Convergence of Iterative Difference Schemes for Two and Three Dimensional Nonlinear Parabolic Systems

In this paper, we study the general difference schemes of the boundary value problem for the nonlinear parabolic systems with two and three space dimensions. To solve the nonlinear difference schemes, we construct an iterative sequence from the solutions or the linearized difference schemes. We shall prove the convergence of the difference solutions for the iterative difference schemes to the solution of the original boundary value problem or the nonlinear parabolic systems.     

求解带非线性边界条件的反应扩散方程数值解的组合迭代方法

1　引　　言我们首先考虑如下抛物型方程ｕｔ-ＤΔｕ =ｆ(ｘ ,ｔ ,ｕ) (ｔ∈ ( 0 ,Ｔ],ｘ∈Ω ) ｕ/ ν+ βｕ =ｇ(ｘ ,ｔ ,ｕ) (ｔ∈ ( 0 ,Ｔ],ｘ∈ Ω )ｕ(ｘ ,0 ) =ψ(ｘ) (ｘ∈Ω )( 1 .1 )其中Ｔ为正常数 ,Ω 是ＲＰ 空间的有界区域 记ＱＴ=Ω × ( 0 ,Ｔ],ＳＴ= Ω × ( 0 ,Ｔ],假设在ＱＴ上Ｄ≡ｄ(ｘ ,ｔ) >0 ,在ＳＴ 上β≡β(ｘ ,ｔ)≥ 0 又设 ｆ(ｘ ,ｔ,ｕ) ,ｇ(ｘ ,ｔ,ｕ)为关于ｕ的非线性函数 ,且对ｘ ,ｔ各参数满足Ｈ¨ｏｌｄｅｒ连续条件 将 ( 1 .1 )离散化之后我们得到相应的有限差分系统 ,当 ｇ(ｘ ,ｔ,ｕ)为ｕ的线性…     

Block monotone iterative methods for numerical solutions of nonlinear elliptic equations

Summary. Two block monotone iterative schemes for a nonlinear algebraic system, which is a finite difference approximation of a nonlinear elliptic boundary-value problem, are presented and are shown to converge monotonically either from above or from below to a solution of the system. This monotone convergence result yields a computational algorithm for numerical solutions as well as an existence-comparison theorem of the system, including a sufficient condition for the uniqueness of the solution. An advantage of the block iterative schemes is that the Thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one-dimensional problems. The block iterative schemes are compared with the point monotone iterative schemes of Picard, Jacobi and Gauss-Seidel, and various theoretical comparison results among these monotone iterative schemes are given. These comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes. Application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given. Received August 1, 1993 / Revised version received November 7, 1994     

Existence and iteration of positive solutions for high-order fractional differential equations with integral conditions on a half-line

In this paper, existence and iteration of positive solutions for a class of high-order fractional differential equations with integral conditions on a half-line is investigated. By means of a monotone iterative technique, we obtain not only the existence of positive solutions for the problems, but also establish iterative schemes for approximating the solutions.     

EXISTENCE AND ITERATION OF POSITIVE SYMMETRIC SOLUTIONS TO A MULTI-POINT BOUNDARY VALUE PROBLEM

In this paper,we consider the existence of symmetric solutions to a nonlinear second order multi-point boundary value problem,and establish corresponding iterative schemes based on the monotone iterative method.     

半直线上Riemann-Liouville型奇异分数阶微分方程边值问题的单调迭代方法

运用不动点定理和单调迭代方法研究半直线上Riemann-Liouville型奇异分数阶微分方程边值问题的正解的存在性.在没有上、下解存在的假设下建立了边值问题存在两个正解的结果,构造了逼近正解的迭代格式,该迭代格式便于应用.     

二阶非线性q-差分微分方程两点边值问题两个迭代正解的存在性*

研究了二阶非线性q-差分微分方程两点边值问题,给出了系统两个正解存在的充分条件. 与其他文献中使用的不动点定理不同,文章不仅证明了该系统正解的存在性, 而且还利用单调迭代技巧给出了逼近正解的迭代格式.     

Monotone iteration method for differential equations involving integral boundary conditions on the half line

By applying the monotone iterative methods, we obtain the existence of monotone positive solutions for integral boundary value problems of differential equations on the half line, and establish the iterative schemes for approximating the solutions. Our approach is based on the fixed point theorem and the monotone iterative technique. The existence of lower and upper solutions is not needed.     

无穷区间上具有p-Laplacian算子的脉冲多点边值问题单调迭代正解的存在性

利用单调迭代方法得到了无穷区间上具有p-Laplacian算子的脉冲微分方程多点边值问题单调迭代正解的存在性,同时也给出了解的相应迭代序列.     

Banach空间中均衡问题和弱相对非扩展映射的强收敛定理及其应用(英文)

本文在Banach空间中设计了一些新的杂交迭代算法用以逼近一类均衡问题解集和弱相对非扩展映射不动点集或极大单调算子零点集的公共元.得到了一些强收敛的结论,并将它们推广到逼近一类均衡问题解集和有限个弱相对非扩展映射公共不动点集或有限个极大单调算子公共零点集的公共元的情形.最后,展示了本文的迭代算法在最优化问题上的应用.     

