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Journal of Applied Mathematics and Physics 2024年第3期12卷 877-896页

作者： Khalid Abd Elrazig Awad Alla Elnour Department of Mathematics Preparatory Deanship Najran University Najran Kingdom of Saudi Arabia

To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.

关键词： First-Order Differential Equations Picard Method Taylor Series Method numerical solutions numerical Examples MATLAB Software

numerical solutions OF DISCONTINUOUS BOUNDARY VALUE PROBLEMS FOR GENERAL ELLIPTIC COMPLEX EQUATIONS OF FIRST ORDER

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Acta Mathematica Scientia 2000年第2期20卷 162-168页

作者：黄沙闻国椿 Department of mathematics Hebei Teacher's University Shijiazhuang 050016 China Department of Mathematics Peking University Beijing 100871 China

In this paper, authors discuss the numerical methods of general discontinuous boundary value problems for elliptic complex equations of first order, They first give the well posedness of general discontinuous boundary value problems, reduce the discontinuous boundary value problems to a variation problem, and then find the numerical solutions of above problem by the finite element method. Finally authors give some error-estimates of the foregoing numerical solutions.

关键词： elliptic complex equations numerical solutions boundary value problem

A New Method for Solving Nonlinear Partial Differential Equations Based on Liquid Time-Constant Networks

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Journal of Systems Science & Complexity 2024年第2期37卷 480-493页

作者： SUN Jiuyun DONG Huanhe FANG Yong College of Mathematics and Systems Science Shandong University of Science and TechnologyQingdao 266590China

In this paper,physics-informed liquid networks(PILNs)are proposed based on liquid time-constant networks(LTC)for solving nonlinear partial differential equations(PDEs).In this approach,the network state is controlled via ordinary differential equations(ODEs).The significant advantage is that neurons controlled by ODEs are more expressive compared to simple activation *** addition,the PILNs use difference schemes instead of automatic differentiation to construct the residuals of PDEs,which avoid information loss in the neighborhood of sampling *** this method draws on both the traveling wave method and physics-informed neural networks(PINNs),it has a better physical ***,the KdV equation and the nonlinear Schr¨odinger equation are solved to test the generalization ability of the *** the best of the authors’knowledge,this is the first deep learning method that uses ODEs to simulate the numerical solutions of PDEs.

关键词： Nonlinear partial differential equations numerical solutions physics-informed liquid networks physics-informed neural networks

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Lie group analysis, numerical and non-traveling wave solutions for the (2+1)-dimensional diffusion–advection equation with variable coefficients

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Chinese Physics B 2014年第3期23卷 71-76页

作者： Vikas Kumar R.K.Gupta Ram Jiwari Department of Mathematics D.A.V.College Pundari School of Mathematics and Computer Applications Thapar University

In this paper, the variable-coefficient diffusion-advection （DA） equation, which arises in modeling various physical phenomena, is studied by the Lie symmetry approach. The similarity reductions are derived by determining the complete sets of point symmetries of this equation, and then exact and numerical solutions are reported for the reduced second-order nonlinear ordinary differential equations. Further, an extended （Gl/G）-expansion method is applied to the DA equation to construct some new non-traveling wave solutions.

关键词： diffusion-advection equation Lie group analysis numerical solutions extended (G'/G)-expansion method

A Stochastic Study of the Fractional Order Model of Waste Plastic in Oceans

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Computers, Materials & Continua 2022年第11期73卷 4441-4454页

作者： Muneerah Al Nuwairan Zulqurnain Sabir Muhammad Asif Zahoor Raja Maryam Alnami Hanan Almuslem Department of Mathematics and Statistics College of ScienceKing Faisal UniversityP.O.Box 400Al-Ahsa31982Saudi Arabia Department of Mathematics and Statistics Hazara UniversityMansehraPakistan Future Technology Research Center National Yunlin University of Science and TechnologyDouliouYunlin64002Taiwan

In this paper,a fractional order model based on the management of waste plastic in the ocean(FO-MWPO)is numerically *** mathematical form of the FO-MWPO model is categorized into three components,waste plastic,Marine debris,and *** stochastic numerical solvers using the Levenberg-Marquardt backpropagation neural networks(LMQBP-NNs)have been applied to present the numerical solutions of the FO-MWPO *** competency of the method is tested by taking three variants of the FO-MWPO model based on the fractional order *** data ratio is provided for training,testing and authorization is 77%,12%,and 11%*** exactness of LMQBP-NNs is observed by using the comparative performances of the obtained and the Adams-BashforthMoulton *** verify the competence,validity,capability,exactness,and consistency of LMQBP-NNs,the performances have been obtained using the regression,state transitions,error histograms,correlation and mean square error.

