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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 27 2019
Journal Name
Journal Of Accounting And Financial Studies ( Jafs )
The application of the dimensions of TQM and its impact in enhancing the satisfaction of the taxpayers: Applied Study in the General Commission for Taxes
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The concept of TQM is based on one of the concepts that combine administrative and innovative methods. The aim of the research is to demonstrate the dimensions of TQM in enhancing the satisfaction of the taxpayers through a survey of a sample of officials in the General Authority for Taxation and 50 officials. In the collection of data and information, and the results were analyzed using the SPSS program to find the most important compounds and factors in he method of analysis.

The research problem was represented by the non-application of the General Authority for Taxation to the entrances and modern practices in the administrative work. The results of some of the complications that accompany the tax accounting process, which af

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Crossref
Publication Date
Wed Sep 30 2020
Journal Name
Journal Of Accounting And Financial Studies ( Jafs )
Adopting International Financial Reporting Standard No. (9) Financial Instruments - Recognition and Measurement of Accounting for Shares and its Reflection on the Financial Statements of Companies
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Many financial institutions invest their surplus funds in stocks, either to obtain dividends or for trading purposes and to obtain profits from the difference between the cost and the selling price, and investment in shares represents an important part of the financial position of financial institutions applying to the common accounting system of banks and insurance companies, in addition to their impact It is clear on the result of the activity of these institutions.The aim of the research is to define what the shares and their types are, and to indicate the accounting treatments needed to move towards the process of adopting the International Financial Reporting Standard No. (9) and its reflection on its financial statements. I

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Publication Date
Tue Oct 01 2019
Journal Name
International Journal Of Drug Delivery Technology
Simple and Rapid Method For Estimate of Propranolol With Bi (III) Via Long-Distance Chasing Photometer (NAG-ADF-300-2) Utilization Continuous Flow Injection Analysis
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A simple, sensitive and rapid method was used for the estimate of: Propranolol with Bi (III) to prove the efficiency, reliability and repeatability of the long distance chasing photometer (NAG-ADF-300-2) using continuous flow injection analysis. The method is based on a reaction between propranolol and Bi (III) in an aqueous medium to obtain a yellow precipitate. Optimum parameters were studied to increase the sensitivity for the developed method. A linear range for calibration graph was 0.1-25 mmol/L for cell A and 1-40 mmol/L for cell B, and LOD 51.8698 ng/200 µL and 363.0886 ng /200 µL , respectively to cell A and cell B with correlation coefficient (r) 0.9975 for cell A, 0.9966 for cell B, RSD% was lower than 1%, (n = 8) for the

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Publication Date
Wed Oct 02 2024
Journal Name
International Development Planning Review
THE EFFECT OF EXERCISES USING A MINI SQUASH COURT ON IMPROVING SOME MOTOR ABILITIES AND LEARNING SOME BASIC SKILLS FOR PLAYERS AGED 10-12 YEARS
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Publication Date
Wed Mar 12 2025
Journal Name
The Alarba'een Journal
Analyzing and tracking the data of the millions sized gatherings for the Arba'in visit and proposing alternative ways to relieve congestion using spatial analysis algorithms
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The city of Karbala is one of the most important holy places for visitors and pilgrims from the Islamic faith, especially through the Arabian visit, when crowds of millions gather to commemorate the martyrdom of Imam Hussein. Offering services and medical treatments during this time is very important, especially when the crowds head to their destination (the holy shrine of Imam Hussein (a.s)). In recent years, the Arba'in visit has witnessed an obvious growth in the number of participants. The biggest challenge is the health risks, and the preventive measures for both organizers and visitors. Researchers identified various challenges and factors to facilitating the Arba'in visit. The purpose of this research is to deal with the religious an

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Publication Date
Sun Feb 22 2026
Journal Name
Clinical Drug Investigation
A Field-Based Cost-Effectiveness Study of Aflibercept and Bevacizumab in Treat-and-Extend Versus Pro Re Nata Protocols for Diabetic Macular Edema in Iraq
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Publication Date
Sun Jun 27 2021
Journal Name
Iraqi National Journal Of Nursing Specialties
Assessment of Mothers’ Knowledge about Their Children with Sickle Cell Anemia and Non-Pharmacological Approaches to Pain Management in Basra Center for Hereditary Blood Diseases
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Objective(s): To assess mothers’ knowledge about their children with sickle cell anemia and non-Pharmacological approaches to pain management and found some relationship between mothers knowledge and their demographic data of age, level of education, and occupation.
Methodology: A descriptive design used in the present study established was for a period from September 19th, 2020 to March 30th, 2021. The study was conducted on a non-probability (purposive) sample of (30) mother their children with sickle cell anemia was chosen. The data were analyzed through the application of descriptive and inferential statistical approaches which are applied by using SPSS version 22.0.
Results: The findings of the study indicated that moderate

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Publication Date
Sun Sep 01 2013
Journal Name
Journal Of Economics And Administrative Sciences
Measurement of economic returns and environmental and social for green sector trade A prospective study in institutions of Basra to focus on the electricity sector
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        Environmental risk growing Become challenge "and a matter of controversy and concern to many of those concerned with environment, social, economic, and the same happens with the administrative rather than in isolation for this movement, as the issues of climate change Disturbed and troubled him, especially after what caused the risk of destruction, and irresponsibility , chaos, and the futility of resources, crops, fields,  nature and homes and reactors, and after what happened in Japan from the scourge of "Hurricane tsunami " and earthquakes successive accompanied him and what became of him by surprise catastrophic affected the economy and the univ

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Crossref
Publication Date
Sun Sep 06 2009
Journal Name
Baghdad Science Journal
Synthesis of several Urea plast Resins using Different Aldehydes Preparation of some Alkyde Resins and Empolyment of the prepared Resins as Additives for AZO Dyes
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Urea formaldehyde resin was prepared by using basic media by yield 95%. The Remaining of ureaplasts resin were prepared in acetic acid media by high yield. Alkyde resins were prepared by condensation polymerization by react Succinic, Maleic, Phthalic anhydrides with Ethylene glycol or Glycerol. Select samples of the prepared alkyde resins were mixed with Azo dyes in special ratio. The mixtures were used as coatings for wood, and compaised with pure dyes. The Coating that some alkyde resins showed better adhesion from using dyes alone. Preparation of wood coating by mixing ureaplast resins and alkyde resins with Azo dyes in special ratios. The coating showed better adhesion, brighter colors and better resistance to heat from Preceding coat

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Crossref
Publication Date
Thu Mar 10 2022
Journal Name
International Journal Of Early Childhood Special Education
The effect of reciprocal style exercises in developing some physical abilities in learning the performance of female players for the effectiveness of the long jump
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The importance of the research lies in knowing the effect of the exercises of the reciprocal method in developing some physical abilities in learning the performance of the players for the effectiveness of the long jump in an economical manner in terms of time and effort and knowing their positive impact and the extent of their impact in creating the required learning for students, and the research aims to prepare reciprocal style exercises in developing some abilities The researchers used the experimental method in the pre and post test for the experimental and control groups to suit the nature of the research, and the research community was identified for the long jump players, the Specialized School for Talent Care in the 2022 sports sea

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