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THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS
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作者 曲伟 钱涛 +1 位作者 梁应德 李澎涛 《Acta Mathematica Scientia》 SCIE CSCD 2024年第2期567-582,共16页
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 an... 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. 展开更多
关键词 reproducing kernel Hilbert space DICTIONARY sparse representation approximation to the identity fractional heat equations
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A Sparse Kernel Approximate Method for Fractional Boundary Value Problems
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作者 Hongfang Bai ieng tak leong 《Communications on Applied Mathematics and Computation》 EI 2023年第4期1406-1421,共16页
In this paper,the weak pre-orthogonal adaptive Fourier decomposition(W-POAFD)method is applied to solve fractional boundary value problems(FBVPs)in the reproducing kernel Hilbert spaces(RKHSs)W_(0)^(4)[0,1] and W^(1)[... In this paper,the weak pre-orthogonal adaptive Fourier decomposition(W-POAFD)method is applied to solve fractional boundary value problems(FBVPs)in the reproducing kernel Hilbert spaces(RKHSs)W_(0)^(4)[0,1] and W^(1)[0,1].The process of the W-POAFD is as follows:(i)choose a dictionary and implement the pre-orthogonalization to all the dictionary elements;(ii)select points in[0,1]by the weak maximal selection principle to determine the corresponding orthonormalized dictionary elements iteratively;(iii)express the analytical solution as a linear combination of these determined dictionary elements.Convergence properties of numerical solutions are also discussed.The numerical experiments are carried out to illustrate the accuracy and efficiency of W-POAFD for solving FBVPs. 展开更多
关键词 Weak pre-orthogonal adaptive Fourier decomposition(W-POAFD) Weak maximal selection principle Fractional boundary value problems(FBVPs) Reproducing kernel Hilbert space(RKHS)
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