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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 Mar 29 2021
Journal Name
Journal Of Engineering
Extracting Four Solar Model Electrical Parameters of Mono-Crystalline Silicon (mc-Si) and Thin Film (CIGS) Solar Modules using Different Methods
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Experimental measurements were done for characterizing current-voltage and power-voltage of two types of photovoltaic (PV) solar modules; monocrystalline silicon (mc-Si) and copper indium gallium di-selenide (CIGS). The conversion efficiency depends on many factors, such as irradiation and temperature. The assembling measures as a rule cause contrast in electrical boundaries, even in cells of a similar kind. Additionally, if the misfortunes because of cell associations in a module are considered, it is hard to track down two indistinguishable photovoltaic modules. This way, just the I-V, and P-V bends' trial estimation permit knowing the electrical boundaries of a photovoltaic gadget with accuracy. This measure

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Crossref (5)
Crossref
Publication Date
Wed Mar 30 2022
Journal Name
Journal Of Economics And Administrative Sciences
Planning the Production of the Electrical Distribution Converter (400KV/11) Using Time Series Methods and Goal Programming in the Fuzzy Environment
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This Paper aims to plan the production of the electrical distribution converter (400 KV/11) for one month at Diyala Public Company and with more than one goal for the decision-maker in a fuzzy environment. The fuzzy demand was forecasting using the fuzzy time series model. The fuzzy lead time for raw materials involved in the production of the electrical distribution converter (400 KV/11) was addressed using the fuzzy inference matrix through the application of the matrix in Matlab, and since the decision-maker has more than one goal, so a mathematical model of goal programming was create, which aims to achieve two goals, the first is to reduce the total production costs of the electrical distribution converter (400 KV/11) and th

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Crossref
Publication Date
Wed Dec 18 2019
Journal Name
Baghdad Science Journal
The Calculation and Analysis of the Total Electron Content Over Different Latitudes and Seasons Using the Numerical Trapezoidal and Simpson Methods
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It has been shown in ionospheric research that calculation of the total electron content (TEC) is an important factor in global navigation system. In this study, TEC calculation was performed over Baghdad city, Iraq, using a combination of two numerical methods called composite Simpson and composite Trapezoidal methods. TEC was calculated using the line integral of the electron density derived from the International reference ionosphere IRI2012 and NeQuick2 models from 70 to 2000 km above the earth surface. The hour of the day and the day number of the year, R12, were chosen as inputs for the calculation techniques to take into account latitudinal, diurnal and seasonal variation of TEC. The results of latitudinal variation of TE

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Publication Date
Sun Dec 30 2018
Journal Name
Baghdad Science Journal
Comparative morphological and histological study of the pecten oculi in two species of Iraqi birds (Falco tinnunculus L. and Streptopelia decaocto F.)
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Study showed structure of pecten oculi in the Kestrel Falco tinnunculus L.was
Pleated type and consisted of 17 folds which were thick. While in the Collared Dove
Streptopelia decaocto F. was Vaned type and consisted of 13 folds and it described
thin. The illustrated histological study of pecten oculi folds in the Kestrel and the
Collared Dove was composed of large number of capillaries, large blood vessels and
pigment cells which were few in Kestrel compare with the Collared Dove. The bridge
in the Kestrel and the Collared Dove pecten oculi was consisted of connective tissue,
many pigment cells, and contains on little capillaries and it linked the membrane to
the internal limiting membrane of the retina in the Kes

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Crossref
Publication Date
Thu Jun 15 2017
Journal Name
Journal Of Baghdad College Of Dentistry
Effects of Two Different Colorant Solutions on the Color Stability of Bleached Enamel in Association with CPP-ACPF: An In Vitro Study
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Background: One of the drawbacks of vital teeth bleaching is color stability. The aim of the present study was to evaluate the effects of tea and tomato sauce on the color stability of bleached enamel in association with the application of MI Paste Plus (CPP-ACPF). Materials and Methods: Sixty enamel samples were bleached with 10% carbamide peroxide for two weeks then divided into three groups (A, B and C) of 20 samples each. After bleaching, the samples of each group were subdivided into two subgroups (n=10). While subgroups A1, B1 and C1 were kept in distilled water, A2, B2, and C2 were treated with MI Paste Plus. Then, the samples were immersed in different solutions as follow: A1 and A2 in distilled water (control); B1 and B2 in black

