分数阶发展方程时间步长格式的离散极大正则性
Discrete maximal regularity of time-stepping schemes for fractional evolution equations.
作者信息
Jin Bangti, Li Buyang, Zhou Zhi
机构信息
1Department of Computer Science, University College London, Gower Street, London, WC1E 6BT UK.
2Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong.
出版信息
Numer Math (Heidelb). 2018;138(1):101-131. doi: 10.1007/s00211-017-0904-8. Epub 2017 Jul 22.
In this work, we establish the maximal [Formula: see text]-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order [Formula: see text], [Formula: see text], in time. These schemes include convolution quadratures generated by backward Euler method and second-order backward difference formula, the L1 scheme, explicit Euler method and a fractional variant of the Crank-Nicolson method. The main tools for the analysis include operator-valued Fourier multiplier theorem due to Weis (Math Ann 319:735-758, 2001. doi:10.1007/PL00004457) and its discrete analogue due to Blunck (Stud Math 146:157-176, 2001. doi:10.4064/sm146-2-3). These results generalize the corresponding results for parabolic problems.
在这项工作中，我们为一个分数阶演化模型的几种时间步长格式建立了最大([公式：见正文])-正则性，该模型在时间上涉及阶数为([公式：见正文])，([公式：见正文])的分数阶导数。这些格式包括由向后欧拉方法和二阶向后差分公式生成的卷积求积、L1格式、显式欧拉方法以及克兰克 - 尼科尔森方法的分数阶变体。分析的主要工具包括魏斯（《数学年刊》319:735 - 758，2001。doi:10.1007/PL00004457）的算子值傅里叶乘子定理及其由布伦克（《数学研究》146:157 - 176，2001。doi:10.4064/sm146 - 2 - 3）给出的离散类似物。这些结果推广了抛物型问题的相应结果。
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