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Numerical analysis for Klein‐Gordon equation with time‐space fractional derivatives
Author(s) -
Zhang Jun,
Wang JinRong,
Zhou Yong
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6147
Subject(s) - mathematics , fractional calculus , numerical analysis , mathematical analysis , nonlinear system , polynomial , space (punctuation) , spectral method , klein–gordon equation , linguistics , philosophy , physics , quantum mechanics
We present and analyze two numerical schemes for solving a nonlinear Klein‐Gordon equation with time‐space fractional derivatives. Numerical methods are base on finite difference scheme in fractional derivative and Fourier‐spectral method in spatial variable. It is proved that the linearized method is conditionally stable while the nonlinearized one is unconditionally stable. In addition, the error estimate shows that the linearized method is in the order of O ( Δ t + N β − r ) , and the nonlinearized method converge with the order O ( Δ t 3 − α + N β − r ) , where Δ t , N , β , and r are, respectively, step of time, polynomial degree, the fractional derivative in space, and regularity of u . Some numerical experiments are performed to demonstrate the theoretical results.