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.     

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.     

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...     

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.     

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     

求解参数识别反问题的同伦正则化方法

提出了求解参数识别反问题的同伦正则化方法,给出了相应的收敛性定理.数值结果表明该方法是一种快速的大范围收敛方法.     

奇摄动非线性边值问题

本文讨论了一类奇摄动非线性边值问题 .利用伸长变量和边界层校正法 ,得到了问题解的形式渐近展开式 .再用微分不等式理论 ,证明了解的一致有效性     

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.     

Shooting Method for Nonlinear Singularly Perturbed Boundary-Value Problems

Asymptotic formulas, as  ɛ→ 0⁺  , are derived for the solutions of the nonlinear differential equation  ɛ u" + Q ( u ) = 0  with boundary conditions   u (-1) = u (1) = 0  or   u '(-1) = u '(1) = 0  . The nonlinear term Q ( u ) behaves like a cubic; it vanishes at   s _-, 0, s ₊  and nowhere else in  [ s _-, s ₊]  , where   s _- < 0 < s ₊  . Furthermore,   Q '( s _±) < 0, Q '(0) > 0  and the integral of Q on the interval [ s _-, s ₊] is zero. Solutions to these boundary-value problems are shown to exhibit internal shock layers, and the error terms in the asymptotic approximations are demonstrated to be exponentially small. Estimates are obtained for the number of internal shocks that a solution can have, and the total numbers of solutions to these problems are also given. All results here are established rigorously in the mathematical sense.     

一个改进的奇异摄动边值问题解的六次样条方法(英文)

基于六次样条插值,文章针对奇异摄动边值问题给出一个改进的六次样条方法.该方法的精度在所有内部结点达到八阶,在边界点达到五阶,通过一致收敛性分析和数值试验表明所给方法是有效的.     

奇摄动非线性边值问题

§ 1 .　Introduction　　AclassofintegraldifferentialequationsDirichletboundaryvalueproblemsforordinarydif ferentialequationandellipticequationarediscussedin [1]and [2 ]respectively .Andin [3]akindofnonlocalproblemsforsingularlyperturbedreactiondiffusionsystemsarestudied.Inthispaper,whatisworthpointingoutisaclassofnonlinearboundaryvalueproblemsdiscussed,applyingthemethodofcompositeexpandandthetheoryofdifferentialinequalities.εy″ =f(x ,y ,Tεy ,ε) y′ +g(x ,y ,Tεy,ε) ,0 相似文献   

二阶非线性微分方程边值问题的同伦分析解

研究了结合变量替换应用同伦分析方法,去求解二阶非线性微分方程的两点边值问题,并得到了逼近解析解的函数级数形式.给出了应用同伦分析方法求解二阶非线性问题的三个实例,显示了同伦分析方法可以比较有效地求解非线性问题.     

非线性波方程的初边值问题

本文利用Leray-Schauder不动点定理证明了非线性波方程utt-[a0 a2(ux)^β]uxx-a1uxxtt=f(x,t,ux,ut,uxt)的初边值问题广义解的存在唯一性。     

一类非线性奇摄动Dirichlet边值问题的匹配解法

利用匹配渐近展开法,讨论一类形如εy″+(xn-k)(y′+ym)=0的非线性奇摄动方程的Dirichlet边值问题,并且通过对参数k的五种不同取值的分类探讨,得到了该问题必有左边界层、右边界层或内部层之一的结论(其中左、右边界层又各分为两种类型).进而给出该问题解的零次渐近展开式,推广并改进了已有的结果.     

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.     

奇摄动非线性系统Robin边值问题

本文研究了非线性系统奇摄动问题：ε2y"－（x，y，y）＝0，0＜x＜1，0＜ε≤1，y（0）－py'（0）＝A，p＞0，y（1）＝B，其中y，f，A，B为n维向量．在相应的假设下，利用代数型边界层函数，证明了该问题存在一个解y（x，ε），并利用微分不等式方法得到了其解的渐近估计．