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ASYMPTOTIC METHOD FOR SINGULAR PERTURBATION PROBLEM OF ORDINARY DIFFERENCE EQUATIONS
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作者 吴启光 苏煜城 孙志忠 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1989年第3期221-230,共10页
This paper is taken up for the following difference equation problem (Pe);where e is a small parameter, c1, c2,constants and functions of k and e . Firstly, the case with constant coefficients is considered. Secondly,... This paper is taken up for the following difference equation problem (Pe);where e is a small parameter, c1, c2,constants and functions of k and e . Firstly, the case with constant coefficients is considered. Secondly, a general method based on extended transformation is given to handle (P.) where the coefficients may be variable and uniform asymptotic expansions are obtained. Finally, a numerical example is provided to illustrate the proposed method. 展开更多
关键词 ASYMPTOTIC method FOR SINGULAR PERTURBATION PROBLEM OF ORDINARY difference equationS
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NEGATIVE NORM LEAST-SQUARES METHODS FOR THE INCOMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS 被引量:2
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作者 高少芹 段火元 《Acta Mathematica Scientia》 SCIE CSCD 2008年第3期675-684,共10页
The purpose of this article is to develop and analyze least-squares approximations for the incompressible magnetohydrodynamic equations. The major advantage of the least-squares finite element method is that it is not... The purpose of this article is to develop and analyze least-squares approximations for the incompressible magnetohydrodynamic equations. The major advantage of the least-squares finite element method is that it is not subjected to the so-called Ladyzhenskaya-Babuska-Brezzi (LBB) condition. The authors employ least-squares functionals which involve a discrete inner product which is related to the inner product in H^-1(Ω). 展开更多
关键词 The incompressible MHDs equation negative norm VORTICITY least-squares mixed finite element method
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THE GROWTH OF DIFFERENCE EQUATIONS AND DIFFERENTIAL EQUATIONS
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作者 Zongxuan CHEN Ranran ZHANG +1 位作者 Shuangting LAN Chuangxin CHEN 《Acta Mathematica Scientia》 SCIE CSCD 2021年第6期1911-1920,共10页
In this paper,we mainly apply a new,asymptotic method to investigate the growth of meromorphic solutions of linear higher order difference equations and differential equations.We delete the condition(1.6)of Theorems E... In this paper,we mainly apply a new,asymptotic method to investigate the growth of meromorphic solutions of linear higher order difference equations and differential equations.We delete the condition(1.6)of Theorems E and F,yet obtain the same results for Theorems E and F.We also weaken the condition(1.4)of Theorems C and D. 展开更多
关键词 asymptotic method difference equations differential equations
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New Asymptotical Stability and Uniformly Asymptotical Stability Theorems for Nonautonomous Difference Equations
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作者 Limin Zhang Chaofeng Zhang 《Applied Mathematics》 2016年第10期1023-1031,共9页
New theorems of asymptotical stability and uniformly asymptotical stability for nonautonomous difference equations are given in this paper. The classical Liapunov asymptotical stability theorem of nonautonomous differ... New theorems of asymptotical stability and uniformly asymptotical stability for nonautonomous difference equations are given in this paper. The classical Liapunov asymptotical stability theorem of nonautonomous difference equations relies on the existence of a positive definite Liapunov function that has an indefinitely small upper bound and whose variation along a given nonautonomous difference equations is negative definite. In this paper, we consider the case that the Liapunov function is only positive definite and its variation is semi-negative definite. At these weaker conditions, we put forward a new asymptotical stability theorem of nonautonomous difference equations by adding to extra conditions on the variation. After that, in addition to the hypotheses of our new asymptotical stability theorem, we obtain a new uniformly asymptotical stability theorem of nonautonomous difference equations provided that the Liapunov function has an indefinitely small upper bound. Example is given to verify our results in the last. 展开更多
关键词 Nonautonomous difference equations New Asymptotical Stability Theorem New Uniformly Asymptotical Stability Theorem Liapunovs Direct method
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Analysis of Extended Fisher-Kolmogorov Equation in 2D Utilizing the Generalized Finite Difference Method with Supplementary Nodes
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作者 Bingrui Ju Wenxiang Sun +1 位作者 Wenzhen Qu Yan Gu 《Computer Modeling in Engineering & Sciences》 SCIE EI 2024年第10期267-280,共14页
In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolso... In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme.Following temporal discretization,the generalized finite difference method(GFDM)with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node.These supplementary nodes are distributed along the boundary to match the number of boundary nodes.By incorporating supplementary nodes,the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation.To demonstrate the efficacy of our approach,we present three numerical examples showcasing its performance in solving this nonlinear problem. 展开更多
