南京信息工程大学校庆科技活动月—特邀美国伍斯特理工学院郝朝鹏博士来我校作学术报告

发布单位:数学与统计学院创建者:蔡惠华发布时间:2019-04-23浏览量:976

报告题目:Regularity and spectral method for two-dimensional fractional diffusion equations with fractional Laplacian

报告地点:尚贤楼706会议室

报告时间:2019年4月25日(周四)下午14: 00--15: 00

主持人:王廷春副教授

 

报告人简介:郝朝鹏,博士,毕业于东南大学数学学院,现在美国伍斯特理工学院攻读第二博士学位;先后多次访问普渡大学、伍斯特理工学院。主要研究领域为偏微分方程数值分析、分数阶微分方程的高精度与快速数值方法、随机常微分方程和随机偏微分方程的蒙特卡洛数值模拟及数值理论研究、计算流体力学等。在SIAM J. Sci. Comput.、J. Comput. Phys.J. Sci. Comput等高水平SCI期刊发表学术论文12篇。

 

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                                                                                                数学与统计学院

2019年4月23日

报告简介:Fractional Laplacian, one of nonlocal operators, has attracted increasing attention recently since they are found to be more accurate than local operators to model real world phenomenon with long range interactions in many fields, e.g., fluid dynamics, finance in phase transitions, material science etc. However, the difficulty lies in: how to efficiently solve partial differential equations rested on nonlocal operators and justify the convergence order of the algorithms when they are applied to these models. We investigate a spectral Galerkin method for the two-dimensional fractional diffusion-reaction equations on a disk. We first prove sharp regularity estimates of solutions in the weighted Sobolev space. With the regularity estimates, then we obtain optimal convergence orders of the spectral Galerkin method for the fractional diffusion-reaction equations.  Numerical results are presented to verify the theoretical analysis. For the equations defined on general domains, we propose a very cheap method, the fictitious domain method, to solve fractional diffusion equations efficiently. We first consider the equation on a disk and then embed the irregular domain into the disk. In this way we can transform the diffusion equations into variable coefficient, reaction type diffusion equations. Then we use the established spectral method for the equation on a disk to solve the transformed ones. Numerical results on various piece-wise smooth domains will be presented.  And the convergence order and numerical accuracy will be discussed.