Finite difference method for inhomogeneous fractional Dirichlet problem

2021/01/27 by Jing Sun, Weihua Deng, Sun, Jing +3
Mathematics · #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2101.11378

openalex publication_date 2021/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We make the split of the integral fractional Laplacian as (-Δ)^s u=(-Δ)(-Δ)^s-1u, where s∈(0,(1)/(2))∪((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 O(h^1+α-2s) convergence rate is obtained when the solution u∈ C^1,α(Ω^δ_n), where n is the dimension of the space, α∈(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.

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