A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

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.

Funding

Fundo para o Desenvolvimento das Ciências e da Tecnologia(FDCT/0123/2018/A3); Universidade de Macau(MYRG2018-00168-FST)