A local iteration scheme for nonlinear two-dimensional steady-state heat conduction: a BEM approach

An efficient algorithm is proposed to solve the steady-state nonlinear heat conduction equation using the boundary element method (BEM). Nonlinearity of the heat conduction equation arises from nonlinear boundary conditions and temperature dependence of thermal conductivity. Using Kirchhoff's transformation, the case of temperature dependence of thermal conductivity can be transformed to the nonlinear boundary conditions case. Applying the BEM technique, the resulting matrix equation becomes nonlinear. The nonlinearity, however, only involves the boundary nodes that have nonlinearboundary conditions. The proposed local iterative scheme reduces the entire BEM matrix equation to a smaller matrix equation whose rank is the same as the number of boundary nodes with nonlinear boundary conditions. The Newton-Raphson iteration scheme is used to solve the reduced nonlinear matrix equation. The local iterative scheme is first applied to two one-dimensional problems (analytical solutions are possible) with different nonlinear boundary conditions. It is then applied to a two-region problem. Finally, the local iterative scheme is applied to two cavity problems in which radiation plays a role in the heat transfer.     

Analytical solution to a generalized spectral problem in a method of recalculating boundary conditions for the biharmonic equation

An iterative algorithm with an efficient preconditioner for the numerical solution to an elasticity problem in the approximation of plate theory with mixed boundary conditions is proposed and substantiated. Exact constants of energy equivalence for optimization of the iterative method are obtained. Inversion of the preconditioner is equivalent to the double inversion of a discrete analog of the Laplace operator with Dirichlet boundary conditions.     

A block monotone iterative method for numerical solutions of nonlinear elliptic boundary value problems

The aim of this article is to develop a new block monotone iterative method for the numerical solutions of a nonlinear elliptic boundary value problem. The boundary value problem is discretized into a system of nonlinear algebraic equations, and a block monotone iterative method is established for the system using an upper solution or a lower solution as the initial iteration. The sequence of iterations can be computed in a parallel fashion and converge monotonically to a maximal solution or a minimal solution of the system. Three theoretical comparison results are given for the sequences from the proposed method and the block Jacobi monotone iterative method. The comparison results show that the sequence from the proposed method converges faster than the corresponding sequence given by the block Jacobi monotone iterative method. A simple and easily verified condition is obtained to guarantee a geometric convergence of the block monotone iterations. The numerical results demonstrate advantages of this new approach. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

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.     

Preconditioning in the numerical solution to a Dirichlet problem for the biharmonic equation

An iterative algorithm for the numerical solution of the biharmonic equation with boundary conditions of the first kind (a clamped plate) is investigated. At every step of this iterative method, it is necessary to solve two Dirichlet problems for a Poisson equation. Constants of energy equivalence for the optimization of the iterative method are obtained.     

New results for the second order impulsive integro-differential equations with nonlinear boundary conditions

This paper considers the second order integro-differential equations with impulses. Some sufficient conditions for the existence of solutions are proposed by using monotone iterative method and Schauder fixed point theorem. Moreover, new concepts of lower and upper solutions are introduced for nonlinear boundary value problems.     

一维非线性Schrödinger 方程的两个无条件收敛的守恒紧致差分格式

本文对一维非线性 Schrödinger 方程给出两个紧致差分格式, 运用能量方法和两个新的分析技 巧证明格式关于离散质量和离散能量守恒, 而且在最大模意义下无条件收敛. 对非线性紧格式构造了 一个新的迭代算法, 证明了算法的收敛性, 并在此基础上给出一个新的线性化紧格式. 数值算例验证 了理论分析的正确性, 并通过外推进一步提高了数值解的精度.     

Boundary value problems for first order differential equations of mixed type

We apply the monotone iterative method to formulate existence results for integro-differential equations with nonlinear boundary conditions. Some results for integro-differential inequalities are also given. Examples illustrate obtained results.     

Direct shooting method for the solution of boundary-value problems

A new iterative method is developed to solve the boundary-value problems for ordinary nonlinear differential equations. The method requires that only the original system of differential equations is solved once in each iteration. The initial conditions for a new iteration are evaluated directly from the given boundary conditions and the initial and boundary conditions obtained in the previous iteration, thus avoiding the necessity of solving a system of algebraic equations. The convergence proofs of the method are given. Examples of the application of the method are presented and discussed.     

A truncated conjugate gradient method with an inexact Gauss-Newton technique for solving nonlinear systems

In this paper, a truncated conjugate gradient method with an inexact Gauss-Newton technique is proposed for solving nonlinear systems.?The iterative direction is obtained by the conjugate gradient method solving the inexact Gauss-Newton equation.?Global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions. Finally, some numerical results are presented to illustrate the effectiveness of the proposed algorithm.     

