边值问题的概率数值方法

该文运用概率理论研究了一种有界和无界区域上边值问题的数值方法.其主要思想如下: 先写出所求边值问题解的随机表达方式, 再构造一个辅助球, 并且通过区域边界上的剖分将问题离散化,最后利用漂移布朗族的强马尔可夫性和它的球面首中时、首中位置的分布, 获得离散问题的解.     

Recurrence Method for Solving Singularly Perturbed Boundary Value Problems

We base on Taylor series expansions to construct the numerical method for solving singularly perturbed boundary value problems. We use the trapezoid method to approximate the integrals and obtain three‐term recurrence relationship. The efficiency of the proposed method is demonstrated by test problems. The numerical result is found in a good agreement with exact solution.     

Bernstein Polynomial Collocation Method for Elliptic Boundary Value Problems

We present a summary of recent developments in application of Bernstein polynomials to solution of elliptic boundary value problems with a pseudospectral method. Solution is approximated using Benstein polynomial interpolant defined at points of the extrema of Chebyshev polynomials i.e. the Chebyshev-Gauss-Lobatto (CGL) nodes. This approach brings impovement comparing to the Bernstein interpolation at equidistant nodes we used previously [1]. We show that this approach leads to spectral convergence and accuracy comparable to that of pseudospectral methods with orthogonal polynomials (Chebyshev, Legendre). The algorithm is implemented in open source library bernstein-poly , which is available online. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Linear Rational Pseudospectral Method for Boundary Value Problems

We consider the version of the pseudospectral method for solving boundary value problems which replaces the differential operator with a matrix constructed from the elementary differentiation matrices whose elements are the derivatives of the Lagrange fundamental polynomials at the collocation points. The iterative solution of the resulting system of equations then requires the recurrent application of that differentiation matrix. Since global polynomial interpolation on the interval only gives useful approximants for points which accumulate in the vicinity of the extremities, the matrix is ill-conditioned. To reduce this drawback, we use Kosloff and Tal-Ezer's suggestion to shift the collocation points closer to equidistant by a conformal map. However, instead of applying their change of variable setting, we extend to stationary equations the linear rational collocation method introduced in former work on partial differential equations. Numerically about as efficient, this does not require any new coding if one starts from an efficient program for the polynomial differentiation matrices.This revised version was published online in October 2005 with corrections to the Cover Date.     

关于边值问题的一种新的再生核数值算法

利用一种新的再生核算法讨论了二阶微分方程边值问题的数值解,并证明了解的收敛性.算法中,定义了再生核空间,避开了施密特正交化过程,借助于逆矩阵给出了近似解的表达式.通过数值算例,分析了数值解的逼近效果,并与泰勒级数法进行了比较,说明了方法的实用性和有效性.     

On a Method for Solving Boundary Value Problems of Aerohydrodynamics

Journal of Applied and Industrial Mathematics - Under consideration is the well-known boundary value problem of aerohydrodynamics in which it is required to find the velocity distribution of fluid...     

A Power Series Method for Treating Mixed Boundary Value Problems

A technique for treating certain mixed boundary value problemsis presented, and illustrated by its application to a probleminvolving short surface waves in infinitely long canals. Twocases are treated, namely canals of segmental and triangularcrosssections. The method consists of expanding a power series in a workingplane (a semi-circle), and results, in the examples given, inan infinite set of equations, although satisfactory numericalvalues may be obtained by truncation.     

Boundary Value Problems for Odes

We use the method of quasilinearization to boundary value problems of ordinary differential equations showing that the corresponding monotone iterations converge to the unique solution of our problem and this convergence is quadratic.     

Monotone Iterative Method for Set-valued Quasilinear Elliptic Boundary Value Problems

This paper deals with the solutions of the following differential inclusion problem：     

一种解非线性三阶多点边值问题的数值算法

主要研究了一类带有多点边值条件的非线性三阶微分方程的求解方法.利用迭代技巧和再生核(RKM)理论相结合来求解此类问题,同时给出了一些算例来说明方法的有效性.     

