THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS

doi:10.1007/s10473-024-0211-2

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (2): 567-582.doi: 10.1007/s10473-024-0211-2

THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS

Wei QU¹, Tao QIAN², Ieng Tak LEONG³, Pengtao LI^4,*

1. College of Sciences, China Jiliang University, Hangzhou 310018, China;
2. Macau Center for Mathematical Science, Macau University of Science and Technology, Macau 999078, China;
3. Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau 999078, China;
4. School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China

收稿日期:2022-11-20 修回日期:2023-01-08 出版日期:2024-04-25 发布日期:2024-04-16
通讯作者: *Pengtao LI, E-mail: ptli@qdu.edu.cn
作者简介:Wei QU, E-mail: quwei2math@qq.com; Tao QIAN, E-mail: tqian@must.edu.mo; Ieng Tak LEONG, E-mail: itleong@um.edu.mo
基金资助:
Science and Technology Development Fund of Macau SAR (FDCT0128/2022/A, 0020/2023/RIB1, 0111/2023/AFJ, 005/2022/ALC), the Shandong Natural Science Foundation of China (ZR2020MA004), the National Natural Science Foundation of China (12071272), the MYRG 2018-00168-FST, and Zhejiang Provincial Natural Science Foundation of China (LQ23A010014).

THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS

Wei QU¹, Tao QIAN², Ieng Tak LEONG³, Pengtao LI^4,*

1. College of Sciences, China Jiliang University, Hangzhou 310018, China;
2. Macau Center for Mathematical Science, Macau University of Science and Technology, Macau 999078, China;
3. Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau 999078, China;
4. School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China

Received:2022-11-20 Revised:2023-01-08 Online:2024-04-25 Published:2024-04-16
Contact: *Pengtao LI, E-mail: ptli@qdu.edu.cn
About author:Wei QU, E-mail: quwei2math@qq.com; Tao QIAN, E-mail: tqian@must.edu.mo; Ieng Tak LEONG, E-mail: itleong@um.edu.mo
Supported by:
Science and Technology Development Fund of Macau SAR (FDCT0128/2022/A, 0020/2023/RIB1, 0111/2023/AFJ, 005/2022/ALC), the Shandong Natural Science Foundation of China (ZR2020MA004), the National Natural Science Foundation of China (12071272), the MYRG 2018-00168-FST, and Zhejiang Provincial Natural Science Foundation of China (LQ23A010014).

摘要/Abstract

摘要： This study introduces a pre-orthogonal adaptive Fourier decomposition (POAFD) to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre (generalized Poisson equation). As a first step, the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence, and, as a second step, makes use of the semigroup or the reproducing kernel property of each of the expanding entries. Experiments show the effectiveness and efficiency of the proposed series solutions.

关键词: reproducing kernel Hilbert space, dictionary, sparse representation, approximation to the identity, fractional heat equations

Abstract: This study introduces a pre-orthogonal adaptive Fourier decomposition (POAFD) to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre (generalized Poisson equation). As a first step, the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence, and, as a second step, makes use of the semigroup or the reproducing kernel property of each of the expanding entries. Experiments show the effectiveness and efficiency of the proposed series solutions.

Key words: reproducing kernel Hilbert space, dictionary, sparse representation, approximation to the identity, fractional heat equations

中图分类号:

65M80

Wei QU, Tao QIAN, Ieng Tak LEONG, Pengtao LI. THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS[J]. 数学物理学报(英文版), 2024, 44(2): 567-582.

Wei QU, Tao QIAN, Ieng Tak LEONG, Pengtao LI. THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS[J]. Acta mathematica scientia,Series B, 2024, 44(2): 567-582.

参考文献

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[12] Qian T. Reproducing kernel sparse representations in relation to operator equations. Complex Anal Oper Theory, 2020, 14: 1-15
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THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS

THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS

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