关键词 Generalized finite difference method nonlinear extended Fisher-Kolmogorov equation Crank-Nicolson scheme
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Efficient Finite Difference/Spectral Method for the Time Fractional Ito Equation Using Fast Fourier Transform Technic
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作者 Dakang Cen Zhibo Wang Seakweng Vong 《Communications on Applied Mathematics and Computation》 EI 2023年第4期1591-1600,共10页
A finite difference/spectral scheme is proposed for the time fractional Ito equation.The mass conservation and stability of the numerical solution are deduced by the energy method in the L^(2)norm form.To reduce the c... A finite difference/spectral scheme is proposed for the time fractional Ito equation.The mass conservation and stability of the numerical solution are deduced by the energy method in the L^(2)norm form.To reduce the computation costs,the fast Fourier transform technic is applied to a pair of equivalent coupled differential equations.The effectiveness of the proposed algorithm is verified by the first numerical example.The mass conservation property and stability statement are confirmed by two other numerical examples. 展开更多
关键词 Time fractional Ito equation Finite difference method Spectral method STABILITY
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Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation
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作者 Joseph Hunter Zheng Sun Yulong Xing 《Communications on Applied Mathematics and Computation》 EI 2024年第1期658-687,共30页
This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either... This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints. 展开更多
关键词 Linearized Korteweg-de Vries(KdV)equation Implicit-explicit(IMEX)Runge-Kutta(RK)method STABILITY Courant-Friedrichs-Lewy(CFL)condition Finite difference(FD)method Local discontinuous Galerkin(DG)method
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Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions 被引量:4
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作者 Boling GUO Qiang XU Zhe YIN 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2016年第3期403-416,共14页
An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation (FPE) with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for ... An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation (FPE) with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for two special cases, i.e., a continued seep- age flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples. 展开更多
关键词 fractional percolation equation (FPE) Riemann-Liouville derivative frac-tional boundary condition finite difference method stability and convergence Toeplitzmatrix
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Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations 被引量:3
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作者 罗志强 陈志敏 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2013年第8期931-944,共14页
A three-dimensional (3D) predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is ... A three-dimensional (3D) predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically. 展开更多
关键词 three-dimensional (3D) nonlinear potential flow equation predictor-corrector finite difference method staggered grid nested iterative method 3D sloshing
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Solution of shallow-water equations using least-squares finite-element method 被引量:3
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作者 S.J. Liang J,-H, Tang M.-S. Wu 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2008年第5期523-532,共10页
A least-squares finite-element method (LSFEM) for the non-conservative shallow-water equations is presented. The model is capable of handling complex topography, steady and unsteady flows, subcritical and supercriti... A least-squares finite-element method (LSFEM) for the non-conservative shallow-water equations is presented. The model is capable of handling complex topography, steady and unsteady flows, subcritical and supercritical flows, and flows with smooth and sharp gradient changes. Advantages of the model include: (1) sources terms, such as the bottom slope, surface stresses and bed frictions, can be treated easily without any special treatment; (2) upwind scheme is no needed; (3) a single approximating space can be used for all variables, and its choice of approximating space is not subject to the Ladyzhenskaya-Babuska-Brezzi (LBB) condition; and (4) the resulting system of equations is symmetric and positive-definite (SPD) which can be solved efficiently with the preconditioned conjugate gradient method. The model is verified with flow over a bump, tide induced flow, and dam-break. Computed results are compared with analytic solutions or other numerical results, and show the model is conservative and accurate. The model is then used to simulate flow past a circular cylinder. Important flow charac-teristics, such as variation of water surface around the cylinder and vortex shedding behind the cylinder are investigated. Computed results compare well with experiment data and other numerical results. 展开更多
关键词 least-square finite-element method Shallow-water equations DAM-BREAK Vortex shedding
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Least-squares finite-element method for shallow-water equations with source terms 被引量:2
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作者 Shin-Jye Liang Tai-Wen Hsu 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2009年第5期597-610,共14页
