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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 Sep 01 2025
Journal Name
Results In Engineering
Storage characteristics of solar thermal collector integrated with latent heat material: A comparative experimental study of two types of paraffin wax under varying incident radiation and different water flow rates
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Publication Date
Sun Mar 15 2026
Journal Name
Sciences Journal Of Physical Education
Using two types of electronic questions by applying the Quiz Creator program in the method of guided discovery and its effect on learning and maintaining a motor chain on the throat
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Publication Date
Fri Sep 01 2023
Journal Name
Journal Of Medicine And Life
Potential anti-obesity effects of two-graded doses of Iraqi <i>Hibiscus tiliaceus</i> leaves extract, alone and in combination with orlistat, on high-fat diet-induced obesity in male rats
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Publication Date
Sun Jan 01 2017
Journal Name
Iraq Journal Of Agricultural Research
THE COMPARATIVE STUDY OF THE TOTAL COSTS AND NET REVENUE OF DIFFERENT TILLAGE SYSTEMS USING TWO TYPES OF SEEDLINGS IN THE PRODUCTION OF BARLEY GREEN VARIETY SAMEER IN THE GYPSUM SOILS
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Publication Date
Sun Mar 15 2026
Journal Name
Journal Of Baghdad College Of Dentistry
The Effect of Plasma Treatment on Shear Bond Strength of High Impact Acrylic Resin Denture Base Lined with Two Types of Soft Lining Materials after Immersion in Distilled Water and Denture Cleanser
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Background: In dentistry, dentist takes the advantages of soft lining materials due to the viscoelastic properties. The major problem is the adhesion of the soft liner with the denture base material. Materials and Methods: Heat cured of high impact acrylic resin specimens prepared with dimensions 75x13x13mm for shear bond strength test, soft lining material (Refit and Mollosil) with a 3-mm thickness and used to join each two acrylic blocks. Also four specimens with the same previous dimensions utilized for chemical and physical surface analysis. The specimens grouped as control (without plasma) and experiment (with oxygen plasma) treated high impact acrylic specimens. Results: Plasma treatment increased the shear bond strength for both Refi

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Publication Date
Mon Dec 25 2017
Journal Name
Iraqi Journal Of Pharmaceutical Sciences ( P-issn 1683 - 3597 E-issn 2521 - 3512)
Extraction, Identification and Isolation of B-sitosterol from Iraqi Wild Awsaj plant (Lycium barbarum) Using UAE (Probe and Bath) and two isolation technique (HPTLC and PHPLC): B-sitosterol in Iraqi Awsaj plant
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Awsaj (Lycium barbarum) is a plant belong to family Solanaceae serves as a good source of bioactive compounds like phytosterols which have many important biological activity. Literature survey available so far revealed that there was no studies about Iraqi wild Awsaj phytosterols especially B-sitosterol, there for the objective of this study was to examine the efficiency of ultrasound assisted extraction (probe and bath) as compared to the conventional (Soxhlet) extraction method for extraction of phytosterols especially B-sitosterol from fruits, leaves, stems and roots of Iraqi wild Awsaj plant. This goal was achieved by comparing the extraction mass yield, also by a quick and easy approach for identification and quantification of bioac

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Publication Date
Mon Mar 07 2022
Journal Name
Journal Of Educational And Psychological Researches
Awareness of Students at Imam Muhammad bin Saud Islamic University of the Requirements of Married Life in the Light of Social Changes and Methods of Deepening It in Accordance with an Islamic Educational Vision
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This study aims to identify the level of students’ awareness at Imam Muhammad bin Saud University of the requirements of married life in the light of social changes and suggested methods to deepen this awareness (according to the Islamic educational vision) from their own perspective. In this study, the researcher used the descriptive approach with a survey research method, depending on questionnaires to collect data, which he applied to students of College of Sharia in Imam Muhammad bin Saud Islamic University, as well as, students in the fields of Sociology, Social Work, and Psychology in the College of Social Sciences. The findings of the study revealed that students are aware of the requirements of married life concerning mutual ri

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Publication Date
Sun Oct 01 2017
Journal Name
مجلة القادسية للعلوم الرياضية
The impact of leverage in two different environments on the development of weapons and arms explosion strength on some physiological indicators and the accuracy of the performance of the service and crushing volleyball skills
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ھدف البحث الـــــى : ١ -إعداد تدریبات القوة الارتدادیة في وسطین متباینین على بعض المؤشرات الفسیولوجیة لتطویر القوة الانفجاریة ودقة مھارتي الأرسال والضرب الساحق بالكرة الطائرة . ٢ -التعرف على تأثیر تدریبات القوة الارتدادیة في وسطین متباینین على بعض المؤشرات الفسیولوجیة لتطویر القوة الانفجاریة.. ٣ -التعرف على تأثیر تدریبات القوة الارتدادیة في وسطین متباینین على دقة مھارتي الأرسال والضرب الساحق بالكرة الطائرة

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Publication Date
Wed Jun 01 2022
Journal Name
International Journal Of Research In Social Sciences And Humanities
THE EFFECT OF SPECIAL EXERCISES IN THE DEVELOPMENT OF THE MOMENTARY BIOMECHANICAL STRENGTH OF THE SKILL OF THE TWO REAR AIR CORES ON THE PARALLEL DEVICE IN THE TECHNICAL GYMNASTICS OF THE EMERGING PLAYERS
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Gymnastics is one of the most interesting games in terms of organizing various world championships and tournaments. Their basic skills and their technical development, especially those with high difficulty levels that rely on basic and complex skills, are the basic skills of gymnastics are the main pillar to be developed and therefore high-difficulty skills will not depend on basic skills, which should reach the stage of the automatic mechanism. Retention) in order to move on to training and developing another skill for the difficulty of movements as well as the smooth flow of performance, ie the degrees of difficulty of each skill specific to your gym and The researcher used the experimental method and The research community determin

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Publication Date
Sun Jan 02 2022
Journal Name
Journal Of The College Of Languages (jcl)
The Role Of Historical Memory In Promoting The Concept Of Belonging To The Homeland In A Novel "Mazurka For Two Dead Men" Of The Spanish Novelist Camilo Jose Cela: El Papel De La Memoria Histórica En El Apoyo Del Concepto De Pertenencia A La Patria , En La Novela “Mazurca Para Dos Muertos”, De Camilo José Cela
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       The present study discusses the significant role of the historical memory in all the Spanish society aspects of life. When a novelist takes the role and puts on the mask of one of the novel’s protagonists or hidden characters, his memory of the events becomes the keywords of accessing the close-knit fabric of society and sheds lights on deteriorating social conceptions in  a backwards social reality that rejects all new progressive ideas and  modernity. Through concentrating on the society flawing aspects and employing everything of his stored memory, the author uses sarcasm to criticize and change such old deteriorating reality conceptions.   

   &nbs

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