无拉力Winkler地基上自由边矩形Reissner板的弯曲

本文提出了一种求解无拉力Winkler地基上自由边矩形Reissner板受任意载荷的弯曲问题的解析方法.通过适当设定满足可导条件的Fourier级数加补充项形式的挠度函数和剪力函数,把给定边界条件下的微分方程化成最简形式的无穷代数方程组.对于常规的Winkler地基,可直接求解;而对于无拉力Winkler地基,方程组为一组弱非线性代数方程组.使用迭代法容易得到解.     

Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using Gauss product quadrature rules

The Gauss product quadrature rules and collocation method are applied to reduce the second-kind nonlinear two-dimensional Fredholm integral equations (FIE) to a nonlinear system of equations. The convergence of the proposed numerical method is proved under certain conditions on the kernel of the integral equation. An iterative method for approximating the solution of the obtained nonlinear system is provided and its convergence is proved. Also, some numerical examples are presented to show the efficiency and accuracy of the proposed method.     

Numerical solution of vector Sturm–Liouville problems with Dirichlet conditions and nonlinear dependence on the spectral parameter

A numerical-analytical iterative method is proposed for solving generalized self-adjoint regular vector Sturm–Liouville problems with Dirichlet boundary conditions. The method is based on eigenvalue (spectral) correction. The matrix coefficients of the equations are assumed to be nonlinear functions of the spectral parameter. For a relatively close initial approximation, the method is shown to have second-order convergence with respect to a small parameter. Test examples are considered, and the model problem of transverse vibrations of a hinged rod with a variable cross section is solved taking into account its rotational inertia.     

ON MONOTONE ITERATIVE METHOD FOR FIRST ORDER IMPULSIVE INTEGRO-DIFFERENTIAL BOUNDARY VALUE PROBLEMS IN BANACH SPACE

1 IntroductionThe theory of impulsive differential equation has been emerging as animportant area of investigation in recent years (see [11).Let E be a real Banach space, P is a cone of E which defines a partialordering in E: x 5 y1 x, y 6 E if and only if y -- x e P (see [2]). In this paper,we consider the fOllowing periodic boundary value problem (BVP in short) forimpulsive integro-differential equation in Ewhere f 6 C[J x E x E x E,E], J = [0,2r], 0 < t1 < t2 <'' < t. < 2T, Ii 6C[…     

A modified variable time step method for solving ice melting problem

A modified numerical method was used by authors for solving 1D Stefan problem. In this paper a modified method is proposed with difference formulae and different methods of calculating the variable time step, which are deduced from Taylor series expansions of different conditions at the boundary. Also an extrapolation formula for the solution at the first point at the right of the computational domain is proposed. The numerical results are compared with those obtained from other methods.     

An iterative method for solving an inverse problem for a first-order nonlinear partial differential equation with estimates of guaranteed accuracy and the number of steps

For a partial differential equation simulating population dynamics, the inverse problem of determining its nonlinear right-hand side from an additional boundary condition is studied. This inverse problem is reduced to a functional equation, for which the existence and uniqueness of a solution is proven. An iterative method for solving this inverse problem is proposed. The accuracy of the method is estimated, and restrictions on the number of steps are obtained.     

一类含有p-拉普拉斯算子的二阶边值问题的单调迭代正解（英文）

研究了一类含有p-拉普拉斯算子的微分方程积分边值问题.运用迭代技巧,给出了这一类边值问题的单调正解,值得感兴趣的是微分方程中的非线性项含有一阶导数.     

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…     

热方程非线性边界条件问题的有限差分方法

This article is concerned with the numerical solution to a parabolic equation with a kind of nonlinear boundary conditions. A difference scheme is constructed by the method of reduction of order on uniform mesh to solve the problem. It is proved that the difference scheme is uniquely solvable and uncon-ditionaUy convergent with the convergence order 2 in both space and time in an energy norm. An effective iterative algorithm is given and a numerical example is presented to demonstrate the theoretical results.     

A FINITE DIFFERENCE METHOD FOR THE HEAT EQUATION WITH A NONLINEAR BOUNDARY CONDITION

This article is concerned with the numerical solution to a parabolic equation with a kind of nonlinear boundary conditions. A difference scheme is constructed by the method of reduction of order on uniform mesh to solve the problem. It is proved that the difference scheme is uniquely solvable and uncon-ditionaUy convergent with the convergence order 2 in both space and time in an energy norm. An effective iterative algorithm is given and a numerical example is presented to demonstrate the theoretical results.