meaning construction, that it gains an active role in the construction and production of the connotation that influences and is influenced by the actor, where the communication channels are open between the two sides, consequently a kind of watching and joint interaction happens between them. Thus, it has become necessary for the actor to create a suitable environment for the onlookers in order for it to be an essential part of the show system. The

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Publication Date
Thu Aug 01 2019
Journal Name
Journal Of Economics And Administrative Sciences
Development of methods of treatment of medical waste in the hospitals of Baghdad health Department / (Rusafa survey)
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Purpose: To contribute to the development of an appropriate program for the management of medical waste based on clear-cut principles in order to reach the overall goal of improving the public health and environment of the population in our country.

Design / Approach / Introduction: The research is based on the analytical descriptive approach as a method of study in the field of data collection using a check list and analysis of the data through the use of some statistical treatments.

 Results: The need is to establish a medical waste management in hospitals and follow international standards in all stages of waste management from sorting, collection, transportation and treat

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Publication Date
Mon Feb 14 2022
Journal Name
Journal Of Educational And Psychological Researches
Comparison between Rush Model Parameters to Completed and Lost Data by Different Methods of Processing Missing Data
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The current study aims to compare between the assessments of the Rush model’s parameters to the missing and completed data in various ways of processing the missing data. To achieve the aim of the present study, the researcher followed the following steps: preparing Philip Carter test for the spatial capacity which consists of (20) items on a group of (250) sixth scientific stage students in the directorates of Baghdad Education at Al–Rusafa (1st, 2nd and 3rd) for the academic year (2018-2019). Then, the researcher relied on a single-parameter model to analyze the data. The researcher used Bilog-mg3 model to check the hypotheses, data and match them with the model. In addition

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Publication Date
Sat Oct 01 2022
Journal Name
Journal Of Applied Geophysics
Predicting dynamic shear wave slowness from well logs using machine learning methods in the Mishrif Reservoir, Iraq
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Publication Date
Tue Jun 15 2021
Journal Name
Al-academy
The artistic and performing methods of the sass dance in Baghdad "An Analytical Study": أمير علي رضا
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The research study includes shedding light on the sas dance in Iraq, as it is one of the popular legacies that are rich in the Mesopotamian civilization and because of the different practices it has in the different occasions on which these dances are held. Which is one of the kinetic arts widespread among the classes of Iraqi society. The researcher explained its features and artistic content that characterizes Iraqi music. The (methodological framework) reviewed the justification, importance and purpose of the research, and the limits of the research that included the (National Troupe of Folk Art) as a human limit to discover the artistic methods of the Sas dance and then define the terms of the search. The "theoretical framework" cont

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