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Methods of Va.ia.ions a.d Their a.plica.ions

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Journa. of a.plied Ma.hema.ics a.d Physics 2023年第5期11卷 1263-1286页

作者： arij a. awin a.la. a. Ela.heq a.dulmta.b M. Hussen a.i M. awin Depa.tment of Ma.hema.ics College of Science University of Tripoli Tripoli Libya Depa.tment of Ma.hema.ics Navajo Technical University New Mexico USA

In this review a.ticle, we begin with reviewing Ca.culus of va.ia.ions giving few exa.ples on its use to solve a.la.ge number of problems in geometry, physics, a.d other bra.ches of knowledge. a.terwa.ds, we direct our a.tention to different methods of va.ia.ions which evolved during the la.t century a.d which include their use in eigenva.ue problems a.d in finite difference methods a.d those a.opted in cla.sica. a.d qua.tum mecha.ics. The methods used in eva.ua.ing products a.d quotients of functiona.s a.e a.so discussed a.ong with va.ia.iona. itera.ion methods. La.er on, a.good number of a.plica.ions in different a.ea. a.e presented a.d discussed;then a.concluding discussion is given.

关键词： Va.ia.ion Itera.ion Ca.culus Method a.plica.ions Technique Extreme

Fermi Function a.d Its a.plica.ions

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Journa. of a.plied Ma.hema.ics a.d Physics 2023年第1期11卷 135-146页

作者： a.ma.a. Ela.wel Na.a. a. a.ja.a.i Ma.ia. Ba.ba.h a.i M. awin Depa.tment of Ma.hema.ics Faculty of Education University of Tripoli Tripoli Libya Depa.tment of Ma.hema.ics Faculty of Science University of Tripoli Tripoli Libya

In this pa.er, we give a.definition of the Fermi function, or the so-ca.led Woods-Sa.on potentia., a.well-known potentia. in nuclea. physics;then, we give a.few of its a.plica.ions a. exa.ples. Some importa.t integra.s, which involve this function, a.e computed discussing the integra.ility a.d convergence of these integra.s. Following, we derive formula. tha. encounter the a.ove-mentioned function to get nuclea. a.d genera.ized moments;the ra.ia. Fourier tra.sforma.ion is a.so exposed. Some rela.ed a.plica.ions a.e then given tha. use such importa.t integra.s;in pa.ticula., we give the computa.ion in conjunction with the problem of getting the optica.-model potentia. for hea.y-ion intera.tions a. intermedia.e energies. Fina.ly, we conclude with importa.t rema.ks to do with the evolution of the subject.

关键词： Fermi Function Integra.s Optica. Model Potentia. Nuclea. Intera.tions Convergence

On Continued Fra.tions a.d Their a.plica.ions

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Journa. of a.plied Ma.hema.ics a.d Physics 2022年第1期10卷 142-159页

作者： Za.iya.M. Ibra. Efa. a. a.ja.la.i a.i M. awin Depa.tment of Ma.hema.ics Faculty of Science University of Tripoli Tripoli Libya

Continued fra.tions constitute a.very importa.t subject in ma.hema.ics. Their importa.ce lies in the fa.t tha. they ha.e very interesting a.d bea.tiful a.plica.ions in ma.y fields in pure a.d a.plied sciences. This review a.ticle will revea. some of these a.plica.ions a.d will reflect the bea.ty behind their uses in ca.cula.ing roots of rea. numbers, getting solutions of a.gebra.c Equa.ions of the second degree, a.d their uses in solving specia. ordina.y differentia. Equa.ions such a. Legendre, Hermite, a.d La.uerre Equa.ions;moreover a.d most importa.t, their use in physics in solving Schrodinger Equa.ion for a.certa.n potentia.. a.compa.ison will a.so be given between the results obta.ned via.continued fra.tions a.d those obta.ned through the use of well-known numerica. methods. a.va.ces in the subject will be discussed a. the end of this review a.ticle.

关键词： Continued Fra.tion Equa.ion Numerica. Method Roots Series Finite

Specia. Trilinea. Functions with Few a.plica.ions

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Journa. of a.plied Ma.hema.ics a.d Physics 2021年第11期9卷 2698-2705页

作者： a.sha.a. Juroud Ba.hir W. Sha.if a.i M. awin Depa.tment of Ma.hema.ics University of Tripoli Tripoli Libya

Specia. bilinea. functions (SBF) proved to be a.plica.le in ma.y situa.ions a.d for a.good number of problems. Hence it is importa.t to genera.ize them to a.higher degree by expa.ding previous work. In the beginning, we give a.quick review of SBF [or qua.roms of second degree a.d dimension 2 x n];then we give a.few a.plica.ions ba.ed on previously published resea.ch concentra.ing on their use in eva.ua.ing some specia. functions a.d where we present the eva.ua.ion of Chebyshev polynomia.s a. a.new work. Following tha., we define specia. trilinea. functions (STF) of three n-tuples vectors, which a.e the genera.iza.ion of SBF. Fina.ly, a.few a.plica.ions, such a. ta.ing the product of three polynomia.s of degree n, a.e given stressing the fa.t tha. the process of ta.ing the product of three integers using STF techniques, pra.tica.ly, ta.es pla.e in a.very efficient wa. a.d with no mentioned effort. a.short discussion on the future of the subject constitutes the conclusion of our a.ticle.

关键词： Bilinea. Trilinea. Function Vectors Polynomia. Specia. Qua.rom

Remova. of a.iline Blue from a.ueous Solutions Using Ce1-xBixCrO3 (x = 0, 0.5, 1)

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Open Journa. of Inorga.ic Non 2018年第1期8卷 1-10页

作者： La.ib a. awin Ma.moud a. El-Ra.s a.dunna.er M. Etorki Na.a. a. Moha.ed Wesa. a. Ma.hlof Depa.tment of Chemistry Faculty of Science University of Tripoli Tripoli Libya Depa.tment of Chemistry School of Basic Sciences Libyan Academy Tripoli Libya

The remova. of a.iline blue dye from a.ueous solutions using the a.site doped perovskite Ce1-xBixCrO3 (x = 0, 0.5, 1) wa. investiga.ed. The perovskite oxides were synthesised using Sol-Gel method a.d cha.a.terised by conventiona. powder X-ra. diffra.tion technique. The X-ra. diffra.tion mea.urements suggested tha. doping with Bismuth Ion influences both the crysta. structure a.d the pa.ticle size of the oxides, a.d consequently a.fects the a.sorption properties. It wa. found tha. both CeCrO3 a.d Ce0.5Bi0.5CrO3 compounds a.e orthorhombic a.d ha.e a.proxima.e pa.ticle size of 87 a.d 36 nm respectively, wherea. BiCrO3 oxide ha. rhombohedra. spa.e group symmetry a.d the pa.ticle sizes a.e less tha. 49 nm. The ba.ch mode study demonstra.ed tha. the remova. ca.a.ities of a.iline Blue a. 150 min a.d pH = 4.3 for Ce0.5Bi0.5CrO3, CeCrO3 a.d BiCrO3 a.e 779.67, 705.45 a.d 440.18 mg/g respectively. The results reflect the influence of the a.site doping on the a.sorption properties of the oxides. The remova. of a.iline Blue wa. found to be nega.ively correla.ed with tempera.ure a.d pH.

关键词： Remova. a.iline Blue Perovskite Oxides