Numerical solution of shallow-water equations (SWE) has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of s... Numerical solution of shallow-water equations (SWE) has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of source terms in momentum equations, such as the bottom slope and friction of bed, compounds the difficulties further. In this paper, a least-squares finite-element method for the space discretization and θ-method for the time integration is developed for the 2D non-conservative SWE including the source terms. Advantages of the method include: the source terms can be approximated easily with interpolation functions, no upwind scheme is needed, as well as the resulting system equations is symmetric and positive-definite, therefore, can be solved efficiently with the conjugate gradient method. The method is applied to steady and unsteady flows, subcritical and transcritical flow over a bump, 1D and 2D circular dam-break, wave past a circular cylinder, as well as wave past a hump. Computed results show good C-property, conservation property and compare well with exact solutions and other numerical results for flows with weak and mild gradient changes, but lead to inaccurate predictions for flows with strong gradient changes and discontinuities. 展开更多
关键词 Shallow-water equations Source terms least-squares finite-element method DAM-BREAK C-property
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Computing Bifurcation Diagrams of Steady State Kuramoto Sivashinsky Equation by Difference Method 被引量:1
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作者 LI Chang pin (College of Sciences, Shanghai University) 《Advances in Manufacturing》 SCIE CAS 1999年第3期248-250,共3页
Utilizing difference formulae, we obtained the discrete systems of steady state Kuramoto Sivashinsky (K S) equation. Applied Newton's method and continuation technology to the systems, the bifurcated solutio... Utilizing difference formulae, we obtained the discrete systems of steady state Kuramoto Sivashinsky (K S) equation. Applied Newton's method and continuation technology to the systems, the bifurcated solutions are derived, and the bifurcation diagrams are constructed. All the results are successful and satisfactory. 展开更多
关键词 BIFURCATION K S equation difference method
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The Exact Formulation of the Inverse of the Tridiagonal Matrix for Solving the 1D Poisson Equation with the Finite Difference Method 被引量:2
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作者 Serigne Bira Gueye 《Journal of Electromagnetic Analysis and Applications》 2014年第10期303-308,共6页
A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. T... A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. This is the first time that the inverse of this remarkable matrix is determined directly and exactly. Thus, solving 1D Poisson equation becomes very accurate and extremely fast. This method is a very important tool for physics and engineering where the Poisson equation appears very often in the description of certain phenomena. 展开更多
关键词 1D POISSON equation Finite difference method TRIDIAGONAL Matrix INVERSION Thomas Algorithm GAUSSIAN ELIMINATION Potential Problem
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Finite difference streamline diffusion method using nonconforming space for incompressible time-dependent Navier-Stokes equations 被引量:1
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作者 陈刚 冯民富 何银年 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2013年第9期1083-1096,共14页
This paper proposes a new nonconforming finite difference streamline diffusion method to solve incompressible time-dependent Navier-Stokes equations with a high Reynolds number. The backwards difference in time and th... This paper proposes a new nonconforming finite difference streamline diffusion method to solve incompressible time-dependent Navier-Stokes equations with a high Reynolds number. The backwards difference in time and the Crouzeix-Raviart (CR) element combined with the P0 element in space are used. The result shows that this scheme has good stabilities and error estimates independent of the viscosity coefficient. 展开更多
关键词 Navier-Stokes equation high Reynolds number Ladyzhenskaya-Babugka- Brezzi (LBB) condition finite difference streamline diffusion method discrete Gronwall's inequality
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AN ACCURATE SOLUTION OF THE POISSON EQUATION BY THE FINITE DIFFERENCE-CHEBYSHEV-TAU METHOD
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作者 Hani I. Siyyam (Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid_Jordan) (Communicated by DAI Shi_qiang) 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2001年第8期935-939,共5页
A new finite difference-Chebyshev-Tau method for the solution of the two-dimensional Poisson equation is presented. Some of the numerical results are also presented which indicate that the method is satisfactory and c... A new finite difference-Chebyshev-Tau method for the solution of the two-dimensional Poisson equation is presented. Some of the numerical results are also presented which indicate that the method is satisfactory and compatible to other methods. 展开更多
关键词 Poisson equation Chebyshev polynomials Tau method finite difference method
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Sloshing simulation of standing wave with time-independent finite difference method for Euler equations
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作者 罗志强 陈志敏 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2011年第11期1475-1488,共14页
