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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
Tue Jan 01 2019
Journal Name
J. Mech. Cont.& Math. Scis
The Use of Non-Parametric Methods to Estimate Density Functions of Copulas
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Publication Date
Tue Jul 20 2021
Journal Name
Polymer Bulletin
Influence methods of preparation on the thermal stability of polyimide/silica dust
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Publication Date
Sun Mar 01 2020
Journal Name
International Journal Of Pharmaceutical Research
Eco-friendly and Efficient Composition, Diagnosis Theoretical, kinetic studies, Antibacterial and Anticancer Activities of Mixed Some Metal Complexes of Tridentate Schiff base Ligand
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Abstract: Mixed ligand Mn(II), Co(II), Ni(II), Cu (II), Zn(II), and Cd(II) complexes with (TMAP) Schiff base ligand and (8HQ) have been composition and analyzed. Diagnosis by, melting point, solubility, Electronic, mass and IR-spectroscopic studies, conductivity elemental, thermoanalytical analysis displayed the forming of mononuclear complexes. Spectral studies results suggest an octahedral system or the metal (II) mixed complexes. The detainments of molar conductance of the mixed complexes in DMF coincide to electrolytic nature of the mixed complexes, consequently, these complexes could be subedited as [M(TMAP)(8Q)(H2O)]nX.yH2O (M=Co(II) and Cu(II) complexes(where n = 1, y = 0 ); [M(TMAP)(8Q)(H2O)]nX.yH2O (M = (where n = 1, y = 1 for Ni(

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Publication Date
Fri Jan 01 2021
Journal Name
International Journal Of Pharmaceutical Research
Eco-friendly and Efficient Composition, Diagnosis Theoretical, kinetic studies, Antibacterial and Anticancer Activities of Mixed Some Metal Complexes of Tridentate Schiff base Ligand
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Publication Date
Sat Jun 01 2024
Journal Name
Case Studies In Chemical And Environmental Engineering
Congo red removal from aqueous solution by electrocoagulation- electro-oxidation combined system with Al and Cu–Mn–Ni nano composite as efficient electrodes
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Removing Congo red (CR) is critical in wastewater treatment. We introduce a combination of electrocoagulation (EC) and electro-oxidation (EO) to address the elimination of CR. We also discuss the deposition of triple oxides (Cu–Mn–Ni) simultaneously on both anodic and cathodic graphite electrodes at constant current density. These electrodes efficiently worked as anodes in the EC-EO system. The EC-CO combination eliminated around 98 % of the CR dye and about 95 % of the Chemical Oxygen demand (COD), and similar results were obtained with the absence of NaCl. Thus, EC-EO is a promising technique to remove CR in an environmentally friendly pathway.

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Publication Date
Fri Dec 30 2011
Journal Name
Iraqi Journal Of Chemical And Petroleum Engineering
Extraction of Valuable Metals From Spent Hydrodesulfurization Catalyst By Two Stage Leaching Method
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Spent hydrodesulfurization (Co-Mo/γ-Al2O3) catalyst generally contains valuable metals like molybdenum (Mo), cobalt (Co), aluminium (Al) on a supporting material, such as γ-Al2O3. In the present study, a two stages alkali/acid leaching process was conducted to study leaching of cobalt, molybdenum and aluminium from Co-Mo/γ-Al2O3 catalyst. The acid leaching of spent catalyst, previously treated by alkali solution to remove molybdenum, yielded a solution rich in cobalt and aluminium.

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Publication Date
Tue Dec 20 2022
Journal Name
Bulletin Of The Iraq Natural History Museum (p-issn: 1017-8678 , E-issn: 2311-9799)
FIRST RECORD OF TWO PARASITOID WASPS OF THE FAMILY CHALCIDIDAE (HYMENOPTERA) IN IRAQ
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The family Chalcididae (Order: Hymenoptera) is known as one of the large chalcidoid wasps with some distinct morphological characters. The first occurrence of two parasitoid species belonging to this family was reported in the Al-Husayniya district Karbala Province, Iraq; which are: Brachymeria podagrica (Fabricius, 1787) and Chalcis myrifex (Sulzer, 1776). Both species were collected by using the sweeping net from orchards during July 2020.

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Publication Date
Wed Mar 30 2016
Journal Name
College Of Islamic Sciences
Graphical significance To (if) and (if) the two cops in Surat al-Maidah
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Many writers and scientists hardly differentiate between the two instruments of use
(If) and (if) to convergence in the meaning, as they perform the meaning of the penalty and condition, as the answer relates to his condition, but when linguistic meditation we find a great difference in use in terms of meaning, so this research came to reveal these meanings and was extrapolated applied in Surat al-Maidah By counting the verses in which (if) and (if) the cops were given and looking at their contexts, the connotation of the cops in most places was within the context of probability, which is the origin of its use, given in twenty places of Surat al-Ma'idah, and it had multiple connotations according to the contexts of the verses received

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Publication Date
Mon Mar 30 2020
Journal Name
Iraqi Journal Of Chemical And Petroleum Engineering
Simulation of Two Phase Flow Mixing Co – Current in T Junction Using Comsol
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The analysis, behavior of two-phase flow incompressible fluid in T-juction is done by using "A Computational Fluid Dynamic (CFD) model" that application division of different in industries. The level set method was based in “Finite Element method”. In our search the behavior of two phase flow (oil and water) was studed. The two-phase flow is taken to simulate by using comsol software 4.3. The multivariable was studying such as velocity distribution, share rate, pressure and the fraction of volume at various times.  The velocity was employed at the inlet (0.2633, 0.1316, 0.0547 and 0.0283 m/s) for water and (0.1316 m/s) for oil, over and above the pressure set at outlet as a boundary condition. It was observed through the program

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Publication Date
Tue Jan 17 2017
Journal Name
International Journal Of Computer Applications
Improving of Photovoltaic Cell Performance by Cooling using Two different Types of Fins
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