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A superlinear convergence scheme for the multi‐term and distribution‐order fractional wave equation with initial singularity
Author(s) -
Huang Jianfei,
Zhang Jingna,
Arshad Sadia,
Tang Yifa
Publication year - 2021
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22773
Subject(s) - mathematics , singularity , discretization , mathematical analysis , fractional calculus , convergence (economics) , quadrature (astronomy) , partial differential equation , rate of convergence , engineering , electrical engineering , economics , economic growth , channel (broadcasting)
In this paper, a superlinear convergence scheme for the multi‐term and distribution‐order fractional wave equation with initial singularity is proposed. The initial singularity of the solution of the multi‐term time fractional partial differential equation often generate a singular source, it increases the difficulty to numerically solve the equation. Thus, after discretizing the spatial distribution‐order derivative by the midpoint quadrature, an integral transformation is applied to deal with the temporal direction for obtaining a temporal superlinear convergence scheme based on the uniform mesh. Then, the fully discrete scheme is constructed by using Crank–Nicolson technique and L1 approximation in time, and fractional centered difference approximation in space. The convergence and stability of the proposed scheme are rigorously analyzed. Finally, numerical experiments are presented to support the theoretical results of our scheme.