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A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin–Bona–Mahony‐type equation with nonsmooth solutions
Author(s) -
Lyu Pin,
Vong Seakweng
Publication year - 2020
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.22441
Subject(s) - mathematics , discretization , fractional calculus , norm (philosophy) , nonlinear system , mathematical analysis , gravitational singularity , type (biology) , convergence (economics) , unconditional convergence , derivative (finance) , rate of convergence , ecology , channel (broadcasting) , physics , electrical engineering , engineering , quantum mechanics , political science , compact convergence , financial economics , law , economics , biology , economic growth
To recover the full accuracy of discretized fractional derivatives, nonuniform mesh technique is a natural and simple approach to efficiently resolve the initial singularities that always appear in the solutions of time‐fractional linear and nonlinear differential equations. We first construct a nonuniform L2 approximation for the fractional Caputo's derivative of order 1 < α < 2 and present a global consistency analysis under some reasonable regularity assumptions. The temporal nonuniform L2 formula is then utilized to develop a linearized difference scheme for a time‐fractional Benjamin–Bona–Mahony‐type equation. The unconditional convergence of our scheme on both uniform and nonuniform (graded) time meshes are proven with respect to the discrete H 1 ‐norm. Numerical examples are provided to justify the accuracy.