THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS

doi:10.1007/s10473-024-0211-2

Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (2): 567-582.doi: 10.1007/s10473-024-0211-2

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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).

Abstract

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

CLC Number:

65M80

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.

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References

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THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS

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