The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional (2D) tank are studied. The irregular tank is mapped onto a fixed square domain th... The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional (2D) tank are studied. The irregular tank is mapped onto a fixed square domain through proper mapping functions. A staggered mesh system is employed in a 2D tank to calculate the elevation of the transient fluid. A time-independent finite difference method, which is developed by Bang- fuh Chen, is used to solve the Euler equations for incompressible and inviscid fluids. The numerical results agree well with the analytic solutions and previously published results. The sloshing profiles of surge and heave motion with initial standing waves are presented. The results show very clear nonlinear and beating phenomena. 展开更多
关键词 Euler equation finite difference method numerical simulation Crank- Nicolson scheme
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A Computational Study with Finite Difference Methods for Second Order Quasilinear Hyperbolic Partial Differential Equations in Two Independent Variables
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作者 Pavlos Stampolidis Maria Ch. Gousidou-Koutita 《Applied Mathematics》 2018年第11期1193-1224,共32页
In this paper we consider the numerical method of characteristics for the numerical solution of initial value problems (IVPs) for quasilinear hyperbolic Partial Differential Equations, as well as the difference scheme... In this paper we consider the numerical method of characteristics for the numerical solution of initial value problems (IVPs) for quasilinear hyperbolic Partial Differential Equations, as well as the difference scheme Central Time Central Space (CTCS), Crank-Nicolson scheme, ω scheme and the method of characteristics for the numerical solution of initial and boundary value prob-lems for the one-dimension homogeneous wave equation. The initial deriva-tive condition is approximated by different second order difference quotients in order to examine which gives more accurate numerical results. The local truncation error, consistency and stability of the difference schemes CTCS, Crank-Nicolson and ω are also considered. 展开更多
关键词 Finite difference method CTCS method CRANK-NICOLSON method ω-method Numerical method of Characteristics Wave equation
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Solution of 1D Poisson Equation with Neumann-Dirichlet and Dirichlet-Neumann Boundary Conditions, Using the Finite Difference Method
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作者 Serigne Bira Gueye Kharouna Talla Cheikh Mbow 《Journal of Electromagnetic Analysis and Applications》 2014年第10期309-318,共10页
An innovative, extremely fast and accurate method is presented for Neumann-Dirichlet and Dirichlet-Neumann boundary problems for the Poisson equation, and the diffusion and wave equation in quasi-stationary regime;usi... An innovative, extremely fast and accurate method is presented for Neumann-Dirichlet and Dirichlet-Neumann boundary problems for the Poisson equation, and the diffusion and wave equation in quasi-stationary regime;using the finite difference method, in one dimensional case. Two novels matrices are determined allowing a direct and exact formulation of the solution of the Poisson equation. Verification is also done considering an interesting potential problem and the sensibility is determined. This new method has an algorithm complexity of O(N), its truncation error goes like O(h2), and it is more precise and faster than the Thomas algorithm. 展开更多
关键词 1D POISSON equation Finite difference method Neumann-Dirichlet Dirichlet-Neumann Boundary Problem TRIDIAGONAL Matrix Inversion Thomas Algorithm
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Solution of a One-Dimension Heat Equation Using Higher-Order Finite Difference Methods and Their Stability
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作者 M. Emran Ali Wahida Zaman Loskor +1 位作者 Samia Taher Farjana Bilkis 《Journal of Applied Mathematics and Physics》 2022年第3期877-886,共10页
One-dimensional heat equation was solved for different higher-order finite difference schemes, namely, forward time and fourth-order centered space explicit method, backward time and fourth-order centered space implic... One-dimensional heat equation was solved for different higher-order finite difference schemes, namely, forward time and fourth-order centered space explicit method, backward time and fourth-order centered space implicit method, and fourth-order implicit Crank-Nicolson finite difference method. Higher-order schemes have complexity in computing values at the neighboring points to the boundaries. It is required there a specification of the values of field variables at some points exterior to the domain. The complexity was incorporated using Hicks approximation. The convergence and stability analysis was also computed for those higher-order finite difference explicit and implicit methods in case of solving a one dimensional heat equation. The obtained numerical results were compared with exact solutions. It is found that backward time and fourth-order centered space implicit scheme along with Hicks approximation performed well over the other mentioned higher-order approaches. 展开更多
关键词 Heat equation Boundary Condition Higher-Order Finite difference methods Hicks Approximation
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SECOND-ORDER ACCURATE DIFFERENCE METHOD FOR THE SINGULARLY PERTURBED PROBLEM OF FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS
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作者 王国英 陈明伦 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1990年第5期463-468,共6页
In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.
关键词 SECOND-ORDER ACCURATE difference method FOR THE SINGULARLY PERTURBED PROBLEM OF FOURTH-ORDER ORDINARY differENTIAL equationS
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