A simple yet effective numerical method using orthogonal hybrid functions consisting of piecewise constant orthogonal sample-and-hold functions and piecewise linear orthogonal triangular functions is proposed to solve numerically fractional order non-stiff and stiff differential-algebraic equations. The complementary generalized one-shot operational matrices, which are the foundation for the developed numerical method, are derived to estimate the Riemann-Liouville fractional order integral in the new orthogonal hybrid function domain. It is theoretically and numerically shown that the numerical method converges the approximate solutions to the exact solution in the limit of step size tends to zero. Numerical examples are solved using the proposed method and the obtained results are compared with the results of some popular semi-analytical techniques used for solving fractional order differential-algebraic equations in the literature. Our results are in good accordance with the results of those semi-analytical methods in case of non-stiff problems and our method provides valid approximate solution to stiff problem (fractional order version of Chemical Akzo Nobel problem) which those semi-analytical methods fails to solve.
Instituto de Matemáticas and Departamento de Matematica Aplicada, Universidad de Granada（格拉纳达大学数学研究所与应用数学系）；School of Mathematics, Georgia Institute of Technology（佐治亚理工学院数学学院）
A Stiff Order Condition Theory for Runge-Kutta Methods Applied to Semilinear ODEs
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；New Jersey Institute of Technology（新泽西理工学院）；Indian Institute of Technology Kanpur（坎普尔印度理工学院）；Temple University（天普大学）
University of Houston（休斯顿大学）；Moscow State University（莫斯科国立大学）；RWTH-Aachen University（亚琛工业大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所，国家数学与交叉研究中心）
Nonconforming finite element spaces for $HΛ^k$ in $\mathbb{R}^n$
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；University of Chinese Academy of Sciences（中国科学院大学）
A cell-centred finite volume approximation for second order partial derivative operators with full matrix on unstructured meshes in any space dimension
Discrete Modified Projection Methods for Urysohn Integral Equations with Green's Function Type Kernels
在本文中，我们考虑离散版本的修正投影方法求解Urysohn积分方程的核的类型的格林函数。对于$r \geq 0,$，选择关于均匀划分的次数为$\leq r $的分段多项式的空间作为近似空间。我们定义了一个离散的正交投影到这个空间，并取代Urysohn积分算子的Nyström近似。我们得到的离散版本的收敛顺序表明选择数值积分保持连续修正投影法的收敛顺序。最后给出了具体算例的数值结果。
