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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 Aug 25 2012
Journal Name
Wireless Personal Communications
Multi-Objective Evolutionary Algorithm Based on Decomposition for Energy Efficient Coverage in Wireless Sensor Networks
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Publication Date
Thu Jun 01 2023
Journal Name
Chemical Physics Impact
An efficient Ag decorated CeO2 synergetic catalyst for improved catalytic reduction of lethal 4-nitrophenol
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Catalytic reduction is considered an effective approach for the reduction of toxic organic pollutants from the environment, but finding an active catalyst is still a big challenge. Herein, Ag decorated CeO2 catalyst was synthesized through polyol reduction method and applied for catalytic reduction (conversion) of 4-nitrophenol (4-NP) to 4-aminophenol (4-AP). The Ag decorated CeO2 catalyst displayed an outstanding reduction activity with 99% conversion of 4-NP in 5 min with a 0.61 min−1 reaction rate (k). A number of structural characterization techniques were executed to investigate the influence of Ag on CeO2 and its effect on the catalytic conversion of 4-NP. The outstanding catalytic performances of the Ag-CeO2 catalyst can be assigne

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Publication Date
Fri May 15 2026
Journal Name
Network
Toward Efficient Virtual Cell-Based Topology Management and Adaptive Routing for Underwater Wireless Sensor Networks
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Underwater Wireless Sensor Networks (UWSNs) play a vital role in ocean monitoring and exploration. However, harsh underwater conditions and frequent topology changes caused by node and sink mobility pose significant challenges for reliable routing. Conventional routing protocols that depend on global route reconstruction and static paths generate excessive control overhead and degrade performance in large-scale underwater environments. In this paper, we propose an energy-efficient virtual cell-based mobile-sink adaptive routing (VC-MAR) protocol for UWSNs. The sensing field is logically partitioned into a three-dimensional grid of virtual cells, where a cell-gateway is elected in each cell to construct a low-overhead routing backbon

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Publication Date
Wed Feb 16 2022
Journal Name
Journal Of Economics And Administrative Sciences
Solving Resource Allocation Model by Using Dynamic Optimization Technique for Al-Raji Group Companies for Soft Drinks and Juices
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In this paper, the problem of resource allocation at Al-Raji Company for soft drinks and juices was studied. The company produces several types of tasks to produce juices and soft drinks, which need machines to accomplish these tasks, as it has 6 machines that want to allocate to 4 different tasks to accomplish these tasks. The machines assigned to each task are subject to failure, as these machines are repaired to participate again in the production process. From past records of the company, the probability of failure machines at each task was calculated depending on company data information. Also, the time required for each machine to complete each task was recorded. The aim of this paper is to determine the minimum expected ti

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Publication Date
Tue Jun 01 2021
Journal Name
Baghdad Science Journal
Three-Dimensional Nonlinear Integral Operator with the Modelling of Majorant Function
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In this paper, the process for finding an approximate solution of nonlinear three-dimensional (3D) Volterra type integral operator equation (N3D-VIOE) in R3 is introduced. The modelling of the majorant function (MF) with the modified Newton method (MNM) is employed to convert N3D-VIOE to the linear 3D Volterra type integral operator equation (L3D-VIOE). The method of trapezoidal rule (TR) and collocation points are utilized to determine the approximate solution of L3D-VIOE by dealing with the linear form of the algebraic system. The existence of the approximate solution and its uniqueness are proved, and illustrative examples are provided to show the accuracy and efficiency of the model.

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Publication Date
Sun Aug 03 2014
Journal Name
Journal Of Advances In Mathematics
On types of Delay in Delay Differential equation
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Publication Date
Wed Mar 10 2021
Journal Name
Baghdad Science Journal
Estimation of Bladder pressure by using poiseuilles equation
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Equation Boizil used to Oatae approximate value of bladder pressure for 25 healthy people compared with Amqas the Alrotinahh ways used an indirect the catheter Bashaddam and found this method is cheap and harmless and easy

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Publication Date
Fri Aug 01 2014
Journal Name
International J. Of Math. Sci. & Engg. Appls.
NEUTRAL DELAY DIFFERENTIAL EQUATION WITH ONE LARGE DELAY
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Publication Date
Sat Dec 10 2016
Journal Name
Misan Journal For Physical Education Sciences
Designing and Standardizing two tests for motor coordination timing for youth basketball players
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Designing and Standardizing two tests for motor coordination timing for youth basketball players Research submitted by Prof. Faris sami & asst. prof. Wasan hanoon ali & asst. prof. Feras muttasher Baghdad University-College Of Physical Education and Sport Sciences Motor coordination in basketball is considered one of the most important factors for success in skill performance accuracy and speed due to the defensive and offensive situations of the game. The problem of the research lies in the lack of tests that can specify the growth of motor coordination through which the relative change for a number of players can be noticed due to practice and training. The subjects of the research were (30) young league players of National Center for gif

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Publication Date
Tue Jun 20 2023
Journal Name
Baghdad Science Journal
Numerical Solutions for the Nonlinear PDEs of Fractional Order by Using a New Double Integral Transform with Variational Iteration Method
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This paper considers a new Double Integral transform called Double Sumudu-Elzaki transform DSET. The combining of the DSET with a semi-analytical method, namely the variational iteration method DSETVIM, to arrive numerical solution of nonlinear PDEs of Fractional Order derivatives. The proposed dual method property decreases the number of calculations required, so combining these two methods leads to calculating the solution's speed. The suggested technique is tested on four problems. The results demonstrated that solving these types of equations using the DSETVIM was more advantageous and efficient

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