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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 Jun 30 2007
Journal Name
Iraqi Journal Of Chemical And Petroleum Engineering
Corrosion Inhibition of Carbon Steel under Two Phase Flow (Water-Petroleum) Simulated by Turbulently Agitated System
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The corrosion of carbon steel in single phase (water with 0.1N NaCl ) and two immiscible phases (kerosene-water) using turbulently agitated system is investigated. The experiments are carried out for Reynolds number (Re) range of 38000 to 95000 corresponding  to rotational velocities from 600 to 1400 rpm using circular disk turbine agitator at 40 0C. In two-phase system test runs are carried out in aqueous phase (water) concentrations of 1 % vol., 5 % vol., 8% vol., and 16% vol. mixed with kerosene at various Re. The effect of Reynolds number (Re), percent of dispersed phase, dispersed drops diameter, and number of drops per unit volume on the corrosion rate is investigated and discussed. Test runs are carried out using two types of

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Publication Date
Sun Mar 18 2018
Journal Name
Mustansiria Dental Journal
The Effect of Pepsi Cola Beverage on Surface Roughness of Two Composite Resins (In Vitro study)
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An acidic environment causes surface changes of resin composites. Filler particlesize and filler distribution also have a direct effect on these surface changes. This invitro study evaluated the influence of Pepsi Cola drink on the surface roughness ofComposan LCM and Composan Ceram over time. Sixteen disc shaped specimens(10mm diameter, 2mm thickness) of each resin composite were fabricated, therebyforming two groups (n= 8). Surface roughness (Ra) was analyzed after 24 hrs beforeexposure to beverage. The specimens were submitted to a five minutes immersion inPepsi Cola three times daily interrupted by immersion in deionized distilled water (37C˚). Surface roughness measurements were done at 10, 30, and 60 days intervals. Datawere

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Publication Date
Fri Mar 15 2019
Journal Name
Al-khwarizmi Engineering Journal
Experimental and Theoretical Study of the Energy Flow of a Two Stages Four Generators Adsorption Chiller
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This work is concerned with a two stages four beds adsorption chiller utilizing activated carbon-methanol adsorption pair that operates on six separated processes. The four beds that act as thermal compressors are powered by a low grade thermal energy in the form of hot water at a temperature range of 65 to 83 °C.  As well as, the water pumps and control cycle consume insignificant electrical power. This adsorption chiller consists of three water cycles. The first water cycle is the driven hot water cycle. The second cycle is the cold water cycle to cool the carbon, which adsorbs the methanol. Finally, the chilled water cycle that is used to overcome the building load. The theoretical results showed that average cycle cooling power

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Publication Date
Wed Jan 15 2020
Journal Name
Iraqi Journal Of Laser
Optimizing the Secure Key Rate of a Single Sequential Quantum Repeater with Two Different Quantum Memories
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Quantum channels enable the achievement of communication tasks inaccessible to their
classical counterparts. The most famous example is the distribution of secret keys. Unfortunately, the rate
of generation of the secret key by direct transmission is fundamentally limited by the distance. This limit
can be overcome by the implementation of a quantum repeater. In order to boost the performance of the
repeater, a quantum repeater based on cut-off with two different types of quantum memories is suggestd,
which reduces the effect of decoherence during the storage of a quantum state.

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Publication Date
Wed Sep 14 2016
Journal Name
Journal Of Baghdad College Of Dentistry
Effect of Mouth Rinses on Surface Roughness of Two Methacrylate-Based and Siloraine-Based Composite Resins
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Background: Various fluids in the oral environment can affect the surface roughness of resin composites. This in vitro study was conducted to determine the influence of the mouth rinses on surface roughness of two methacrylate-based resin (nanofilled and packable composite) and siloraine-based resin composites.

Materials and methods: Disc-shaped specimens (12 mm in diameter and 2mm in height) were prepared from three types of composi

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Publication Date
Sun Jan 01 2023
Journal Name
Journal Of Oral Medicine And Oral Surgery
Reliability of two difficulty indexes in predicting the surgical extraction difficulty of impacted mandibular third molars
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Introduction: The aim of this study was to compare the reliability and the agreement of the Pederson and Pernambuco difficulty indexes in predicting the surgical extraction difficulty of the impacted mandibular third molars. Materials and methods: A prospective observational cohort study was conducted on 83 patients who had undergone surgical removal of impacted mandibular third molar. The difficulty of extraction was determined preoperatively according to the total scores obtained from the Pederson and Pernambuco difficulty indexes, the operative difficulty was determined by the surgical technique and the duration of extraction. The accuracy of prediction of the surgical difficulty and

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Publication Date
Fri Jan 01 2016
Journal Name
Aip Conference Proceedings
Analysis of performance measures with single channel fuzzy queues under two class by using ranking method
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Publication Date
Tue Sep 01 2020
Journal Name
Iraqi Journal Of Physics
Study of the nuclear structure of halo nuclei 23O and 24F using the two-body model
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The nuclear structure included the matter, proton and neutron densities of the ground state, the nuclear root-mean-square (rms) radii and elastic form factors of one neutron 23O and 24F halo nuclei have been studied by the two body model of  within the harmonic oscillator (HO) and Woods-Saxon (WS) radial wave functions. The calculated results show that the two body model within the HO and WS radial wave functions succeed in reproducing neutron halo in these exotic nuclei. Moreover, the Glauber model at high energy has been used to calculated the rms radii and reaction cross section of these nuclei.

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Publication Date
Tue Sep 01 2015
Journal Name
Ama, Agricultural Mechanization In Asia, Africa And Latin America
Effect of three honeycomb interplant distances on yield and it components of two cultivars of bean
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A factorial experiment (2× 3) in randomized complete block design (RCBD) with three replications was conducted to examine the effect of honeycomb selection method using three interplant distances on yield and its components of two cultivars of bean, Bronco and Strike. Interplant distances used were 75× 65 cm, 90× 78 cm, and 105× 91 cm (row× plant) represent short (high plant density), intermediate (intermediate plant density), and wide (low plant density) distance, respectively. Parameters used for selection were number of days from planting to the initiation of first flower, number of nodes formed prior to first flower, and number of main branches. Results showed significant superiority of the Bronco cultivar represented in the number

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Publication Date
Thu May 31 2018
Journal Name
International Journal Of Control And Automation
Power Flow Control of Iraqi International Super Grid with Two-Terminal HVDC Techniques Using PSS/E
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