Some modifications of the quasilinearization method with higher-order convergence for solving nonlinear BVPs

In this paper, modifications of the quasilinearization method with higher-order convergence for solving nonlinear differential equations are constructed. A general technique for systematically obtaining iteration schemes of order m (?>?2) for finding solutions of highly nonlinear differential equations is developed. The proposed iterative schemes have convergence rates of cubic, quartic and quintic orders. These schemes were further applied to bifurcation problems and to obtain critical parameter values for the existence and uniqueness of solutions. The accuracy and validity of the new schemes is tested by finding accurate solutions of the one-dimensional Bratu and Frank-Kamenetzkii equations.     

Successive iteration and positive solutions for boundary value problem of nonlinear fractional q-difference equation

In this paper, we investigate the positive solutions for a class of nonlinear q-fractional boundary value problem. We not only obtain the existence and uniqueness of positive solutions, but also establish the iterative schemes for approximating the solutions, which is benefit for computation and application.     

General variational inequalities and nonexpansive mappings

In this paper, we suggest and analyze some three-step iterative schemes for finding the common elements of the set of the solutions of the Noor variational inequalities involving two nonlinear operators and the set of the fixed points of nonexpansive mappings. We also consider the convergence analysis of the suggested iterative schemes under some mild conditions. Since the Noor variational inequalities include variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as an refinement and improvement of the previously known results.     

Existence and iteration of monotone positive solutions for an elastic beam equation with a corner

This paper is concerned with the existence of monotone positive solutions for an elastic beam equation with a corner. The boundary conditions mean that the beam is embedded at one end and fastened with a sliding clamp at the other end. By applying monotone iterative techniques, we not only obtain the existence of positive solutions, but also establish iterative schemes for approximating the solutions.     

Banach空间中集值混和变分不等式问题解的迭代算法

介绍了集值映象的伪单调定义，并在Banach空间中构造了集值混和变分不等式问题近似解的迭代算法．应用伪单调映象定义，证明了该迭代算法收敛于集值混和变分不等式问题的近似解．特别值得注意的是：在文章中对集值映象没有Lipschitz连续性假设．     

On the numerical solution of Korteweg-de Vries equation by the iterative splitting method

In this paper, we apply the method of iterative operator splitting on the Korteweg-de Vries (KdV) equation. The method is based on first, splitting the complex problem into simpler sub-problems. Then each sub-equation is combined with iterative schemes and solved with suitable integrators. Von Neumann analysis is performed to achieve stability criteria for the proposed method applied to the KdV equation. The numerical results obtained by iterative splitting method for various initial conditions are compared with the exact solutions. It is seen that they are in a good agreement with each other.     

Existence and iteration of monotone positive solutions for a third-order two-point boundary value problem

The existence of nondecreasing positive solutions for the nonlinear third-order twopoint boundary value problem u′″（t） ＋ q（t）f（t,u（t）,u′（t）） = 0, 0 〈 t 〈 1, u（0） = u″（0） = u′（1） = 0 is studied. The iterative schemes for approximating the solutions are obtained by applying a monotone iterative method.     

Mann-Type Steepest-Descent and Modified Hybrid Steepest-Descent Methods for Variational Inequalities in Banach Spaces

In this paper, we propose three different kinds of iteration schemes to compute the approximate solutions of variational inequalities in the setting of Banach spaces. First, we suggest Mann-type steepest-descent iterative algorithm, which is based on two well-known methods: Mann iterative method and steepest-descent method. Second, we introduce modified hybrid steepest-descent iterative algorithm. Third, we propose modified hybrid steepest-descent iterative algorithm by using the resolvent operator. For the first two cases, we prove the convergence of sequences generated by the proposed algorithms to a solution of a variational inequality in the setting of Banach spaces. For the third case, we prove the convergence of the iterative sequence generated by the proposed algorithm to a zero of an operator, which is also a solution of a variational inequality.     

Monotone iterations for numerical solutions of reaction-diffusion-convection equations with time delay

In this article we use the monotone method for the computation of numerical solutions of a nonlinear reaction-diffusion-convection problem with time delay. Three monotone iteration processes for a suitably formulated finite-difference system of the problem are presented. It is shown that the sequence of iteration from each of these iterative schemes converges from either above or below to a unique solution of the finite-difference system without any monotone condition on the nonlinear reaction function. An analytical comparison result among the three processes of iterations is given. Also given is the application of the iterative schemes to some model problems in population dynamics, including numerical results of a model problem with known analytical solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 339–351, 1998     

无穷区间上具有p-Laplacian算子边值问题的正解

研究了一类在无穷区间上具有p-Laplacian算子的边值问题的迭代正解.利用单调迭代方法得到问题的迭代正解存在性的充分条件,同时得到了解的相应迭代序列,最后给出例子证明所得结论.