One of the most attractive subjects in applied sciences is to obtain exact or approximate solutions for different types of linear and nonlinear systems.Systems of ordinary differential equations like systems of second...One of the most attractive subjects in applied sciences is to obtain exact or approximate solutions for different types of linear and nonlinear systems.Systems of ordinary differential equations like systems of second-order boundary value problems(BVPs),Brusselator system and stiff system are significant in science and engineering.One of the most challenge problems in applied science is to construct methods to approximate solutions of such systems of differential equations which pose great challenges for numerical simulations.Bernstein polynomials method with residual correction procedure is used to treat those challenges.The aim of this paper is to present a technique to approximate solutions of such differential equations in optimal way.In it,we introduce a method called residual correction procedure,to correct some previous approximate solutions for such systems.We study the error analysis of our given method.We first introduce a new result to approximate the absolute solution by using the residual correction procedure.Second,we introduce a new result to get appropriate bound for the absolute error.The collocation method is used and the collocation points can be found by applying Chebyshev roots.Both techniques are explained briefly with illustrative examples to demonstrate the applicability,efficiency and accuracy of the techniques.By using a small number of Bernstein polynomials and correction procedure we achieve some significant results.We present some examples to show the efficiency of our method by comparing the solution of such problems obtained by our method with the solution obtained by Runge-Kutta method,continuous genetic algorithm,rational homotopy perturbation method and adomian decomposition method.展开更多
The buoyant Marangoni convection heat transfer in a differentially heated cavity is numerically studied. The cavity is filled with water-Ag, water-Cu, water-Al2O3, and water-TiO2 nanofiuids. The governing equations ar...The buoyant Marangoni convection heat transfer in a differentially heated cavity is numerically studied. The cavity is filled with water-Ag, water-Cu, water-Al2O3, and water-TiO2 nanofiuids. The governing equations are based on the equations involving the stream function, vorticity, and temperature. The dimensionless forms of the governing equations are solved by the finite difference （FD） scheme consisting of the alternating direction implicit （ADI） method and the tri-diagonal matrix algorithm （TDMA）. It is found that the increase in the nanoparticle concentration leads to the decrease in the flow rates in the secondary cells when the convective thermocapillary and the buoyancy force have similar strength. A critical Marangoni number exists, below which increasing the Marangoni number decreases the average Nusselt number, and above which increasing the Marangoni number increases the average Nusselt number. The nanoparticles play a crucial role in the critical Marangoni number.展开更多
Intense visible emissions from dysprosium(Dy3+) ions doped glasses became prospective for diverse technological applications. In this paper, physical, optical and structural properties of magnesium borate glasses dope...Intense visible emissions from dysprosium(Dy3+) ions doped glasses became prospective for diverse technological applications. In this paper, physical, optical and structural properties of magnesium borate glasses doped with varied concentrations of DyOwere examined. Such glasses were synthesised by melt quenching method and characterized at room temperature using several analytical techniques.Luminescence and absorption spectra(in the visible region) of as-quenched samples were used to evaluate the physical and optical properties. XRD pattern confims the amorphous state of as-quenched samples. The Fourier transform infrared(FTIR) spectra of glasses reveal various bonding vibrations assigned to different functional groups. UV-vis-NIR spectra disclose eight absorption bands accompanied by a band for hypersensitive transition positioned at 1260 nm(~6 H→~6 F). The values of direct and indirect optical energy band gap of the studied glasses are decreased with the increase of Dyion contents. The photoluminescence spectra of all glasses under the excitation of 380 nm display two prominent emission bands centred at 497 nm(~4 F→~6 H, blue) and 587 nm(~4 F→~6 H, green).The achieved intense luminescence from the proposed glass composition may be beneficial for solidstate laser applications.展开更多
文摘One of the most attractive subjects in applied sciences is to obtain exact or approximate solutions for different types of linear and nonlinear systems.Systems of ordinary differential equations like systems of second-order boundary value problems(BVPs),Brusselator system and stiff system are significant in science and engineering.One of the most challenge problems in applied science is to construct methods to approximate solutions of such systems of differential equations which pose great challenges for numerical simulations.Bernstein polynomials method with residual correction procedure is used to treat those challenges.The aim of this paper is to present a technique to approximate solutions of such differential equations in optimal way.In it,we introduce a method called residual correction procedure,to correct some previous approximate solutions for such systems.We study the error analysis of our given method.We first introduce a new result to approximate the absolute solution by using the residual correction procedure.Second,we introduce a new result to get appropriate bound for the absolute error.The collocation method is used and the collocation points can be found by applying Chebyshev roots.Both techniques are explained briefly with illustrative examples to demonstrate the applicability,efficiency and accuracy of the techniques.By using a small number of Bernstein polynomials and correction procedure we achieve some significant results.We present some examples to show the efficiency of our method by comparing the solution of such problems obtained by our method with the solution obtained by Runge-Kutta method,continuous genetic algorithm,rational homotopy perturbation method and adomian decomposition method.
基金Project supported by the Fundamental Research Grant Scheme of the Ministry of Education of Malaysia(No.FRGS/1/2014/SG04/UKM/01/1)the Dana Impak Perdana of Universiti Kebangsaan Malaysia(No.DIP-2014-015)
文摘The buoyant Marangoni convection heat transfer in a differentially heated cavity is numerically studied. The cavity is filled with water-Ag, water-Cu, water-Al2O3, and water-TiO2 nanofiuids. The governing equations are based on the equations involving the stream function, vorticity, and temperature. The dimensionless forms of the governing equations are solved by the finite difference （FD） scheme consisting of the alternating direction implicit （ADI） method and the tri-diagonal matrix algorithm （TDMA）. It is found that the increase in the nanoparticle concentration leads to the decrease in the flow rates in the secondary cells when the convective thermocapillary and the buoyancy force have similar strength. A critical Marangoni number exists, below which increasing the Marangoni number decreases the average Nusselt number, and above which increasing the Marangoni number increases the average Nusselt number. The nanoparticles play a crucial role in the critical Marangoni number.
基金Project supported by the Ministry of Higher Education Malaysia and Universiti Teknologi Malaysia through Research University Grant Scheme(17H79,18H68,03G72)International Doctoral Fellowship(IDF)and UTM Zamalah Scholarship
文摘Intense visible emissions from dysprosium(Dy3+) ions doped glasses became prospective for diverse technological applications. In this paper, physical, optical and structural properties of magnesium borate glasses doped with varied concentrations of DyOwere examined. Such glasses were synthesised by melt quenching method and characterized at room temperature using several analytical techniques.Luminescence and absorption spectra(in the visible region) of as-quenched samples were used to evaluate the physical and optical properties. XRD pattern confims the amorphous state of as-quenched samples. The Fourier transform infrared(FTIR) spectra of glasses reveal various bonding vibrations assigned to different functional groups. UV-vis-NIR spectra disclose eight absorption bands accompanied by a band for hypersensitive transition positioned at 1260 nm(~6 H→~6 F). The values of direct and indirect optical energy band gap of the studied glasses are decreased with the increase of Dyion contents. The photoluminescence spectra of all glasses under the excitation of 380 nm display two prominent emission bands centred at 497 nm(~4 F→~6 H, blue) and 587 nm(~4 F→~6 H, green).The achieved intense luminescence from the proposed glass composition may be beneficial for solidstate laser applications.
