ALTERNATING DIRECTION IMPLICIT OSC SCHEME FOR THE TWO-DIMENSIONAL FRACTIONAL EVOLUTION EQUATION WITH A WEAKLY SINGULAR KERNEL

数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (6): 1689-1711.

ALTERNATING DIRECTION IMPLICIT OSC SCHEME FOR THE TWO-DIMENSIONAL FRACTIONAL EVOLUTION EQUATION WITH A WEAKLY SINGULAR KERNEL

张海湘^1,2, 杨雪花^1,2, 徐大³

1 Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, China;
2 School of Science, Hunan University of Technology, Zhuzhou 412007, China;
3 Department of Mathematics, Hunan Normal University, Changsha 410081, China

收稿日期:2017-08-31 修回日期:2018-03-06 出版日期:2018-12-25 发布日期:2018-12-28
通讯作者: Xuehua YANG E-mail:hunanshidayang@163.com
作者简介:Haixiang ZHANG,E-mail:hassenzhang@163.com;Da XU,E-mail:daxu@hunnu.edu.cn
基金资助:
This research was supported by National Nature Science Foundation of China (11701168, 11601144 and 11626096), Hunan Provincial Natural Science Foundation of China (2018JJ3108, 2018JJ3109 and 2018JJ4062), Scientific Research Fund of Hunan Provincial Education Department (16K026 and YB2016B033), and China Postdoctoral Science Foundation (2018M631403).

ALTERNATING DIRECTION IMPLICIT OSC SCHEME FOR THE TWO-DIMENSIONAL FRACTIONAL EVOLUTION EQUATION WITH A WEAKLY SINGULAR KERNEL

Haixiang ZHANG^1,2, Xuehua YANG^1,2, Da XU³

1 Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, China;
2 School of Science, Hunan University of Technology, Zhuzhou 412007, China;
3 Department of Mathematics, Hunan Normal University, Changsha 410081, China

Received:2017-08-31 Revised:2018-03-06 Online:2018-12-25 Published:2018-12-28
Contact: Xuehua YANG E-mail:hunanshidayang@163.com
Supported by:
This research was supported by National Nature Science Foundation of China (11701168, 11601144 and 11626096), Hunan Provincial Natural Science Foundation of China (2018JJ3108, 2018JJ3109 and 2018JJ4062), Scientific Research Fund of Hunan Provincial Education Department (16K026 and YB2016B033), and China Postdoctoral Science Foundation (2018M631403).

摘要/Abstract

摘要： In this paper, a new kind of alternating direction implicit (ADI) Crank-Nicolson-type orthogonal spline collocation (OSC) method is formulated for the two-dimensional fractional evolution equation with a weakly singular kernel arising in the theory of linear viscoelasticity. The novel OSC method is used for the spatial discretization, and ADI Crank-Nicolson-type method combined with the second order fractional quadrature rule are considered for the temporal component. The stability of proposed scheme is rigourously established, and nearly optimal order error estimate is also derived. Numerical experiments are conducted to support the predicted convergence rates and also exhibit expected super-convergence phenomena.

关键词: fractional equation, orthogonal spline collocation scheme, alternating direction implicit, stability, convergence

Abstract: In this paper, a new kind of alternating direction implicit (ADI) Crank-Nicolson-type orthogonal spline collocation (OSC) method is formulated for the two-dimensional fractional evolution equation with a weakly singular kernel arising in the theory of linear viscoelasticity. The novel OSC method is used for the spatial discretization, and ADI Crank-Nicolson-type method combined with the second order fractional quadrature rule are considered for the temporal component. The stability of proposed scheme is rigourously established, and nearly optimal order error estimate is also derived. Numerical experiments are conducted to support the predicted convergence rates and also exhibit expected super-convergence phenomena.

Key words: fractional equation, orthogonal spline collocation scheme, alternating direction implicit, stability, convergence

张海湘, 杨雪花, 徐大. ALTERNATING DIRECTION IMPLICIT OSC SCHEME FOR THE TWO-DIMENSIONAL FRACTIONAL EVOLUTION EQUATION WITH A WEAKLY SINGULAR KERNEL[J]. 数学物理学报(英文版), 2018, 38(6): 1689-1711.

Haixiang ZHANG, Xuehua YANG, Da XU. ALTERNATING DIRECTION IMPLICIT OSC SCHEME FOR THE TWO-DIMENSIONAL FRACTIONAL EVOLUTION EQUATION WITH A WEAKLY SINGULAR KERNEL[J]. Acta mathematica scientia,Series B, 2018, 38(6): 1689-1711.

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ALTERNATING DIRECTION IMPLICIT OSC SCHEME FOR THE TWO-DIMENSIONAL FRACTIONAL EVOLUTION EQUATION WITH A WEAKLY SINGULAR KERNEL

ALTERNATING DIRECTION IMPLICIT OSC SCHEME FOR THE TWO-DIMENSIONAL FRACTIONAL EVOLUTION EQUATION WITH A WEAKLY SINGULAR KERNEL

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