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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 May 01 2022
Journal Name
Journal Of Engineering
Parametric Study of a Two-Phase Closed Thermosyphon Loop
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A theoretical and experimental investigation was carried out to study the behavior of a two-phase closed thermosyphon loop (TPCTL) during steady-state operation using different working fluids. Three working fluids were investigated, i.e., distilled water, methanol, and ethanol. The TPCTL was constructed from an evaporator, condenser, and two pipelines (riser and downcomer). The driving force is the difference in pressure between the evaporator and condenser sections and the fluid returns to the heating section by gravity. In this study, the significant parameters used in the experiments were filling ratios (FR%) of 50%, 75%, and 100% and heat-input range at the evaporator section of 215-860.2 W. When the loop reached to

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Publication Date
Wed Jun 01 2022
Journal Name
Iraqi Journal Of Physics
Effects of Hydrogen Peroxide Concentration on Properties of Black Silicon Fabricated by Two-Step Silver-Assisted Wet Chemical Etching for Photovoltaics
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Crystalline silicon (c-Si) has low optical absorption due to its high surface reflection of incident light. Nanotexturing of c-Si which produces black silicon (b-Si) offers a promising solution. In this work, effect of H2O2 concentrations towards surface morphological and optical properties of b-Si fabricated by two-step silver-assisted wet chemical etching (Ag-based two-step MACE) for potential photovoltaic (PV) applications is presented. The method involves a 30 s deposition of silver nanoparticles (Ag NPs) in an aqueous solution of AgNO3:HF (5:6) and an optimized etching in HF:H2O2:DI H2O solution under 0.62 M, 1.85 M, 2.47 M, and 3.7 M concentrations of H2O<

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Publication Date
Sun Oct 01 2017
Journal Name
Journal Of Economics And Administrative Sciences
Under Different Priors &Two Loss Functions To Compare Bayes Estimators With Some of Classical Estimators For the Parameter of Exponential Distribution
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المستخلص:

          في هذا البحث , استعملنا طرائق مختلفة لتقدير معلمة القياس للتوزيع الاسي كمقدر الإمكان الأعظم ومقدر العزوم ومقدر بيز في ستة أنواع مختلفة عندما يكون التوزيع الأولي لمعلمة القياس : توزيع لافي  (Levy) وتوزيع كامبل من النوع الثاني وتوزيع معكوس مربع كاي وتوزيع معكوس كاما وتوزيع غير الملائم (Improper) وتوزيع

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Publication Date
Thu Jan 01 2015
Journal Name
Journal Of Traumatic Stress Disorders &amp; Treatment
Randomised Controlled Study Comparing Two Psychological Therapies for Posttraumatic Stress Disorder (PTSD): Emotional Freedom Techniques (EFT) vs. Narrative Exposure Therapy (NET)
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Publication Date
Mon Oct 01 2018
Journal Name
Journal Of Educational And Psychological Researches
The ability of solving a mathematical problem and its relation to system thinking among fifth preparatory students
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The research seeks to examine the ability of fifth preparatory students in solving a mathematical problem in relation to system thinking. To this end, the researcher chose (140) fifth preparatory students from four-different secondary schools in Kirkuk city for the academic year (2016-2017). Two tests were adopted to collect study data: a test of (5) items about skills in solving math problem designed by (Al-raihan, 2006); and a test of system thinking skills designed by the researcher himself consisted of (14) items. It was divided into four skills (analyzing the main system to subsystems, eliminating all inner gaps of system, identifying the inner connection of system, and reorganizing the system). The findings indicated a good ability

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Publication Date
Tue Feb 01 2022
Journal Name
Baghdad Science Journal
Solving Whitham-Broer-Kaup-Like Equations Numerically by using Hybrid Differential Transform Method and Finite Differences Method
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This paper aims to propose a hybrid approach of two powerful methods, namely the differential transform and finite difference methods, to obtain the solution of the coupled Whitham-Broer-Kaup-Like equations which arises in shallow-water wave theory. The capability of the method to such problems is verified by taking different parameters and initial conditions. The numerical simulations are depicted in 2D and 3D graphs. It is shown that the used approach returns accurate solutions for this type of problems in comparison with the analytic ones.

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Publication Date
Wed Sep 15 2021
Journal Name
Al-academy
Applying the substance-field model mechanism to problem solving in industrial product design: محمد علي حسين القيسي
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  Problem solving methods and mechanisms contribute to facilitating human life by providing tools to solve simple and complex daily problems. These mechanisms have been essential tools for professional designers and design students in solving design problems.
This research dealt with one of those mechanisms, which is the (the substance-field model model), as it has been mentioning that this mechanism is characterized by the difficulty of its application, which formed the main research problem. In home gardens (the sub-problem of research), an analysis of this problem was applied and then a solution was found to address it. The researcher used the 3dsmax program to implement the proposed design.
The most important research res

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Publication Date
Sat May 02 2026
Journal Name
Modern Sport
Information Processing Styles, Problem Solving, and Their Relationship with Metamotivation and Perceived Control among Postgraduate Female Students
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The demands of professional, scientific, and academic life have made it necessary to identify the various difficulties faced by postgraduate female students, which lead to problems that hinder their achievement of the required goals. This necessitates directing efforts toward finding solutions to confront and solve problems through appropriate cognitive behavior. Therefore, this study aimed to identify information processing styles and problem-solving and their relationship with metamotivation and perceived control among postgraduate female students (PhD/Master's). To achieve this aim, the descriptive approach using the survey method was adopted due to its suitability to the nature of the research problem. The population was defined

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Publication Date
Thu Nov 08 2018
Journal Name
Iraqi National Journal Of Nursing Specialties
Suggested Technique for Creating Physical Growth Curves Charts for Anthropometric Measurements in Admixed form in a sample of children under two years of age In Diyala Governorate
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Objective: This study aimed to assessing new suggested technique of Physical Growth Curves (PGC) charts in
children under two years old of a non-probability sample.
Methodology: A non-probability sample of size (420) children under two years selected from 12 Primary
Health Care Centers in Diyala governorate during the period from 15th Nov. 2010 to 13th Mar. 2011
according to admix of a different properties together in one chart/or growth curve chart included in at least
weight, Height, and Head circumference.
Results: the results showed different properties that can be admix together in one chart/or growth curve
chart included in at least weight, Height, and Head circumference. And to overtake the problem of the norm

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Publication Date
Tue Oct 01 2024
Journal Name
Renewable And Sustainable Energy Reviews
A critical review on the efficient cooling strategy of batteries of electric vehicles: Advances, challenges, future perspectives
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