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

Volume 11, Issue 5, pp 672--682 http://dx.doi.org/10.22436/jnsa.011.05.08
Publication Date: April 01, 2018 Submission Date: January 25, 2018 Revision Date: February 16, 2018 Accteptance Date: March 03, 2018

Authors

Yanhua Cao - School of Sciences, East China Jiaotong University, Nanchang 330013, China. Zhendong Luo - School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China.


Abstract

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.


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ISRP Style

Yanhua Cao, Zhendong Luo, A reduced-order extrapolating Crank-Nicolson finite difference scheme for the Riesz space fractional order equations with a nonlinear source function and delay, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 5, 672--682

AMA Style

Cao Yanhua, Luo Zhendong, A reduced-order extrapolating Crank-Nicolson finite difference scheme for the Riesz space fractional order equations with a nonlinear source function and delay. J. Nonlinear Sci. Appl. (2018); 11(5):672--682

Chicago/Turabian Style

Cao, Yanhua, Luo, Zhendong. "A reduced-order extrapolating Crank-Nicolson finite difference scheme for the Riesz space fractional order equations with a nonlinear source function and delay." Journal of Nonlinear Sciences and Applications, 11, no. 5 (2018): 672--682


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