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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 Aug 01 2022
Journal Name
Journal Of Physics: Conference Series
The local bifurcation analysis of two preys stage-structured predator model with anti-predator behavior
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Abstract<p>This paper deals with two preys and stage-structured predator model with anti-predator behavior. Sufficient conditions that ensure the appearance of local and Hopf bifurcation of the system have been achieved, and it’s observed that near the free predator, the free second prey and the free first prey equilibrium points there are transcritical or pitchfork and no saddle node. While near the coexistence equilibrium point there is transcritical, pitchfork and saddle node bifurcation. For the Hopf bifurcation near the coexistence equilibrium point have been studied. Further, numerical analysis has been used to validate the main results.</p>
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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 study focuses on the implementation of interfaces for human machine interaction (HMI) control and monitor automatic production line. The automatic production line can performance feeding, transportation, sorting functions. The objectives of this study are implemented two SCADA/HMI system using two different software. TIA portal software is used to build HMI, alarm, and trends in touch panel which is helped an operator to control and monitor the production line. LabVIEW software is used to build HMI and trends in the computer screen and is linked with Microsoft Excel (ME) to generate information table helped to monitor the performance of the pneumatic equipment. The production line can do performance feeding, transportation, sorting fun

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Publication Date
Fri Mar 06 2026
Journal Name
Journal Of Baghdad College Of Dentistry
An Evaluation of Ion Released from Two Brands of Brackets in Three Types of Mouthwashes
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Background: Mouthwashes used widely as ancillary to mechanical oral hygiene methods. Little information provided about the effect of mouthwashes on ions released from orthodontic brackets. Therefore, the present study has been established to evaluate the effect of different mouthwashes on the corrosion resistance and the biocompatibility of two brands of brackets. Materials and Methods: Eighty premolar stainless steel brackets were used (40 brackets from each brand). They were subdivided into four subgroups (n=10) according to immersion media (deionized distilled water, Corsodyl, Listerine and Silca herb mouthwashes). Each bracket was stored in a closely packed glass tube filled with 15ml of the immersion media and incubated for 45 days at

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Publication Date
Fri Mar 31 2017
Journal Name
Al-khwarizmi Engineering Journal
Active Vibration Suppression of Smart Cantilever Beam with Sliding Mode Observer Using Two Piezoelectric Patches
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This paper presents a vibration suppression control design of cantilever beam using two piezoelectric ‎patches. One patch was used as ‎an actuator element, while the other was used as a sensor. The controller design was designed via the balance realization reduction method to elect the reduced order model that is most controllable and observable. ‎the sliding mode observer was designed to estimate six states from the reduced order model but three states are only used in the control law. Estimating a number of states larger than that used is in order to increase the estimation accuracy. Moreover, the state ‎estimation error is proved bounded. An ‎optimal LQR controller is designed then using the ‎estimated states with the slid

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Publication Date
Sat Dec 02 2017
Journal Name
Al-khwarizmi Engineering Journal
Biomechanical Aspects of Shoulder and Hip Articulations: A Comparison of Two Ball and Socket Joints
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The shoulder and hip joints though essentially both are ball and socket joints, show structural variability to serve functional needs.

This study aims at revealing some of the structural and functional properties of each of the two joints regarding the factors that contribute to the stability of any joint in the body, namely: bone, ligament, and muscle.

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Publication Date
Fri Sep 30 2016
Journal Name
Iraqi Journal Of Chemical And Petroleum Engineering
Parameters Affecting the Thermodynamic Efficiency of PEM Single Cell and Stack of Cells (Two Cells)
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In this work, thermodynamic efficiency of individual cell and stack of cells (two cells) has been computed by studying the variation of voltage produced during an operation time of 30 min as a result of the affected parameters:- stoichiometric feed ratio, flow field design on single cell and feed distribution on stack of cells. The experiments were carried out by using two cells, one with serpentine flow field and the other with spiral flow field. These cells were fed with hydrogen and oxygen at low volumetric flow rates from 1 to 2 ml/sec and stoichiometric ratios of fuel (H2) to oxidant (O2) as 1:2, 1:1 and 2:1 respectively. The results showed that

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Publication Date
Fri Oct 01 2021
Journal Name
Journal Of Medical And Surgical Practice
Incidence of Stenosis in Graft Renal Arteries Anastomosis in Kidney Transplantation Using Two Different Techniques
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Summary Kidney transplantation is widely performed nowadays as an optimal treatment of end stage kidney diseases. Complications such as stenosis in graft renal arteries anastomosis may occur. Different suturing techniques are available for renal artery anastomosis. We aimed to compare the incidence of renal artery stenosis of the transplanted kidney when two suture techniques (continuous or interrupted) used for renal artery anastomosis. Therefore, a retrospectively comparative study was conducted on 44 patients managed with kidney transplantation during the years 2009-2011. Patients assigned into two groups; first group included 20 patients namely, continuous suture group, and the second group included 24 patients in whom the allograft art

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Publication Date
Sat Dec 17 2011
Journal Name
Journal Of Engineering
Effect of Steering Wheel Vibration on drivers Hands in a Two-Wheel Drivers Hand tractor
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The paper presents research results of the vibration transmitted from the steering wheel of the tractor with a 2-wheel drive to the driver’s hands. The vibration measurements were carried out on the tractor randomly chosen from the collage of agriculture / university of Baghdad. Before testing the tractor was examined and adjusted following the producer’s recommendations. The vibration levels were measured during the operation tillage at idling and at full load .The field was 31.7 m above level sea. Soil was treated at soil constant moisture (17-20 %) with depth of plowing (17 cm). During operation the weather temperature was measured (15 C) and humidity was ( 27 % ) The vibration level on the steering wheel was measured and analyzed .T

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Publication Date
Wed May 14 2025
Journal Name
Journal Of Applied Mathematics
Role of Wind and Fear on the Dynamic of a Prey and Two Competing Predators
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An essential tool for studying the web is its ability to show how energy moves through an ecosystem. Understanding and elucidating the relationship between species variety and their placement within the inclusive trophic dynamics is also beneficial. A food web ecological model with prey and two rival predators under fear and wind flow conditions is developed in this article. The boundedness and positivity of the system’s solution are established mathematically. The stability and existence constraints of the system’s equilibria are examined. The proposed system’s persistence limitations are established. Additionally, the bifurcation analysis of every potential equilibrium is examined using the Sotomayor theorem. To describe the

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Publication Date
Sat Dec 01 2007
Journal Name
Al-nahrain Journal Of Science
Delayed type of hypersensitivity and its relation with experimental infection with two different Leishmania antigen
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