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Crossref
Publication Date
Fri Oct 02 2020
Journal Name
Biochem. Cell. Arch.
MORPHOLOGICAL DESCRIPTION AND COMPARATIVE HISTOLOGICAL STUDY OF THE LIVER IN TWO IRAQI MAMMALS : WEASEL (HERPESTES JAVANICUS) AND EASTERN GRAY SQUIRREL (SCIURUS CAROLINENSIS
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The results of the current study showed that the liver of H. javanicus appeared as large lobulated organ divided into six distinct lobes, that filled the cranial region and little extended to the middle region of abdominal cavity. On the other hand, liver of S. carolinensis laid against the diaphragm, occupied the cranial region of the abdominal cavity and consisted of five lobes. The liver is surrounded with a thin capsule of dense regular collagenous connective tissue and few numbers of smooth muscles fibers can be seen in the capsule that covered the squirrel liver. The liver parenchyma divided into a large number of interconnected hepatic lobules marked only by the abundant amount of connective tissue bordered the triads, and within the

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Publication Date
Fri Dec 01 2017
Journal Name
Journal Of Economics And Administrative Sciences
Comparing the Sequential Nonlinear least squared Method and Sequential robust M method to estimate the parameters of Two Dimensional sinusoidal signal model:
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Estimation of the unknown parameters in 2-D sinusoidal signal model can be considered as important and difficult problem. Due to the difficulty to find estimate of all the parameters of this type of models at the same time, we propose sequential non-liner least squares method and sequential robust  M method after their development through the use of sequential  approach in the estimate suggested by Prasad et al to estimate unknown frequencies and amplitudes for the 2-D sinusoidal compounds but depending on Downhill Simplex Algorithm in solving non-linear equations for the purpose of obtaining non-linear parameters estimation which represents frequencies and then use of least squares formula to estimate

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Crossref
Publication Date
Fri Nov 11 2022
Journal Name
Journal Of Accounting And Financial Studies ( Jafs )
Ambidexteritg leadership behaviors in organizational excellence Comparative : applied research in the two public companies, Al-Zawraa and the electrical and electronic industries
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Business organizations strive to reach organizational excellence, in instruction to maintain their position in the market, so the research started from the problem of the weakness of the senior leaders in the two companies investigated in the management and sustainability of work, as well as their inability to face challenges that could affect the work of the two companies in the future, especially the lack of organizational of the future and the prediction of those events, so the two companies seek to explore and invest opportunities to achieve organizational excellence. The research aims to know the role of Ambidexterity  leadership behaviors in organizational excellence in my company .and create ideas that will achieve organizati

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Publication Date
Tue May 26 2026
Journal Name
Research On Crops
Effect of organic acid, amino acids and nano-fertilizer on growth, yield and nitrate concentration of lettuce plant under two farming systems
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Scopus (4)
Scopus
Publication Date
Fri Jun 20 2025
Journal Name
Bulletin Of The Iraq Natural History Museum
A NEW RECORD OF TWO NEMATODES, MESORHABDITISFRANSENI FUCHS, 1933 (MESORHABDITIDAE) AND PRATYLENCHUSGOODEYI SHER AND ALLEN, 1953 (PRATYLENCHIDAE) WITH MOLECULAR DESCRIPTION FROM IRAQ
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rhabditid Mesorhabditis franseni Fuchs, 1933 (Family, Mesorhabditidae) and pratylenchid nematode Pratylenchus goodeyi Sher and Allen, 1953 (Family, Pratylenchidae). They were illustrated by molecular aspects. All specimens of both genera were cultured and reproduced for DNA extraction. M. franseni (IRQ.ZAh2 PP528819.1 isolate) was characterized. P. goodeyi (IRQ.ZAh5 PP535537 isolate) was also characterized. Selected specimens of these two species were molecularly characterized using the partial ITS-rRNA gene sequences. The ITS-rRNA sequence of IRQ.ZAh2 PP528819.1 isolate had a range of (98.62%-100%) sequence homology with ITS-rRNA sequence of M. franseni available in NCBI database. While, the ITS-rRNA sequence of IRQ.ZAh5 PP535537 isolate h

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