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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
Rise Velocity and Drag Coefficient of Collapsing Two-Phase Bubble Condensing in an Immiscible Liquid
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Collapse of the vapor  bubble condensing  in an immiscible  is investigated for  n-pentane and n-hexane vapors condensing in cold water and n-pentane in two different compositions of glycerin- water mixture. The rise velocity and the drag coefficient of the two-phase bubble are measured.

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Publication Date
Thu Feb 07 2019
Journal Name
Iraqi Journal Of Laser
Design and Comparative Study of a one and Two Stage Blumlein Circuit TEA Nitrogen Laser
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Two Prototypes of Transversely Excited at atmospheric pressure (TEA) Nitrogen laser systems (One Stage Blumlein Circuit and Two Stage Blumlein Circuit) were fabricated and operated. High voltage power supply with variable operating voltage (0-20 kv) and operating current (1-3A) was built and tested successfully. The gas flow rate of 15 L/ min and 10 L/ min for OSBC and TSBC was used. The performance of the fabricated systems was studied extensively reaching to the optimum operating conditions. The obtained laser output energy for the first system has linear relationship with the applied voltage. The maximum output energy was about (1.14 mJ) with (10.40) ns pulse duration and the half-wave divergence angle was about (0.1455 m rad). In the

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Publication Date
Thu Aug 07 2025
Journal Name
Scientific Reports
Evaluation of how laser photostimulation at two wavelengths alters the antimicrobial potential of Streptomycetes spp
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The purpose of this study was to isolate and identify additional products produced by direct laser irradiation, as well as to ascertain if laser irradiation may stimulate the synthesis of antibiotic compounds in a local Streptomycetes (Strept). Moreover, we postulate the mechanisms by which lasers function within living bacterial cells and suggest that sequential photochemical reactions may transpire following a designated period of irradiation. Thiophene was found as one of the most significant clinical products with antibacterial and anticancer properties. To accomplish these objectives, we selected two isolates: Streptomyces thinghirensis strain S10 (Strept thin), which inherently synthesizes an antibacterial agent, and Streptomyces lien

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Publication Date
Sun Sep 01 2019
Journal Name
Al-khwarizmi Engineering Journal
Pneumatic Control System of Automatic Production Line Using Two Method of SCADA/HMI Implement PLC
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       This work focuses on the implementation of interfaces for human machine interaction (HMI) for control and monitor of automatic production line. The automatic production line which can performance feeding, transportation, sorting functions.      

            The objectives of this work are implemented two SCADA/HMI system using two different software. TIA portal software was used to build HMI, alarm, and trends in touch panel which are helped the operator to control and monitor the production line. LabVIEW software was used to build HMI and trends on the computer screen and was linked with Micros

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Publication Date
Fri Jan 01 2021
Journal Name
Aip Conference Proceedings
Study of rheological and mechanical properties of bitumen blended with two alpha phases of polyamide
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The effect of D phase polyamide (PA6)on the rheological properties, Young Modulus and the thermal expansion coefficient of two blends groups (bitumen-polyamide) were tested. The first group was for bitumen-PA6 blends and the second group for bitumen blended with polymer resulted from the crystallization of PA6-formic acid solution in water(PAFW).The obtained results proved that adding both types of polyamide has led to a rise in toughness and softening point temperature while the penetration Index approached -3 after adding the polyamide. So, all these changes make bitumen-polyamide blends more suitable for use in hot climate regions. The blends properties were explained according to the reaction that takes place between the polyamide and

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Publication Date
Sun Dec 27 2020
Journal Name
Iraqi Journal Of Pharmaceutical Sciences ( P-issn: 1683 - 3597 , E-issn : 2521 - 3512)
Estimation of Beta Two Microglobulins, Fetuin-A, Resistin Serum Level in Iraqi Multiple Myeloma Patients
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Multiple myeloma is hematological disease produces many complications in the bone, kidney, neural and other complications. The study aims to measure serum biomolecules like fetuin-A and resistin and determined the possibility to use these biomarkers as disease predictor. blood samples were isolated from 58 patients and 24 sex and age-matched control, serum then isolated, and proper ELISA kit then used to a determined level of B2 microglobulin, resistin, and fetuin-A. The result demonstrated significant increase in   B2 microglobulin, fetuin-A and resistin in patients compare to control (1.3470.714 vs. 0.9130.253), p = 0.000, (14.00310.352 vs. 9.2594.264), p= 0.005, (1.9673.595 vs. 0.6040.622), p = 0.009, respectively.   These di

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Publication Date
Tue Jun 15 2021
Journal Name
Al-academy
The directing vision and two display technologies in the Iraqi Child Theater: عباس قاسم كاظم
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  The current research contains four chapters. The first topic included the methodological framework that included the problem of research and the need for it, and then the importance of the research, and then the aim of the research, its limits and the definition of terms linguistically, conventionally and procedurally. The second chapter (the theoretical framework) contained two topics, the topic The first is titled: The Outward Vision in Child Theater Performances, while the second topic was titled: Presentation Technology in Child Theater, Chapter Four (Research Procedures), which organized the research community and analyzed its sample, and the fourth chapter contained research results, conclusions and recommendations. The prop

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Crossref
Publication Date
Sun Dec 27 2020
Journal Name
Iraqi Journal Of Pharmaceutical Sciences ( P-issn 1683 - 3597 E-issn 2521 - 3512)
Estimation of Beta Two Microglobulins, Fetuin-A, Resistin Serum Level in Iraqi Multiple Myeloma Patients
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Multiple myeloma is hematological disease produces many complications in the bone, kidney, neural and other complications. The study aims to measure serum biomolecules like fetuin-A and resistin and determined the possibility to use these biomarkers as disease predictor. blood samples were isolated from 58 patients and 24 sex and age-matched control, serum then isolated, and proper ELISA kit then used to a determined level of B2 microglobulin, resistin, and fetuin-A. The result demonstrated significant increase in   B2 microglobulin, fetuin-A and resistin in patients compare to control (1.3470.714 vs. 0.9130.253), p = 0.000, (14.00310.352 vs. 9.2594.264), p= 0.005, (1.9673.595 vs. 0.6040.622), p = 0.009, respectively. &

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Publication Date
Fri Apr 04 2025
Journal Name
European Journal Of Dentistry
The Effectiveness of Conventional and Advanced Aligning Archwires: The Insights of Two Randomized Clinical Trials
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Abstract<p> Objectives This study aimed to compare the clinical effectiveness of four aligning archwires: Superelastic Nickel-Titanium (Superelastic-NiTi), SmartArch, Copper-Nickel-Titanium (Cu-NiTi), and Speed Tubular coaxial-Nickel-Titanium (Tubular coaxial-NiTi), regarding the alignment efficiency, associated perception of pain, and possibility of inducing root resorption.</p><p> Materials and Methods This study includes two randomized clinical trials run in parallel. Patients with 5 to 9 mm of mandibular anterior teeth crowding according to Little's irregularity index (LII) who needed fixed orthodontic appliances without extraction were randomly assigned to four groups of</p> ... Show More
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Publication Date
Fri Jan 01 2021
Journal Name
Aip Conference Proceedings
Study of rheological and mechanical properties of bitumen blended with two alpha phases of polyamide
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