Energy Methods for Problems with Nonhomogeneous Boundary Conditions

This paper employs the weighted energy method to derive estimates for the dynamic behavior of solutions to boundary and initial boundary value problems with nonhomogeneous boundary conditions. In particular, the method is applied to the heat and Laplace equations in a bounded or unbounded region. Extensions to related equations are also studied. Similar estimates but for the spatial behavior is obtained for the heat equation and the backward in time heat equation. Results for blow-up in finite time of solutions to certain nonlinear equations are generalized to include nonhomogeneous boundary conditions, while solutions that vanish on part of the boundary are briefly discussed in the final section.     

Polynomial and rational wave solutions of Kudryashov-Sinelshchikov equation and numerical simulations for its dynamic motions

Polynomial and rational wave solutions of Kudryashov-Sinelshchikov equation and numerical simulations for its dynamic motions are investigated. Conservation flows of the dynamic motion are obtained utilizing multiplier approach. Using the unified method, a collection of exact solitary and soliton solutions of Kudryashov-Sinelshchikov equation is presented. Collocation finite element method based on quintic B-spline functions is implemented to the equation to evidence the accuracy of the proposed method by test problems. Stability analysis of the numerical scheme is studied by employing von Neumann theory. The obtained analytical and numerical results are in good agreement.     

A Tau Method with Perturbed Boundary Conditions for Certain Ordinary Differential Equations

Ortiz recursive formulation of the Lanczos Tau method (TM) is a powerful and efficient technique for producing polynomial approximations for initial or boundary value problems. The method consists in obtaining a polynomial which satisfies (i) a perturbed version of the given differential equation, and (ii) the imposed supplementary conditions exactly. This paper introduces a new form of the TM, (denoted by PTM), for a restricted class of differential equations, in which the differential equations as well as the supplementary conditions are perturbed simultaneously. PTM is compared to the classical TM from the point of view of their errors: it is found that the PTM error is smaller and more oscillatory than that of the TM; we further find that approximations nearly as accurate as minimax polynomial approximations can be constructed by means of the PTM. Detailed formulae are derived for the polynomial approximations in TM and PTM, based on Canonical Polynomials. Moreover, various limiting properties of Tau coefficients are established and it is shown that the perturbation in PTM behaves asymptotically proportional to a Chebyshev polynomial.     

A numerical technique for solution of the MRLW equation using quartic B-splines

Numerical scheme based on quartic B-spline collocation method is designed for the numerical solution of modified regularized long wave (MRLW) equation. Unconditional stability is proved using Von-Neumann approach. Performance of the method is checked through numerical examples. Using error norms L₂ and L_∞ and conservative properties of mass, momentum and energy, accuracy and efficiency of the new method is established through comparison with the existing techniques.     

Stable Boundary Conditions and Discretization for $P_N$ Equations

A solution to the linear Boltzmann equation satisfies an energy bound, which reflects a natural fact: The energy of particles in a finite volume is bounded in time by the energy of particles initially occupying the volume augmented by the energy transported into the volume by particles entering the volume over time. In this paper, we present boundary conditions (BCs) for the spherical harmonic $(P_N)$ approximation, which ensure that this fundamental energy bound is satisfied by the $P_N$ approximation. Our BCs are compatible with the characteristic waves of $P_N$ equations and determine the incoming waves uniquely. Both, energy bound and compatibility, are shown on abstract formulations of $P_N$ equations and BCs to isolate the necessary structures and properties. The BCs are derived from a Marshak type formulation of BC and base on a non-classical even/odd-classification of spherical harmonic functions and a stabilization step, which is similar to the truncation of the series expansion in the $P_N$ method. We show that summation by parts (SBP) finite difference on staggered grids in space and the method of simultaneous approximation terms (SAT) allows to maintain the energy bound also on the semi-discrete level.     

用线方法求解带混合边值条件的抛物方程

研制了分别用显式Euler法、隐式Euler法、Crank-Nicolson格式(梯形方法)求解带第一、第二及混合边值条件的抛物问题的应用软件,通过求解若干抛物问题对该软件作了测试,获得了预期的数值结果,讨论了时间和空间步长的变化对格式计算结果的影响,得到了三种方法的稳定性、收敛精度和计算量.     

A Modified Low-Rank Smith Method for Large-Scale Lyapunov Equations

In this note we present a modified cyclic low-rank Smith method to compute low-rank approximations to solutions of Lyapunov equations arising from large-scale dynamical systems. Unlike the original cyclic low-rank Smith method introduced by Penzl in [20], the number of columns required by the modified method in the approximate solution does not necessarily increase at each step and is usually much lower than in the original cyclic low-rank Smith method. The modified method never requires more columns than the original one. Upper bounds are established for the errors of the low-rank approximate solutions and also for the errors in the resulting approximate Hankel singular values. Numerical results are given to verify the efficiency and accuracy of the new algorithm.     

