TY - JOUR
T1 - Finite Difference Method for Inhomogeneous Fractional Dirichlet Problem
AU - Sun , Jing
AU - Deng , Weihua
AU - Nie , Daxin
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 3
SP - 744
EP - 767
PY - 2022
DA - 2022/07
SN - 15
DO - http://doi.org/10.4208/nmtma.OA-2021-0173
UR - https://global-sci.org/intro/article_detail/nmtma/20814.html
KW - One- and two-dimensional integral fractional Laplacian, Lagrange interpolation,
operator splitting, finite difference, the inhomogeneous fractional Dirichlet problem, error estimates.
AB - <p style="text-align: justify;">We make the split of the integral fractional Laplacian as $$(−∆)^su = (−∆)(−∆)^{s−1}u,$$ where $s ∈ (0,\frac{1}{2}) ∪ (\frac{1}{2}, 1).$ Based on this splitting, we respectively discretize the one- and two-dimensional integral fractional Laplacian with the inhomogeneous Dirichlet
boundary condition and give the corresponding truncation errors with the help of the
interpolation estimate. Moreover, the suitable corrections are proposed to guarantee
the convergence in solving the inhomogeneous fractional Dirichlet problem and an $\mathcal{O}(h ^{1+α−2s})$ convergence rate is obtained when the solution $u ∈ C ^{1,α}(\overline{Ω}^ δ_n),$ where $n$ is the dimension of the space, $α ∈ ({\rm max}(0, 2s − 1), 1], δ$ is a fixed positive constant,
and $h$ denotes mesh size. Finally, the performed numerical experiments confirm the
theoretical results.</p>