关键词： Fractional order ocean Adams-Bashforth-Moulton LevenbergMarquardt backpropagation numerical solutions

A Study of Traveling Wave Structures and numerical Investigation of Two-Dimensional Riemann Problems with Their Stability and Accuracy

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Computer Modeling in Engineering & Sciences 2023年第3期134卷 2193-2209页

作者： Abdulghani Ragaa Alharbi Department of Mathematics College of ScienceTaibah UniversityAl-Madinah Al-Munawarah42353Saudi Arabia

The Riemann wave system has a fundamental role in describing waves in various nonlinear natural phenomena,for instance,tsunamis in the *** paper focuses on executing the generalized exponential rational function approach and some numerical methods to obtain a distinct range of traveling wave structures and numerical results of the two-dimensional Riemann *** stability of obtained traveling wave solutions is analyzed by satisfying the constraint conditions of the Hamiltonian *** simulations are investigated via the finite difference method to verify the accuracy of the obtained *** extract the approximation solutions to the underlying problem,some ODE solvers in FORTRAN software are applied,and outcomes are shown *** stability and accuracy of the numerical schemes using Fourier’s stabilitymethod and error analysis,respectively,to increase the reassurance are investigated.A comparison between the analytical and numerical results is obtained and graphically *** proposed methods are effective and practical to be applied for solving more partial differential equations(PDEs).

关键词： The Riemann wave equation Hamiltonian system solitary solutions numerical solutions stability accuracy

New Soliton Wave solutions to a Nonlinear Equation Arising in Plasma Physics

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Computer Modeling in Engineering & Sciences 2023年第10期137卷 827-841页

作者： M.B.Almatrafi Abdulghani Alharbi Department of Mathematics Faculty of ScienceTaibah UniversitySaudi Arabia

The extraction of traveling wave solutions for nonlinear evolution equations is a challenge in various mathematics,physics,and engineering *** article intends to analyze several traveling wave solutions for themodified regularized long-wave(MRLW)equation using several approaches,namely,the generalized algebraic method,the Jacobian elliptic functions technique,and the improved Q-expansion *** successfully obtain analytical solutions consisting of rational,trigonometric,and hyperbolic *** adaptive moving mesh technique is applied to approximate the numerical solution of the proposed *** adaptive moving mesh method evenly distributes the points on the high error *** method perfectly and strongly reduces the *** compare the constructed exact and numerical results to ensure the reliability and validity of the methods *** better understand the considered equation’s physical meaning,we present some 2D and 3D *** exact and numerical approaches are efficient,powerful,and versatile for establishing novel bright,dark,bell-kink-type,and periodic traveling wave solutions for nonlinear PDEs.

关键词： The modified regularized long wave equation soliton solutions plasma physics numerical solutions

An Analysis of the Characteristics of the Thermal Boundary Layer in Power Law Fluid

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Journal of Thermal Science 2008年第3期17卷 233-237页

作者： Hao ZHANG Xinxin ZHANG Liancun ZHENG Department of Thermal Engineering University of Science and Technology Beijing Beijing 100083 China Department of Mathematics and Mechanics University of Science and Technology Beijing Beijing 100083 China

This paper presents a theoretical analysis of the heat transfer for the boundary layer flow on a continuous moving surface in power law fluid. The expressions of the thermal boundary layer thickness with the different heat conductivity coefficients are obtained according to the theory of the dimensional analysis of fluid dynamics and heat transfer. And the numerical results of CFD agree well with the proposed expressions. The estimate formulas can be successfully applied to giving the thermal boundary layer thickness.

关键词： Thermal boundary layer Power law fluid Dimensional analysis numerical solutions

numerical Study of Thermal Boundary Layer on a Continuous Moving Surface in Power Law Fluids

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Journal of Thermal Science 2007年第3期16卷 243-247页

This paper investigates flow and heat transfer of power law fluids on a continuous moving surface. The temperature distribution is obtained numerically by considering the effect of the power law viscosity on thermal diffusivity and the characteristics of the flow and heat transfer are analyzed. The results show that the distribution of the thermal boundary layer depends not only on the velocity ratio parameter of the plate, but also on the power law index and Prandtl number of fluids.

关键词： Power law fluid thermal boundary layer dimensionless transformation numerical solutions

Dependence of Upper Atmosphere Photochemistry on the Shape of the Diurnal Cycle of the Photolysis Rates

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Advances in Atmospheric Sciences 2006年第2期23卷 207-214页

作者： Sonia MONTECINOS Patricia BARRIENTOS Center of Advanced Studies in Arid Zones (CEAZA) University of La Serena La SerenaChile Department of Physics University of La Frontera TemucoChile

A photochemical model of the atmosphere constitutes a non-linear, non-autonomous dynamical system, enforced by the Earth＇s rotation. Some studies have shown that the region of the mesopause tends towards non-linear responses such as period-doubling cascades and chaos. In these studies, simple approximations for the diurnal variations of the photolysis rates are assumed. The goal of this article is to investigate what happens if the more realistic, calculated photolysis rates are introduced. It is found that, if the usual approximations-sinusoidal and step fiunctions—are assumed, the responses of the system axe similar： it converges to a 2-day periodic solution. If the more realistic, calculated diurnal cycle is introduced, a new 4-day subharmonic appear.

关键词： photochemical models bifurcation theory numerical solutions dynamical systems