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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 Aug 29 2024
Journal Name
International Journal Of Sustainable Development And Planning
Exploring the Transformative Effects of GPS and Satellite Imagery on Urban Landscape Perceptions in Baghdad: A Mixed-Methods Analysis
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Publication Date
Sun Oct 15 2023
Journal Name
Bionatura
Effect of Partial Drip Irrigation Methods on Soil Moisture and Water Potential Distribution, Growth Characteristics and Yield of Maize
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A field experiment was conducted during the autumn of 2021 at the Agricultural Research Department station / Abu Ghraib to evaluate the soil moisture, water potential distribution, and growth factors of maize crops under alternating and constant partial drip irrigation methods. In the experiment, two irrigation systems were used, surface drip irrigation (DI) and subsurface irrigation (SD); under each irrigation system, five irrigation methods were: conventional irrigation (CI), and 75 and 50% of the amount of water of CI of each of the alternating partial irrigation APRI75 and APRI50 and the constant partial irrigation FPRI75 and FPRI50 respectively. The results showed that the water depth for conventional irrigation (C1) was 658.3

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Publication Date
Mon Jun 08 2020
Journal Name
Research Journal Of Chemistry And Environment
High Performance Liquid Chromatographic and Areaunder Curve spectrophotometric Methods forEstimation of Cefixime in Pure and MarketedFormulation: A Comparative Study
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Cefixime is an antibiotic useful for treating a variety ofmicroorganism infections. In the present work, tworapid, specific, inexpensive and nontoxic methods wereproposed for cefixime determination. Area under curvespectrophotometric and HPLC methods were depictedfor the micro quantification of Cefixime in highly pureand local market formulation. The area under curve(first technique) used in calculation of the cefiximepeak using a UV-visible spectrophotometer.The HPLC (2nd technique) was depended on thepurification of Cefixime by a C18 separating column250mm (length of column) × 4.6 mm (diameter)andusing methanol 50% (organic modifier) and deionizedwater 50% as a mobile phase. The isocratic flow withrate of 1 mL/min was applied, the temper

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Publication Date
Mon Mar 02 2009
Journal Name
Albahith Alalami
Types, contents and scientific research methods used in public relations research at the College of Mass Communication 1989 - 2007.
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This study deals with knowing the types of research in public relations studies at the Faculty of Mass Communication for the period from 1989 to 2007 and knowing the methods that researchers used in their studies, as well as knowing the methods and tools of scientific research that were used, and whether one method or tool is used more than others, and knowing Samples used in the analysis. As well as the contents and types of this research The institutions that were subjected to the study, and the preparation of the respondents This research aims to answer the following questions:. A- What are the types of research and methods that were used in university theses that dealt with the topic? Public Relations at the Faculty of Media from 1989

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Publication Date
Wed Mar 30 2022
Journal Name
Journal Of Economics And Administrative Sciences
Comparing Some Methods of Estimating the Parameters and Survival Function of a Log-logistic Distribution with a Practical Application
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The Log-Logistic distribution is one of the important statistical distributions as it can be applied in many fields and biological experiments and other experiments, and its importance comes from the importance of determining the survival function of those experiments. The research will be summarized in making a comparison between the method of maximum likelihood and the method of least squares and the method of weighted least squares to estimate the parameters and survival function of the log-logistic distribution using the comparison criteria MSE, MAPE, IMSE, and this research was applied to real data for breast cancer patients. The results showed that the method of Maximum likelihood best in the case of estimating the paramete

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Publication Date
Sun Jan 01 2023
Journal Name
Revista Iberoamericana De Psicología Del Ejercicio Y El Deporte
THE IMPACT OF ENGINEERING ANXIETY ON STUDENTS: A COMPREHENSIVE STUDY IN THE FIELDS OF SPORT, ECONOMICS, AND TEACHING METHODS
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Publication Date
Thu Aug 01 2019
Journal Name
Journal Of Economics And Administrative Sciences
The use of the Biz method and classical methods in estimating the parameters of the binary logistic regression model
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Abstract

          Binary logistic regression model used in data classification and it is the strongest most flexible tool in study cases variable response binary when compared to linear regression. In this research, some classic methods were used to estimate parameters binary logistic regression model, included the maximum likelihood method, minimum chi-square method, weighted least squares, with bayes estimation , to choose the best method of estimation by default values to estimate parameters according two different models of general linear regression models ,and different s

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Publication Date
Wed May 31 2023
Journal Name
Iraqi Geological Journal
Studying the Effect of Permeability Prediction on Reservoir History Matching by Using Artificial Intelligence and Flow Zone Indicator Methods
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The map of permeability distribution in the reservoirs is considered one of the most essential steps of the geologic model building due to its governing the fluid flow through the reservoir which makes it the most influential parameter on the history matching than other parameters. For that, it is the most petrophysical properties that are tuned during the history matching. Unfortunately, the prediction of the relationship between static petrophysics (porosity) and dynamic petrophysics (permeability) from conventional wells logs has a sophisticated problem to solve by conventional statistical methods for heterogeneous formations. For that, this paper examines the ability and performance of the artificial intelligence method in perme

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Publication Date
Sun Jun 07 2015
Journal Name
Baghdad Science Journal
A comparative study to determine the nature of the eye lens protein in the two types of birds Iraqi
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The study showed significant differences between the average weight lens and the average amount protein in the lens between that Kestrel Falco tinnunculus L. and the Collared Dove Streptopelia decaocto F. , also the study electrical migration of lens proteins having one bundle of crystalline –? in Kestrel compared with three bundles in Collared Dove, two bundles of crystalline – ? in both , and crystalline – ? appeared as one bundle in both birds.

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Publication Date
Tue Sep 01 2020
Journal Name
Baghdad Science Journal
Reconstruction of Three-Dimensional Object from Two-Dimensional Images by Utilizing Distance Regularized Level Algorithm and Mesh Object Generation
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Three-dimensional (3D) reconstruction from images is a most beneficial method of object regeneration by using a photo-realistic way that can be used in many fields. For industrial fields, it can be used to visualize the cracks within alloys or walls. In medical fields, it has been used as 3D scanner to reconstruct some human organs such as internal nose for plastic surgery or to reconstruct ear canal for fabricating a hearing aid device, and others. These applications need high accuracy details and measurement that represent the main issue which should be taken in consideration, also the other issues are cost, movability, and ease of use which should be taken into consideration. This work has presented an approach for design and construc

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