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A Conservative Difference Scheme for the Riesz Space-Fractional Sine-Gordon Equation Zhiyong Xing1,2* and Liping Wen1

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A Conservative Difference Scheme for the Riesz Space-Fractional Sine-Gordon Equation Zhiyong Xing1,2* and Liping Wen1 Xing and Wen Advances in Difference Equations (2018)2018:238 https://doi.org/10.1186/s13662-018-1689-5 R E S E A R C H Open Access A conservative difference scheme for the Riesz space-fractional sine-Gordon equation Zhiyong Xing1,2* and Liping Wen1 *Correspondence: [email protected] Abstract 1School of Mathematics and Computational Science, Xiangtan In this paper, we study a conservative difference scheme for the sine-Gordon University, Xiangtan, P.R. China equation (SGE) with the Riesz space fractional derivative. We rigorously establish the 2Department of Mathematics, conservation property and solvability of the difference scheme. We discuss the Shaoyang University, Shaoyang, P.R. China stability and convergence of the difference scheme in the L∞ 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. Keywords: Space-fractional sine-Gordon equation; Riesz fractional derivative; Conservation law; Newton method; Convergence and stability 1 Introduction The nonlinear SGE ∂2u – u + sin u =0, x =(x ,...,x ) ∈ Rn, (1.1) ∂t2 1 n 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 supra- transmission in nonlinear media [4], and so on. Therefore the investigation for the SGE has attracted attention of some researchers, and many significant achievements have been made. Aremarkablepropertyof(1.1) is the energy conservation: 1 ∂u 2 ∂u 2 + +2P(u) dx =const, 2 R ∂t ∂x where P(u)=1–cos u; it was studied by many researchers [5–9]. The conserved quan- tity is good for the analysis of the nonlinear stability of the numerical schemes proposed, although it is difficult to apply them [10, 11]. Since the fractional calculus is frequently better than the integer calculus in the de- scription of many physical laws, various classical partial differential equations have been extended to the corresponding fractional-order differential equations [12–19]. However, © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Xing and Wen Advances in Difference Equations (2018)2018:238 Page 2 of 22 there are a few works on fractional-order SGEs. Ray [20] combined the modified decom- position method and Fourier transform to approximate the solution of a fractional SGE. In [21], a family of breather-like solutions for the fractional SGE is found numerically by using the approach that is called sometimes the rotating wave approximation. Macías- Díaz [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 also pointed out that numerical simulations for fractional SGE require enormous amount of computer time. In this paper, we consider the space-fractional SGE ∂2u ∂αu – + sin u =0, –∞ < x <+∞,0<t ≤ T, (1.2) ∂t2 ∂|x|α with the boundary and initial conditions ∂u(x,0) u(x,0)=ϕ(x), = ψ(x), –∞ < x <+∞, (1.3) ∂t lim u(x, t)=0, 0<t < T, (1.4) |x|→∞ where 1 < α ