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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
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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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
Thu Feb 01 2018
Journal Name
Journal Of Economics And Administrative Sciences
Estimation of parameters of two-dimensional sinusoidal signal model by employing Deferential Evaluation algorithm and the use of Sequential approach in estimation
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Estimation the unknown parameters of a two-dimensional sinusoidal signal model is an important and a difficult problem , The importance of this model  in modeling Symmetric gray- scale texture image . In this paper, we propose employment Deferential Evaluation algorithm and the use of Sequential approach to estimate the unknown frequencies and amplitudes of the 2-D sinusoidal components when the signal is affected by noise. Numerical simulation are performed for different sample size, and various level of standard deviation to observe the performance of this method in estimate the parameters of 2-D sinusoidal signal model , This model was used for modeling  the Symmetric gray scale texture image and estimating by using

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Publication Date
Fri Oct 09 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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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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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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Publication Date
Sat Feb 03 2018
Journal Name
Chinese Journal Of Physics
A true random number generator based on the photon arrival time registered in a coincidence window between two single-photon counting modules
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True random number generators are essential components for communications to be conconfidentially secured. In this paper a new method is proposed to generate random sequences of numbers based on the difference of the arrival times of photons detected in a coincidence window between two single-photon counting modules

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Publication Date
Mon Oct 21 2019
Journal Name
Biochemical And Cellular Archive
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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(Received 21 May 2019, Revised 30 September 2019, Accepted 21 October 2019)ABSTRACT : The results of the current study showed that the liver of H. javanicus appeared as large lobulated organ dividedinto six distinct lobes, that filled the cranial region and little extended to the middle region of abdominal cavity. On the otherhand, liver of S. carolinensis laid against the diaphragm, occupied the cranial region of the abdominal cavity and consisted of fivelobes. The liver is surrounded with a thin capsule of dense regular collagenous connective tissue and few numbers of smoothmuscles fibers can be seen in the capsule that covered the squirrel liver. The liver parenchyma divided into a large number ofinterconnected hepatic lobules marked

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Publication Date
Thu Dec 15 2022
Journal Name
Journal Of Inorganic And Organometallic Polymers And Materials
Ultrasound-Assisted and One-Pot Synthesis of New Fe3O4/Mo-MOF Magnetic Nano Polymer as a Strong Antimicrobial Agent and Efficient Nanocatalyst in the Multicomponent Synthesis of Novel Pyrano[2,3-d]pyrimidines Derivatives
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Publication Date
Wed Oct 17 2018
Journal Name
Journal Of Economics And Administrative Sciences
Effect Of Using Quantitative Methods Of Demand Forecasting In Improving Of Supply Chain Performance:" Case Study In One Of An Industerial Organization"
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Objecte The study aims to test the effect of using the appropriate quantitative method of demand forecasting in improving the performance of supply chain of the aviation fuel product ( The study sample), One of the products of the Doura refinery (The study site), By testing a set of quantitative methods of demand forecasting using forecasting error measurements, and choosing the least faulty, most accurate and reliable method and adept it in the building  chain.

Is the study of problem through a starting with the fol

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