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.
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
.
DOI: 10.1007/s10473-024-0211-2
[1] Blumenthal R, Getoor R. Some theorems on stable processes. Trans Amer Math Soc, 1960, 95: 263-273
[2] Allen M, Caffarelli L, Vasseur A. A parabolic problem with a fractional time derivative. Arch Ration Mech Anal, 2016, 221: 603-630
[3] Bernardis A, Martín-Reyes F, Stinga P, Torrea J. Maximum principles, extension problem and inversion for nonlocal one-sided equations. J Differ Equ, 2016, 260: 6333-6362
[4] Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Comm Partial Differ Equ, 2007, 32: 1245-1260
[5] Caffarelli L, Salsa S, Silvestre L. Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent Math, 2008, 171: 425-461
[6] Caffarelli L, Stinga P. Fractional elliptic equations, Caccioppoli estimates and regularity. Ann Inst H Poincare Anal Non Linaire, 2016, 33: 767-807
[7] Caffarelli L, Vasseur A.Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann Math, 2010, 171: 1903-1930
[8] Chen Z, Song R. Estimates on Green functions and Poisson kernels for symmetric stable processes. Math Ann, 1998, 312: 465-501
[9] Herrmann R.Fractional Calculus: An Introduction for Physicists. Singapre: World Scientific, 2014
[10] Kwaśnicki M. Ten equivalent definitions of the fractional Laplace operator. Fractional Calculus and Applied Analysis, 2017, 20: 7-51
[11] Li P, Hu R, Zhai Z.Fractional Besov trace/extension type inequalities via the Caffarelli-Silvestre extension. arXiv:2201.00765
[12] Qian T. Reproducing kernel sparse representations in relation to operator equations. Complex Anal Oper Theory, 2020, 14: 1-15
[13] Qu W, Chui C, Deng G, Qian T. Sparse repesentation of approximation to identity. Anal Appl, 2022, 20(4): 815-837
[14] Qian T. Two-dimensional adaptive Fourier decomposition. Math Meth Appl Sci, 2016, 39: 2431-2448
[15] Qian T.Algorithm of adaptive Fourier decomposition. IEEE Trans Signal Proc, 2016, 59: 5899-5906
[16] Ros-Oton X, Serra J. The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary. J Math Pure Appl, 2014, 101: 275-302
[17] Xie L, Zhang X. Heat kernel estimates for critical fractional diffusion operator. Studia Math, 2014, 224: 221-263
