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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
Sat Dec 02 2017
Journal Name
Al-khwarizmi Engineering Journal
Biomechanical Aspects of Shoulder and Hip Articulations: A Comparison of Two Ball and Socket Joints
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The shoulder and hip joints though essentially both are ball and socket joints, show structural variability to serve functional needs.

This study aims at revealing some of the structural and functional properties of each of the two joints regarding the factors that contribute to the stability of any joint in the body, namely: bone, ligament, and muscle.

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Publication Date
Fri Jan 01 2021
Journal Name
Aip Conference Proceedings
Study of rheological and mechanical properties of bitumen blended with two alpha phases of polyamide
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Publication Date
Fri Oct 03 2025
Journal Name
Journal Of Baghdad College Of Dentistry
Salivary Oxidative Stress Markers in Relation to Vascular Disease Risk of Type Two Diabetes Mellitus
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Background: Cardiovascular disease (CVD) is an important complication of type 2 diabetes mellitus (T2DM). Oxidative stress plays a major role in the development of CVD. Saliva has a diagnostic properties aiding in the detection of systemic diseases. This study aimed to assess the association between salivary oxidative stress markers and the risk of vascular disease (VD) in T2DM patients. Materials and Methods: One hundred T2DM patients and fifty apparently healthy males were enrolled in this study. Saliva sample was collected for assessment of oxidative stress markers including: lipid peroxidation plasma thiobarbituric acid-reactive substances (TBARS), uric acid (UA) and total antioxidant capacity (TAC) levels. Arterial stiffness index (ASI

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Publication Date
Sun Dec 14 2014
Journal Name
Molecular Crystals And Liquid Crysta
Synthesis and Mesomorphic Behavior of Two New Homologous Series Containing Azobenzene and 1,3,4-Oxadiazole Units
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Two new nonsymmetrical mesogenic homologous series of terminal substituent ether (series [Vn]) and carboxy (series [VIn]) incorporating azobenzene and 1,3,4-oxadiazole group were synthesized. Both series have been All compounds thus isolated were purified and characterized by elemental analysis, Fourier Transform Infrared Spectroscopy, 1H NMR, along with thermal analysis and texture observation using Differential Scanning Calorimetry (DSC) and Polarizing Optical Microscopy (POM), respectively. All compounds of the first series exhibited liquid crystalline properties. The homologues [V1]-[V3] display a nematic mesophase, the compounds [V4]-[V7] exhibit a dimorphism behavior, nematic (N) and smectic A (SmA) mesophases, the compounds [V8] and

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Publication Date
Tue Jun 15 2021
Journal Name
Al-academy
The directing vision and two display technologies in the Iraqi Child Theater: عباس قاسم كاظم
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  The current research contains four chapters. The first topic included the methodological framework that included the problem of research and the need for it, and then the importance of the research, and then the aim of the research, its limits and the definition of terms linguistically, conventionally and procedurally. The second chapter (the theoretical framework) contained two topics, the topic The first is titled: The Outward Vision in Child Theater Performances, while the second topic was titled: Presentation Technology in Child Theater, Chapter Four (Research Procedures), which organized the research community and analyzed its sample, and the fourth chapter contained research results, conclusions and recommendations. The prop

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Publication Date
Thu Feb 07 2019
Journal Name
Iraqi Journal Of Laser
Design and Comparative Study of a one and Two Stage Blumlein Circuit TEA Nitrogen Laser
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Two Prototypes of Transversely Excited at atmospheric pressure (TEA) Nitrogen laser systems (One Stage Blumlein Circuit and Two Stage Blumlein Circuit) were fabricated and operated. High voltage power supply with variable operating voltage (0-20 kv) and operating current (1-3A) was built and tested successfully. The gas flow rate of 15 L/ min and 10 L/ min for OSBC and TSBC was used. The performance of the fabricated systems was studied extensively reaching to the optimum operating conditions. The obtained laser output energy for the first system has linear relationship with the applied voltage. The maximum output energy was about (1.14 mJ) with (10.40) ns pulse duration and the half-wave divergence angle was about (0.1455 m rad). In the

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Publication Date
Wed May 14 2025
Journal Name
Journal Of Applied Mathematics
Role of Wind and Fear on the Dynamic of a Prey and Two Competing Predators
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An essential tool for studying the web is its ability to show how energy moves through an ecosystem. Understanding and elucidating the relationship between species variety and their placement within the inclusive trophic dynamics is also beneficial. A food web ecological model with prey and two rival predators under fear and wind flow conditions is developed in this article. The boundedness and positivity of the system’s solution are established mathematically. The stability and existence constraints of the system’s equilibria are examined. The proposed system’s persistence limitations are established. Additionally, the bifurcation analysis of every potential equilibrium is examined using the Sotomayor theorem. To describe the

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Publication Date
Mon Aug 01 2022
Journal Name
Journal Of Physics: Conference Series
The local bifurcation analysis of two preys stage-structured predator model with anti-predator behavior
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Abstract<p>This paper deals with two preys and stage-structured predator model with anti-predator behavior. Sufficient conditions that ensure the appearance of local and Hopf bifurcation of the system have been achieved, and it’s observed that near the free predator, the free second prey and the free first prey equilibrium points there are transcritical or pitchfork and no saddle node. While near the coexistence equilibrium point there is transcritical, pitchfork and saddle node bifurcation. For the Hopf bifurcation near the coexistence equilibrium point have been studied. Further, numerical analysis has been used to validate the main results.</p>
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Publication Date
Sat Dec 30 2023
Journal Name
Wasit Journal For Pure Sciences
A Review of Modern Methods of Synthesis 1, 3, 4-Oxadiazole as a Bioactive Compounds
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ABSTRACT: Oxadiazole ring is a heterocyclic molecule with an oxygen and two nitrogen atoms spread throughout its five-membered structure. There are four different isomers that have been discovered, Because of their wide applications in a range of sectors, including medications . Some of these biological activity are; anticonvulsant capacity, anticancer as well, antibacterial, antiviral, antifungal,  antimalarial, antitubercular, anti-asthmatic, antidepressant, antidiabetic, antioxidant, antiparkinsonian, analgesic and  anti-inflammatory, are just some of the therapeutic uses that have drawn attention to drug candidates containing an oxadiazole moiety. This review, we will examine the various methods of oxadiazole synthesis. The mo

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Publication Date
Sun Jun 20 2021
Journal Name
Baghdad Science Journal
Comparison of Some of Estimation methods of Stress-Strength Model: R = P(Y < X < Z)
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In this study, the stress-strength model R = P(Y < X < Z)  is discussed as an important parts of reliability system by assuming that the random variables follow Invers Rayleigh Distribution. Some traditional estimation methods are used    to estimate the parameters  namely; Maximum Likelihood, Moment method, and Uniformly Minimum Variance Unbiased estimator and Shrinkage estimator using three types of shrinkage weight factors. As well as, Monte Carlo simulation are used to compare the estimation methods based on mean squared error criteria.  

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