Stabilized nodal‐based finite element methods for the magnetic induction problem

We consider the time‐dependent magnetic induction model as a step towards the resistive magnetohydrodynamics model in incompressible media. Conforming nodal‐based finite element approximations of the induction model with inf‐sup stable finite elements for the magnetic field and the magnetic pseudo‐pressure are investigated. Based on a residual‐based stabilization technique proposed by Badia and Codina, SIAM J. Numer. Anal. 50 (2012), pp. 398–417, we consider a stabilized nodal‐based finite element method for the numerical solution. Error estimates are given for the semi‐discrete model in space. Finally, we present some examples, for example, for the magnetic flux expulsion problem, Shercliff's test case and singular solutions of the Maxwell problem. Copyright © 2015 John Wiley & Sons, Ltd.     

有限偏差函数与调和函数

分别借助解析函数与调和函数两类函数的Dirichlet积分,利用相关文献给定边界值的拟共形映射极值伸缩商的估计方法,通过有限偏差函数和拟共形映射的关系估计了具有给定边界值的有限偏差函数的极值伸缩商.得到了解析函数的Dirichlet积分在有限偏差函数下具有拟不变性,同时给出有限偏差函数极值伸缩商的下界估计.     

分数阶模糊时滞神经网络模型解的存在唯一性和有限时间稳定性

本文介绍了一类分数阶模糊时滞神经网络模型.利用压缩映射原理,讨论了带时滞的分数阶神经网络模型解的存在性和唯一性,并根据Gronwall不等式结合分数阶微分方程的性质,证明了分数阶神经网络模型平衡点的有限时间稳定性,给出了有限时间稳定性的判断准则.最后,给出数值仿真说明了理论结果的正确性.     

一道Lebesgue积分题目的注记

设E为n维欧氏空间Rn的可测子集,m E+∞,f(x)为E上的非负可测函数,并记Ek=E{x|k≤f(x)k+1}(k=0,1,2,…),利用Lebesgue积分的性质,通过构造反例,指出"级数∑k km Ek收敛"并不是f(x)在E上Lebesgue可积的充分条件.进一步,通过增加条件"f在E上a.e.有限",得到相应的充分必要条件.     

张量积型二元数值积分研究

基于张量积型的二元牛顿插值多项式,构造了矩形网格上的二元求积公式,其对一些二元多项式精确成立,数值算例说明了求积公式的可行性.     

Sobolev方程一个新的H~1-Galerkin混合有限元分析

研究了Sobolev方程的H~1-Galerkin混合有限元方法.利用不完全双二次元Q_2~-和一阶BDFM元,建立了一个新的混合元模式,通过Bramble-Hilbert引理,证明了单元对应的插值算子具有的高精度结果.进一步,对于半离散和向后欧拉全离散格式,分别导出了原始变量u在H~1-模和中间变量p在H(div)-模意义下的超逼近性质.     

一类非线性四阶双曲方程扩展的混合元方法的超收敛分析

对一类非线性四阶双曲方程利用双线性元Q_(1)及Nedelec's元建立一个扩展的协调混合元逼近格式.首先证明了逼近解的存在唯一性.其次,基于上述两个单元的高精度结果,给出了插值和投影之间的误差估计,再利用对时间t的导数转移技巧和插值后处理技术,在半离散和全离散格式下分别导出了原始变量u和中间变量v=-△u在H~1模及中间变量q=▽u,σ=-▽(△u)在(L~2)~2模意义下单独利用插值和投影所无法得到的具有O(h~2)和O(h~2+τ~2)阶的超收敛结果.最后通过数值算例,表明逼近格式是行之有效的.这里,h和τ分别表示空间剖分参数及时间步长.     

Sobolev方程的连续时空有限元方法

本文研究Sobolev方程的连续时空有限元方法.首先建立Sobolev方程的连续时空有限元格式,然后证明了解的存在唯一性和稳定性并给出连续时空有限元解各种范数下的误差估计.最后给出数值算例来验证理论分析的正确性,并进一步说明本文所建立的格式关于时间可以得到比传统有限元方法更高的精度.     

A response to Tisdell on ‘Critical perspectives of the “new” difference equation solution method’ of Rivera-Figueroa and Rivera-Rebolledo

ABSTRACT

Recently, Christopher C. Tisdell questions some claims and results that appear in our article [Rivera-Figueroa A, Rivera-Rebolledo JM. A new method to solve the second-order linear difference equations with constant coefficients. Int J Math Educ Sci Technol. 2016;47(4):636–649.] regarding the novelty and simplicity of a method to solve second-order linear difference equations. He argues that our method was not new because the method already appeared in the mathematical literature, neither was simpler than existing ones. After carefully analysing and comparing the references quoted by Tisdell, we confirm that our method differs from that quoted by Tisdell. Moreover, our approach is simpler because it avoids the uniqueness theorem and the method of variation of parameters, in this context, it offers some clear didactic and epistemological advantages.     

两个Fitting类的例子

给出了两个特殊的群类,即R={H|C_H(F(H))≤F(H)}和S={G|1≠K⊿G~((∞)),G~((∞))=KC_(G(∞))(K)},证明了他们都是Fitting类,这两个Fitting类都和单群理论有密切联系.各自都包含了一些很值得注意的群,对每个Fitting类给出了群G属于该Fitting类的一个充要条件.