A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

Hongfang Bai; Ieng Tak Leong

doi:10.1007/s42967-022-00206-0

Communications on Applied Mathematics and Computation ›› 2022, 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 Bai ¹^,a
, Ieng Tak Leong ²

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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) $W^{4}_0[0,1]$ 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.

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Hongfang Bai, Ieng Tak Leong. A Sparse Kernel Approximate Method for Fractional Boundary Value Problems. Communications on Applied Mathematics and Computation, 2022, 5(4): 1406-1421 DOI:10.1007/s42967-022-00206-0