Abstract
In this paper, we study a conservative difference scheme for the sine-Gordon equation (SGE) with the Riesz space fractional derivative. We rigorously establish the conservation property and solvability of the difference scheme. We discuss the stability and convergence of the difference scheme in the L_{infty} norm. To reduce the computational complexity, we introduce a revised Newton method for implementing the difference scheme. Finally, we provide several numerical experiments to support the theoretical results.
Highlights
The nonlinear sine-Gordon equation (SGE)∂2u ∂t2 – u + sin u = 0, x = (x1, . . . , xn) ∈ Rn, (1.1)arises in many different areas, such as stability of fluid motions, differential geometry, Josephson junctions, models of particle physics [1], the propagation of fluxon [2], the motion of a rigid pendulum attached to a stretched wire [3], the phenomenon of supratransmission in nonlinear media [4], and so on
Since the fractional calculus is frequently better than the integer calculus in the description of many physical laws, various classical partial differential equations have been extended to the corresponding fractional-order differential equations [12,13,14,15,16,17,18,19]
Ray [20] combined the modified decomposition method and Fourier transform to approximate the solution of a fractional SGE
Summary
Arises in many different areas, such as stability of fluid motions, differential geometry, Josephson junctions, models of particle physics [1], the propagation of fluxon [2], the motion of a rigid pendulum attached to a stretched wire [3], the phenomenon of supratransmission in nonlinear media [4], and so on. The conserved quantity is good for the analysis of the nonlinear stability of the numerical schemes proposed, it is difficult to apply them [10, 11]. MacíasDíaz [4] employed an explicit finite difference scheme to simulate a space-fractional SGE. His result supported the fact that nonlinear supratransmission is present in the Riesz space-fractional model. He pointed out that numerical simulations for fractional SGE require enormous amount of computer time. We propose a conservative difference scheme for space-fractional SGEs. Subsequently, we prove that the difference scheme preserves the energy conservation law.
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