A reduced-order extrapolating Crank-Nicolson finite difference scheme for the Riesz space fractional order equations with a nonlinear source function and delay

Cao, Yanhua ¹ ; Luo, Zhendong ²

¹ School of Sciences, East China Jiaotong University, Nanchang 330013, China
² School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 5, p. 672-682.

Résumé

This article mainly studies the order-reduction of the classical Crank-Nicolson finite difference (CNFD) scheme for the Riesz space fractional order differential equations (FODEs) with a nonlinear source function and delay on a bounded domain. For this reason, the classical CNFD scheme for the Riesz space FODE and the existence, stability, and convergence of the classical CNFD solutions are first recalled. And then, a reduced-order extrapolating CNFD (ROECNFD) scheme containing very few degrees of freedom but holding the fully second-order accuracy for the Riesz space FODEs is established by means of proper orthogonal decomposition and the existence, stability, and convergence of the ROECNFD solutions are discussed. Finally, some numerical experiments are presented to illustrate that the ROECNFD scheme is far superior to the classical CNFD one and to verify the correctness of theoretical results. This indicates that the ROECNFD scheme is very effective for solving the Riesz space FODEs with a nonlinear source function and delay.

Détail

DOI : 10.22436/jnsa.011.05.08

Classification : 34K28, 65M12, 35R11, 34K37
Keywords: Crank-Nicolson finite difference scheme, Riesz space fractional order differential equation, existence and stability as well as convergence, reduced-order extrapolating Crank-Nicolson finite difference scheme, proper orthogonal decomposition

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