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

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.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.

On initial conditions for fractional delay differential equations

“…Since the aim of this paper is just to highlight the impact of the initial data on the solution and on the fractional operator, and not to devise highly efficient numerical methods, we consider here basic methods with just a first-order accuracy with respect to the step-size. We refer to the existing literature for specific works concerning numerical methods for FDDEs (e.g., see [16,43,44,45,46]).…”

A reduced-order extrapolating collocation spectral method based on POD for the 2D Sobolev equations

“…It has been applied in many fields including pattern recognition and signal analysis [21], statistical calculations [22], and computational fluid mechanics [23]. For the past few years, it has successfully been used to the order reduction for the Galerkin methods [24,25], the FE methods [26,27], the FD schemes [28][29][30], the FVE methods [31,32], and the reduced basis methods for PDEs [33][34][35]. Nevertheless, the existing POD-based reduced-order methods (see [17][18][19][20][21][22][23][24][25][26][27][29][30][31][32][33][34][35]) are mostly created with the POD bases produced by the classic solutions at all the time nodes on [0, T], before repeatedly finding the order reduction solutions on the same time nodal points.…”

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

“…Though there are lots studies for fractional-order differential equations in recent years (see, e.g., [1][2][3][4]), there are few reports about the reduced-order study for the fractionalorder differential equations except for Ref. [4].…”

“…Though there are lots studies for fractional-order differential equations in recent years (see, e.g., [1][2][3][4]), there are few reports about the reduced-order study for the fractionalorder differential equations except for Ref. [4]. In this paper, by means of 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) as follows.…”