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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 Nov 27 2025
Journal Name
Journal Of Bioscience And Applied Research
Survey of plants for Brenaj, Wasit, Iraq
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The current study showed that the plants were collected from 23 geographical locations in Brenaj, Wasit, Iraq. The region was characterized by a great diversity of wild plants spread densely in this region. The results were as follows: 32 families, 149 species. Asteraceae was the most widespread with 29 species from the group of dicotyledons, followed by the Fabaceae family (19) species, but there are 13 plant families, with one plant species recorded for each plant family. in Brenaj, Wasit included: Aizoaceae, Capparaceae, Convolvulaceae, Frankeniaceae, Molluginaceae, Papaveraceae, Phyllanthaceae, Primulaceae, Rutaceae, Rubiaceae, Verbenaceae, Zygophyllaceae, Urticaceae, while the plant family Poaceae was most widespread in genera and spec

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Publication Date
Wed Jan 01 2025
Journal Name
Journal Of Intelligent Systems And Internet Of Things
Enhancing Convolutional Neural Network for Image Retrieval
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With the continuous progress of image retrieval technology, the speed of searching for the required image from a large amount of image data has become an important issue. Convolutional neural networks (CNNs) have been used in image retrieval. However, many image retrieval systems based on CNNs have poor ability to express image features. Content-based Image Retrieval (CBIR) is a method of finding desired images from image databases. However, CBIR suffers from lower accuracy in retrieving images from large-scale image databases. In this paper, the proposed system is an improvement of the convolutional neural network for greater accuracy and a machine learning tool that can be used for automatic image retrieval. It includes two phases

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Publication Date
Tue Dec 31 2019
Journal Name
Journal Of Economics And Administrative Sciences
Measuring the Range Application of Internal Marketing for HRM Philosophy in the Public Company for Electrical and Electronic Industries
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The reason behind choosing this topic " internal marketing (IM) of human resource management (HRM)" is to highlight the advantages of using IM in the organization framework. The problem of the research paper lies in not paying enough attention to employees genuine needs as they interact with each other in the sake of organization prosper. This research paper can be used as indictor to expose the weaknesses that the organization encounters daily. The current research paper attempts at examining the possibility of developing philosophy of internal marketing of human resources and its most practices, empowering staff, training courses, motivations and recognitions, and within departments communication, in order to reach targeted res

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Publication Date
Sun Mar 01 2020
Journal Name
Iop Conference Series: Materials Science And Engineering
Prepare Maps For Greenhouse Gases With Some Weather Elements For Baghdad City Using Data Observation And Arc-GIS Techniques
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Abstract<p>Air pollution refers to the release of pollutants into the air that are detrimental to human health and the planet as a whole.In this research, the air pollutants concentration measurements such as Total Suspended Particles(TSP), Carbon Monoxides(CO),Carbon Dioxide (CO2) and meteorological parameters including temperature (T), relative humidity (RH) and wind speed & direction were conducted in Baghdad city by several stations measuring numbered (22) stations located in different regions, and were classified into (industrial, commercial and residential) stations. Using Arc-GIS program ( spatial Analyses), different maps have been prepared for the distribution of different pollutant</p> ... Show More
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Scopus (4)
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Publication Date
Sat Feb 01 2020
Journal Name
Journal Of Economics And Administrative Sciences
Applying some hybrid models for modeling bivariate time series assuming different distributions for random error with a practical application
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Abstract

  Bivariate time series modeling and forecasting have become a promising field of applied studies in recent times. For this purpose, the Linear Autoregressive Moving Average with exogenous variable ARMAX model is the most widely used technique over the past few years in modeling and forecasting this type of data. The most important assumptions of this model are linearity and homogenous for random error variance of the appropriate model. In practice, these two assumptions are often violated, so the Generalized Autoregressive Conditional Heteroscedasticity (ARCH) and (GARCH) with exogenous varia

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Publication Date
Sat Dec 01 2012
Journal Name
Journal Of Economics And Administrative Sciences
ESTIMATION OF COEFFICIENTS AND SCALE PARAMETER FOR LINEAR (TYPE 1) EXTREME VALUE REGRESSION MODEL FOR LARGEST VALUES WITH APPLICATIONS
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In this paper we estimate the coefficients and scale parameter in linear regression model depending on the residuals are of type 1 of extreme  value distribution for the largest values . This can be regard as an improvement for the studies with the smallest values . We study two estimation methods ( OLS  & MLE ) where we resort to Newton – Raphson (NR) and Fisher Scoring methods to get MLE estimate because the difficulty of using the usual approach with MLE . The relative efficiency criterion is considered beside to the statistical inference procedures for the extreme value regression model of type 1 for largest values . Confidence interval , hypothesis testing for both scale parameter and regression coefficients

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Publication Date
Mon May 01 2017
Journal Name
Journal Of Arab Studies In Education & Psychology (asep)
Procedural Knowledge for the, Mathematics Departments Students, College of Education for Pure Sciences / Ibn al-Haytham, University of Baghdad
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Publication Date
Mon Feb 04 2019
Journal Name
Journal Of The College Of Education For Women
Self-Efficacy and its Relation to the Cognitive Assessment for the Daily Disturbances for the University of Baghdad Students
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The current research tackles the self-efficacy and its relation to the cognitive assessment for the daily disturbances for the University of Baghdad students. Two criteria have been adopted to achieve the objectives of the research. The sample of this study consists of 200 male and female students who were chosen randomly. The data were analyzed statistically, revealing that the university students owned their own self-efficacy as well as a cognitive assessment for the daily disturbances and they recognized them as self-threatening. The results also indicated the existence of a prediction activity in the field of the cognitive assessment to the daily disturbances selection. In light of the acquired results, the study recommends the neces

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Publication Date
Thu Apr 30 2020
Journal Name
International Journal Of Psychosocial Rehabilitation
The use of physical exercises specialized for with aids of developing explosive power for arms and legs and their effect on free throw and jump shooting skills for advanced basketball players
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The research aims to develop physical exercises with auxiliary training tools that work to develop the explosive power of the arms and legs, and then find out their effect on the accuracy of shooting from free throw and correction from jumping of advanced basketball players, as the researchers found a problem that these players have weakness in the skill of throwing Free throwing and shooting by jumping calculated with two points as a result of adopting unhealthy physical and technical positions, which led to a lack of focus and accuracy, and thus negatively affected the performance technique of free throw and jump shot, as most teams use traditional exercises without the use of auxiliary training tools, and this topic gave researchers the

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Publication Date
Thu Oct 15 2026
Journal Name
Journal Of Physical Education
Standard Criteria for Some Special Abilities as Indicator for Selecting Iraqi Club&#39;s Weightlifters Aged 14 – 16 years Old
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