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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
Mon Jul 27 2026
Journal Name
Al–bahith Al–a'alami
Convincing methods in designing infographic content analytical study
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The research aims to reveal the recent trends used in providing information and to know the persuasive methods used in designing the content of the infographic as well as the nature of persuasive design methods and the topics presented by the infographic in the research sample.   The researcher used the survey method, specifically the survey, by using the content analysis method to analyze the infographic material from the sample selected from the press and news sites that are the subject of the research, based on the method of what was said? How was it said? The researcher relied on the intentional sample, and this sample depends on the researcher selecting the vocabulary of the sample based on experiences and evaluating the c

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Publication Date
Sun Jan 02 2022
Journal Name
Advances In Science And Technology Research Journal
Vein Biometric Recognition Methods and Systems: A Review
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Publication Date
Tue Sep 19 2017
Journal Name
Al-academy
Directed designs the covers of children's magazines methods
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The design processes in general and the design system in particular need to be studied and knowledge. Based on this vision, we find the need for constant research on the design processes, and the subject of the techniques of the designs of the covers of children magazines did not have the view of the researchers to make them the subject of study, For the subject of this research, the problem of research is determined according to the following question:

- What are the methods used in designing children magazine covers in Iraq?

Through this research we seek to find solutions to this question as well as it will serve the cognitive side of a modern study and development of a broader understanding of the subject. Through four

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Publication Date
Sun Jul 25 2021
Journal Name
Journal Of Al-qadisiyah For Computer Science And Mathematics
A Survey paper on Lossy Audio Compression Methods
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During the two last decades ago, audio compression becomes the topic of many types of research due to the importance of this field which reflecting on the storage capacity and the transmission requirement. The rapid development of the computer industry increases the demand for audio data with high quality and accordingly, there is great importance for the development of audio compression technologies, lossy and lossless are the two categories of compression. This paper aims to review the techniques of the lossy audio compression methods, summarize the importance and the uses of each method.

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Publication Date
Mon Sep 30 2019
Journal Name
College Of Islamic Sciences
Grammatical and rhetorical methods   In Surat Al - Iftar
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I started writing the research because of the linguistic and rhetorical methods in the Holy Quran.
This Sura included in all its verses the plural form such as the word planets graves and seas except the word sky was singular as the name of a genus, and this Sura showed the horrors of the Day of Judgment at the world level Included another science is the science of the interview and also came in the repetition and deletion and other linguistic and rhetorical things and concluded this Sura that the goal of breaking the sky and scattering planets and scatter graves is the human himself.
The Koran is a miracle immortal and science is still drawn from him each according to his specialty.

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Publication Date
Tue Jun 23 2020
Journal Name
Baghdad Science Journal
Content Based Image Retrieval (CBIR) by Statistical Methods
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            An image retrieval system is a computer system for browsing, looking and recovering pictures from a huge database of advanced pictures. The objective of Content-Based Image Retrieval (CBIR) methods is essentially to extract, from large (image) databases, a specified number of images similar in visual and semantic content to a so-called query image. The researchers were developing a new mechanism to retrieval systems which is mainly based on two procedures. The first procedure relies on extract the statistical feature of both original, traditional image by using the histogram and statistical characteristics (mean, standard deviation). The second procedure relies on the T-

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Publication Date
Sun Jun 01 2014
Journal Name
Baghdad Science Journal
Solving Optimal Control Linear Systems by Using New Third kind Chebyshev Wavelets Operational Matrix of Derivative
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In this paper, a new third kind Chebyshev wavelets operational matrix of derivative is presented, then the operational matrix of derivative is applied for solving optimal control problems using, third kind Chebyshev wavelets expansions. The proposed method consists of reducing the linear system of optimal control problem into a system of algebraic equations, by expanding the state variables, as a series in terms of third kind Chebyshev wavelets with unknown coefficients. Example to illustrate the effectiveness of the method has been presented.

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Publication Date
Thu May 21 2026
Journal Name
Aip Conference Proceedings
Solving linear fractional Fredholm integro-differential equations using linear programming method via different types of polynomials
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This article introduces a novel procedure to detect an approximate solution to Fredholm fractional integro-differential equations with linear type (LFFIDE) defined using Caputo fractional derivative. The new procedure approximates the solution using three types of polynomials: Laguerre polynomials, Hermite polynomials, and Legendre polynomials, thereafter transforming the problem into a linear programming problem. The approximate solutions are compared using testing examples to examine the efficiency of the suggested approach. Also, a comparison with the other methods using the same polynomials illustrates The effectiveness and consistency of the proposed technique. Finally, the error analysis of the proposed technique and convergent are di

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Publication Date
Thu Nov 17 2016
Journal Name
Plos One
Efficient and Stable Routing Algorithm Based on User Mobility and Node Density in Urban Vehicular Network
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Vehicular ad hoc networks (VANETs) are considered an emerging technology in the industrial and educational fields. This technology is essential in the deployment of the intelligent transportation system, which is targeted to improve safety and efficiency of traffic. The implementation of VANETs can be effectively executed by transmitting data among vehicles with the use of multiple hops. However, the intrinsic characteristics of VANETs, such as its dynamic network topology and intermittent connectivity, limit data delivery. One particular challenge of this network is the possibility that the contributing node may only remain in the network for a limited time. Hence, to prevent data loss from that node, the information must reach the destina

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Publication Date
Fri Mar 01 2013
Journal Name
Journal Of Economics And Administrative Sciences
The construction of Investment Portfolios in the Iraq Stock Exchange: Market Timing Vs. an Efficient Selection
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Abstract

         Uncertainty, the deeply-rooted fact that surrounding the investment environment, especially the stock market which just prices have taken a specific trend until they moved to another one for its up or down. This means that the volatility characteristic of financial market requires the rational investor an argument led towards the adoption of planned acts to gain greater benefit in the goal of wealth maximizing. There is no possibility to achieve this goal without the burden of uncertainty and the risk of systematic fluctuations of investment returns in the financial market after the facts of  efficient diversification have pro

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