A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

doi:10.1007/s42967-022-00206-0

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (4): 1406-1421.doi: 10.1007/s42967-022-00206-0

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A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

Hongfang Bai¹, Ieng Tak Leong²

1 Department of Mathematics, College of Medical Information Engineering, Guangdong Pharmaceutical University, Guangzhou 510006, Guangdong, China;
2 Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, China

收稿日期:2022-01-19 修回日期:2022-06-08 发布日期:2023-12-16
通讯作者: Hongfang Bai,E-mail:byebyenever@163.com;Ieng Tak Leong,E-mail:itleong@um.edu.mo E-mail:byebyenever@163.com;itleong@um.edu.mo

A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

Hongfang Bai¹, Ieng Tak Leong²

1 Department of Mathematics, College of Medical Information Engineering, Guangdong Pharmaceutical University, Guangzhou 510006, Guangdong, China;
2 Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, China

Received:2022-01-19 Revised:2022-06-08 Published:2023-12-16
Contact: Hongfang Bai,E-mail:byebyenever@163.com;Ieng Tak Leong,E-mail:itleong@um.edu.mo E-mail:byebyenever@163.com;itleong@um.edu.mo

摘要/Abstract

摘要： 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) W₀⁴[0, 1] and W¹[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.

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)

Abstract: 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) W₀⁴[0, 1] and W¹[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.

Key words: Weak pre-orthogonal adaptive Fourier decomposition(W-POAFD), Weak maximal selection principle, Fractional boundary value problems(FBVPs), Reproducing kernel Hilbert space(RKHS)

中图分类号:

Hongfang Bai, Ieng Tak Leong. A Sparse Kernel Approximate Method for Fractional Boundary Value Problems[J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1406-1421.

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A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

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