非线性分数次边值问题改进的ADM解法(英文)

We use the modified Adomian decomposition method（ADM） for solving the nonlinear fractional boundary value problem {D（^α₀） + u（x） = f（x, u（x））, 0 < x < 1, 3 < α≤ 4 u（0） = α₀ , u’’ （0） = α₂ u（1） = β₀ , u’’（1） = β₂} （1） where D（₀^α）+u is Caputo fractional derivative and α₀,α₂,β₀,β₂ is not zero at all,and f:[0,1]×R→ R is continuous.The calculated numerical results show reliability and efficiency of the algorithm given.The numerical procedure is tested on linear and nonlinear problems.     

Monotone Iterative Method for Set valued Quasilinear Elliptic Boundary Value Problems

§ 1.ＰｒｏｂｌｅｍａｎｄＡｓｓｕｍｐｔｉｏｎｓ　Ｔｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｓｏｌｕｔｉｏｎｓｏｆｔｈｅｆｏｌｌｏｗｉｎｇｄｉｆｆｅｒｅｎｔｉａｌｉｎｃｌｕｓｉｏｎｐｒｏｂｌｅｍ :Ａｕ∈ｆ(ｘ,ｕ) ,ｘ∈Ω ;ｕ=0 ,ｘ∈ Ω ,(1 )ｗｈｅｒｅＡｕ(ｘ) =-∑Ｎｉ=1Ｄｉ[ａｉ(ｘ ,Ｄｕ(ｘ) ) ] ,Ω ＲＮｉｓａｂｏｕｎｄｅｄｄｏｍａｉｎｗｉｔｈｐｉｅｃｅｗｉｓｅＬｉｐｓｃｈｉｔｚｂｏｕｎｄａｒｙ Ω ,Ｄｕ =(Ｄ1ｕ ,Ｄ2 ｕ ,… ,ＤＮｕ) ,Ｄｉｕ = ｕ ｘｉ,ｉ=1 ,2 ,… ,Ｎ ,ａｎｄｆ:Ω×Ｒ→ 2 Ｒｉｓａ…     

四阶混合非齐次边值问题的Petrov-Galerkin谱方法

研究了矩形区域上的四阶混合非齐次边值问题的Petrov-Galerkin谱方法.利用广义Jacobi多项式对模型问题的精确解进行数值展开,并给出了数值例子.数值结果表明所提算法的有效性和高精度.     

求线性边值问题数值解的Chebyshevτ—方法

提出了一种求解二维线性边值问题的新的τ＿方法·对该问题进行了理论分析和数值求解·结果表明了本文方法的优点和有效性     

On the Gradient Method for Some Nonlinear Elliptic Boundary Value Problems

The infinite-dimensional gradient method is applied to the iterative solution of quasilinear elliptic boundary value problems. Earlier results on uniformly monotone problems are extended to a general case within the scope of Hilbert space well-posedness. Linear convergence is proved in Sobolev norm. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Lie-Group Shooting Method for Solving Multi-dimensional Nonlinear Boundary Value Problems

This paper presents a Lie-group shooting method for the numerical solutions of multi-dimensional nonlinear boundary-value problems, which may exhibit multiple solutions. The Lie-group shooting method is a powerful technique to search unknown initial conditions through a single parameter, which is determined by matching the multiple targets through a minimum of an appropriately defined measure of the mis-matching error to target equations. Several numerical examples are examined to show that the novel approach is highly efficient and accurate. The number of solutions can be identified in advance, and all possible solutions can be numerically integrated by using the fourth-order Runge–Kutta method. We also apply the Lie-group shooting method to a numerical solution of an optimal control problem of the Duffing oscillator.     

Fractional Boundary Value Problems with Integral Boundary Conditions via Topological Degree Method

This paper examines the existence and uniqueness of solutions for the fractional boundary value problems with integral boundary conditions. Banach’s contraction mapping principle and Schaefer’s fixed point theorem have been used besides topological technique of approximate solutions. An example is propounded to uphold our results.     

Symmetric Boundary Value Methods for Second Order Initial and Boundary Value Problems

We introduce symmetric Boundary Value Methods for the solution of second order initial and boundary value problems (in particular Hamiltonian problems). We study the conditioning of the methods and link it to the boundary loci of the roots of the associated characteristic polynomial. Some numerical tests are provided to assess their reliability. Dedicated to the memory of Professor Aldo Cossu