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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
Thu Oct 08 2026
Journal Name
Mustansiriya Medical Journal
The Pattern Of Skin Diseases Among Kindergarten Children In Baghdad. A Comparative Study Between Two Surveys Five Years Apart
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Publication Date
Tue Jun 20 2023
Journal Name
Bulletin Of The Iraq Natural History Museum
MORPHOLOGICAL DESCRIPTION OF TWO LEECH SPECIES (ANNELIDA, HIRUDINEA) WHICH USED IN SOME ALTERNATIVE MEDICINE CLINICS IN BAGHDAD PROVINCE, IRAQ
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The aim of this study is to describe the leech species that are used in some of the alternative medicine clinics in Baghdad Province based on morphometric measurements and colouring pattern of the body. A collection of twenty leeches was provided from some clinics. All specimens were identified and described based on standard available keys. The morphometric characteristics and colouring patterns were recorded and the indicated that these leeches were Hirudo orientalis Utevsky & Trontelj, 2005 and H. verbana Carena, 1820.

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Publication Date
Wed Oct 31 2018
Journal Name
International Journal Of Research In Pharmaceutical Sciences
The impact of systemic hypertension-related heart remodeling on right ventricle mechanics: A two-dimensional echocardiographic speckle tracking study
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Publication Date
Thu Oct 08 2026
Journal Name
Journal Of Baghdad College Of Dentistry
An Assessment of Salivary Leptin and Resistin Levels in Type Two Diabetic Patients with Chronic Periodontitis (A Comparative Study)
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Background: Type 2 diabetes mellitusand chronic periodontitis hold a close relationship that has been the focus of many researches. Currently there is an appreciation to the role of adipose tissue-derived substances "the adipokines" in immune-inflammatory responses; also, there is an interest in using the simple non-invasive saliva in diagnosing and linking oral and general health problems. The current study aims to determine the periodontal health status in the chronic periodontitis patients with and without poorly or well controlled type 2 diabetes mellitus, measure the salivary levels of two adipokines "leptin and resistin", pH and flow rate and then correlate between these clinical periodontal, biochemical and physical parameters in eac

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Publication Date
Sun Jun 07 2015
Journal Name
Baghdad Science Journal
A comparative study to determine the nature of the eye lens protein in the two types of birds Iraqi
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The study showed significant differences between the average weight lens and the average amount protein in the lens between that Kestrel Falco tinnunculus L. and the Collared Dove Streptopelia decaocto F. , also the study electrical migration of lens proteins having one bundle of crystalline –? in Kestrel compared with three bundles in Collared Dove, two bundles of crystalline – ? in both , and crystalline – ? appeared as one bundle in both birds.

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Publication Date
Thu Sep 12 2019
Journal Name
Al-kindy College Medical Journal
Step-up protocol gonadotrophin versus laparoscopic ovarian drilling in clomiphene citrate resistant PCOS infertile women in two Iraqi hospitals
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Background: Polycystic ovarian syndrome is a common endocrine disorder affecting 6-10% of women of reproductive age and the most common cause of anovulatory infertility.

Objective: The aim of the study was to compare the effectiveness, side effects and outcomes of step-up gonadotrophin protocol versus laparoscopic ovarian diathermy (LOD) in infertile patients with clomiphene citrate resistant polycystic ovary syndrome.

Methods:  The sample included women who attended our infertility clinic at Al-Elwiya Maternity Teaching Hospital and Kamal Al-Samarraee for Infertility and IVF Hospital in Baghdad/ Iraq from November 2013 to November 2014.    Eighty case

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Publication Date
Mon Dec 20 2021
Journal Name
Bulletin Of The Iraq Natural History Museum (p-issn: 1017-8678 , E-issn: 2311-9799)
MORPHOLOGY AND MOLECULAR IDENTIFICATION OF THE LARVAL STAGE OF TWO SPECIES FROM THE GENUS CHRYSOBOTHRIS ESCHSCHOLTZ, 1829 (COLEOPTERA, BUPRESTIDAE)
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The genus of Chrysobothris Eschscholtz, 1829 is one of the most diverse and widespread genera of the family Buprestidae of some 700 described species distributed throughout the world. In Iraq, particularly in the Kurdistan region, about 4 species had been recorded so far, many of these species are sympatric, share larval host plants, and are difficult to reliably separate morphologically. The current study investigates species limits and relationships among the recognized species occurring within the Erbil Province; mitochondrial cytochrome C oxidase (COX I) molecular analysis confirmed the monophyly of two Chrysobothris species, Ch. affinis (Fabricius, 1794) and Ch. chrysostigma (Linnaeus, 1758). Implications of the resultant larval mor

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Publication Date
Tue Jan 01 2008
Journal Name
J Bagh College Of Dentistry
A comparative study of the sealing ability of two different obturation techniques with and without the use of sealers.
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Background: The final stage of endodontic therapy is complete obturation of the root canal system to provide as perfect as possible at the cementodentinal junction of the apical foramen. The purpose of this in vitro study was to evaluate the sealing ability of injection molded thermoplasticized gutta percha and lateral condensation techniques with and without the use of sealers. Materials and Methods: Forty freshly extracted adult human maxillary central incisors with complete formed apices were utilized in this study. The teeth were randomly divided into four groups for evaluation of the apical seal. Group (1) lateral condensation gutta percha technique without sealer, (2) lateral condensation gutta percha technique with sealer, (3) Inject

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Publication Date
Mon Jan 01 2018
Journal Name
Euphrates Journal Of Agriculture Science
Effect of Interaction Between Dietary Two Levels of Cumin (Cuminum cyminum) and Ginger (Zingiber officinale) on Japanese Quail performance
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The study was conducted to determine the effect of of cumin (Cuminum cyminum) and ginger (Zingiber officinale) powder on performance in Japanese quails. 630 bird (7 birds for each treatment) nine weeks of age were used. Experiment was continued for 12 weeks divided to six periods and the birds distributed randomly on nine combinations of cumin and ginger, three levels of each plant powder namely, 0, 0.5 and 1%. T1 (Control) contain the slandered diet (0% of cumin and 0% of ginger), T2: (0% of cumin and 0.5% of ginger), T3: (0% of cumin and 1% of ginger), T4: (0.5% of cumin and 0% of ginger), T5: (0.5% of cumin and 0.5% of ginger), T6: (0.5% of cumin and 1% of ginger), T7: (1% of cumin and 0% of ginger), T8: (1% of cumin and 0.5% of ginger),

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Publication Date
Wed Mar 15 2023
Journal Name
Journal Of Baghdad College Of Dentistry
Crystallization firing effect on the marginal discrepancy of the IPS. emax CAD crowns using two different CAD/CAM systems
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Background: Marginal adaptation is critical for long – term success of crown and bridge restoration. Computer aided design / computer aided manufacture (CAD/ CAM) system is gaining more importance in the fabrication of dental restoration. Objective: The aim of this study is to evaluate the effect of crystallization firing on the vertical marginal gap of IPS. emax CAD crowns which fabricated with two different CAD/CAM systems .Materials and Methods: Twenty IPS e.max CAD crowns were fabricated. We had two major groups (A, B) (10 crowns for each group) according to the CAD/CAM system being used: Group A: fabricated with Imes - Icore CAD/CAM system; Group B: fabricated with In Lab Sirona CAD/CAM system. Each group was subdivided into two s

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