Bifurcations and exact solutions for a nonlinear surface wind waves equation

By using the method of dynamical systems, for the nonlinear surface wind waves equation, which is given by Manna, we study its dynamical behavior to determine all exact explicit traveling wave solutions. To guarantee the existence of the aforementioned solutions, all parameter conditions are determined. Our procedure shows that the nonlinear surface wind waves equation has no peakon solution. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical Approximation and Error Analysis for the Timoshenko Beam Equations with Boundary Feedback

In this paper,the numerical approximation of a Timoshenko beam with bound- ary feedback is considered.We derived a linearized three-level difference scheme on uniform meshes by the method of reduction of order for a Timoshenko beam with boundary feedback.It is proved that the scheme is uniquely solvable,unconditionally stable and second order convergent in L_∞norm by using the discrete energy method. A numerical example is presented to verify the theoretical results.     

A Spline Collocation Method for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels

The piecewise polynomial collocation method is discussed to solve linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. Using special graded grids, global convergence estimates are derived. The error analysis is based on certain regularity properties of the solution of the initial value problem.This revised version was published online in October 2005 with corrections to the Cover Date.     

Stokes问题低阶非协调混合元的改进加罚算法(英文)

通过对由经典加罚算法得到的两个解进行线性组合,研究Stokes方程低阶非协调混合元的改进加罚算法.该方法利用较大的罚参数能得到同使用较小参数的经典加罚方法一样的收敛阶.此外,基于单元的特性和插值后处理技巧,得到一些超收敛结果,从而改进以往的文献结果.     

具有特征参数多项式边界条件的Sturm-Liouville 方程的逆结点问题

逆结点问题是通过特征函数的零点重构算子. 本文主要讨论具有特征参数多项式边界条件的 Sturm-Liouville 方程的逆结点问题. 20世纪50年代以后,人们发现在许多工程领域, Sturm-Liouville 问题的谱参数不仅出现在方程中, 而且也出现在边界条件中,因此带参数边界条件的逆结点问题对数学物理方面的研究有重要意义. 本文讨论区间 $[0,1]$ 上边界条件为参数多项式的 Sturm-Liouville 方程的逆结点问题, 并证明在 $[0,b]$ \big($ b\in \big(\frac{1}{2},1\big]$\big) 上结点的稠密子集可唯一确定 $[0,1]$ 上的势函数和边界条件中多项式的未知系数.     

一类具有Dirichlet边界条件的反应扩散方程的爆破解

针对一类具有Dirichlet边界条件的非线性反应扩散方程的爆破问题,通过构造恰当的辅助函数和利用一阶微分不等式技术,给出了解在有限时刻爆破的一个充分条件,并在一定条件下得到了爆破时刻的上界和下界.     

一类具临界指数和等值面边值条件椭圆型方程解的存在性

本文利用临界点理论给出了RN(N≥3)中有界光滑区域上的拟线性椭圆型方程-△pU=|u|p*-2u a(x)|u|p-2u f(x,u),X∈Ω(P*=Np/(N-p),1相似文献   

A collocation approach for the numerical solution of certain linear retarded and advanced integrodifferential equations with linear functional arguments

This article presents a Taylor collocation method for the approximate solution of high‐order linear Volterra‐Fredholm integrodifferential equations with linear functional arguments. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. Some numerical examples, which consist of initial and boundary conditions, are given to show the properties of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.     

复杂边界旋转Navier-Stokes方程的几何方法及二度并行算法

本文提出了一种求解复杂边界旋转Navier-Stokes方程的微分几何方法及其二度并行算法.此方法可用于求解透平机械内部叶片间流动和飞行器外部绕流等复杂流动问题.假设流动区域可以用一系列光滑曲面■_k,k=1,2,…,K分割为一系列子区域(称作流层),通过应用微分几何的方法,三维N-S算子可以分解为两类算子之和:建立在曲面■_k切空间上"膜算子"和曲面■_k法线方向的"挠曲算子",将挠曲算子应用欧拉中心差商来逼近,由此得到建立在■_k上的"2D-3C"N-S方程.求解2D-3C N-S方程并且反复迭代直到收敛.我们得到"二度并行算法",它是2D-3C N-S方程并行算法与k方向的同时并行.这个算法的优点在于,(1)可以改进由于复杂边界造成的不规则三维网格引起的逼近解的精度;(2)为克服边界层的数值效应,在边界层内可以构造很密的流层,形成三维多尺度的网格,是一个很好的边界层算法;(3)这个方法不同于经典的区域分解算法,这里的每个子区域只需要求解一个"2D-3C"N-S方程,而经典区域分解方法要在每个子区域上求解三维问题.     

A discrete collocation method for Fredholm integro-differential equations with weakly singular kernels

A fully discrete version of a piecewise polynomial collocation method is constructed to solve initial or boundary value problems of linear Fredholm integro-differential equations with weakly singular kernels. Using an integral equation reformulation and special graded grids, optimal global convergence estimates are derived. For special values of parameters an improvement of the convergence rate of elaborated numerical schemes is established. Some of our theoretical results are illustrated by numerical experiments.     

Existence and Exponential Decay of Solutions for a Wave Equation with Integral Nonlocal Boundary Conditions of Memory Type

In this paper, we consider a wave equation with integral nonlocal boundary conditions of memory type. First, we establish two local existence theorems by using Faedo–Galerkin method and standard arguments of density. Next, we give a su?cient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.     

A Two-grid Method with Expanded Mixed Element for Nonlinear Reaction-diffusion Equations

Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O(△t + h k+1 + H 2k+2 d/2 ) (k ≥ 1), where k is the degree of the approximating ...