An optimized Crank–Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equation

• DOI: 10.1186/s13662-018-1939-6
• Publication Date: 2019-01-03T00:44:46Z
• Authors: Yanjie Zhou; Zhendong Luo
• Keywords: Fractional-order parabolic-type sine-Gordon equation; Optimized Crank–Nicolson finite difference extrapolating model; QA1-939; Classical Crank–Nicolson finite difference model; Existence, stabilization, and convergence; Proper orthogonal decomposition; Mathematics

Abstract In this paper, by means of a proper orthogonal decomposition (POD) we mainly reduce the order of the classical Crank–Nicolson finite difference (CCNFD) model for the fractional-order parabolic-type sine-Gordon equations (FOPTSGEs). Toward this end, we will first review the CCNFD model for FOPTSGEs and the theoretical results (such as existence, stabilization, and convergence) of the CCNFD solutions. Then we establish an optimized Crank–Nicolson finite difference extrapolating (OCNFDE) model, including very few unknowns but holding the fully second-order accuracy for FOPTSGEs via POD. Next, by a matrix analysis we will discuss the existence, stabilization, and convergence of the OCNFDE solutions. Finally, we will use a numerical example to validate the validity of theoretical conclusions. Moreover, we show that the OCNFDE model is very valid for settling FOPTSGEs.