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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
Wed Feb 01 2023
Journal Name
Petroleum Science And Technology
Lithofacies and electrofacies models for Mishrif Formation in West Qurna oilfield, Southern Iraq by deterministic and stochastic methods (comparison and analyzing)
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Publication Date
Mon Feb 02 2026
Journal Name
Modern Sport
Design and standardization of two scoring tests for the initial selection of people with simple mental disabilities in five-a- Side Soccer
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The five-a- Side Soccer for simple mental disability is as interesting and caring as many other sporting events. Therefore, the researchers believe that their basic skills tests are compatible with then level of the tested individuals in terms of age and mental ability. As a means of selecting those who are qualified to practice this game in a simple way, the problem of research has been identified in the design and standardization of two scoring skills that belong to this category and are approved by the training staff during the initial selection process. The research community is determined by people with mental disabilities who belong to the governmental and private institutes and the Special Olympics Union in Baghdad and for the year 2

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Publication Date
Mon Jul 24 2023
Journal Name
Journal Of University Of Kerbala
The occurrence of two species of the genus Myxobolus Bütschli, 1882 (Myxozoa: Myxosporea) for the first time in Iraq from freshwater fishes.
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The occurrence of two species of the genus Myxobolus Bütschli, 1882 (Myxozoa: Myxosporea) for the first time in Iraq from freshwater fishes.

Publication Date
Wed Sep 01 2021
Journal Name
Baghdad Science Journal
New Method for Estimation Mebeverine Hydrochloride Drugs Preparation by a New Analyser: Ayah 6S.X1(WSLEDs)-T.- Two Solar Cells Complied with C.F.I.A
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A sensitivity-turbidimetric method at (0-180o) was used for detn. of mebeverine in drugs by two solar cell and six source  with C.F.I.A.. The method was based on the formation  of  ion pair for the pinkish banana color  precipitate by the reaction of Mebeverine hydrochloride with Phosphotungstic acid. Turbidity was measured via the reflection of incident light that collides on the surface particles of   precipitated at 0-180o. All variables were optimized. The linearity ranged of Mebeverine hydrochloride was 0.05-12.5mmol.L-1, the L.D. (S/N= 3)(3SB) was 521.92 ng/sample depending on dilution for the minimum concentration , with correlation coefficient r = 0.9966while was R.S.D%

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Publication Date
Mon Feb 02 2026
Journal Name
Revista Iberoamericana De Psicología Del Ejercicio Y El Deporte, Issn 1886-8576, Vol. 17, Nº. 4, 2022, Págs. 165-167
Social-kinetic Cohesion and its Relationship to Goal Direction with its two Dimensions (Performance and Result) for Advanced Handball Players in Iraq
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Autorías: Muhammad Hamza Shihab, Nuha Mohsin Dhahi. Localización: Revista iberoamericana de psicología del ejercicio y el deporte. Nº. 4, 2022. Artículo de Revista en Dialnet.

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Publication Date
Sun Dec 01 2019
Journal Name
Zeitschrift Für Assyriologie Und Vorderasiatische Archäologie
Literary Texts from the Sippar Library I: Two Babylonian Classics
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Abstract<p>Publication and edition of two tablets from the library in the Ebabbar Temple of Sippar, a manuscript of the ‘Babylonian Poem of the Righteous Sufferer’ (<italic>Ludlul bēl nēmeqi</italic>) Tablet III, and a manuscript of the long prayer to Marduk ‘Furious Lord’ (also known as ‘Marduk 1’).</p>
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Publication Date
Tue Dec 26 2017
Journal Name
Al-khwarizmi Engineering Journal
FPGA Realization of Two-Dimensional Wavelet and Wavelet Packet Transform
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The Field Programmable Gate Array (FPGA) approach is the most recent category, which takes the place in the implementation of most of the Digital Signal Processing (DSP) applications. It had proved the capability to handle such problems and supports all the necessary needs like scalability, speed, size, cost, and efficiency.

In this paper a new proposed circuit design is implemented for the evaluation of the coefficients of the two-dimensional Wavelet Transform (WT) and Wavelet Packet Transform (WPT) using FPGA is provided.

In this implementation the evaluations of the WT & WPT coefficients are depending upon filter tree decomposition using the 2-D discrete convolution algorithm. This implementation w

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Publication Date
Mon May 04 2009
Journal Name
Journal Of Al-nahrain University
Solution of two-dimensional fractional order volterra integro-differential equations
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In this paper, our aim is to study variational formulation and solutions of 2-dimensional integrodifferential equations of fractional order. We will give a summery of representation to the variational formulation of linear nonhomogenous 2-dimensional Volterra integro-differential equations of the second kind with fractional order. An example will be discussed and solved by using the MathCAD software package when it is needed.

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Publication Date
Sun Mar 04 2018
Journal Name
Baghdad Science Journal
Biodegradation of Anthracene Compound by Two Species of Filamentous Fungi
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Isolation and identification fungi of Emericella nidulans and Aspergillus flavus from a pinkish and yellowish artificial clay, by using potato dextrose agar (PDA). Results revealed that E. nidulans was the best for degrading anthracene (92.3%) with maximum biomass production (3.7gm/l), compared to A. flavus with the rate of degradation (89%) and biomass production of (1.2gm/l), when methylene blue was used as redox indicator after incubating in a shaker incubator 120rpm at 30Co for 8days. Results indicated that E. nidulans has a high ability of anthracene degradation with the rate of (84%), while A. flavus showed the lower level with (77%) by using HPLC.

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Publication Date
Sat Jan 01 2022
Journal Name
Latin American Journal Of Solids And Structures
Peak Ground Acceleration Models Predictions Utilizing Two Metaheuristic Optimization Techniques
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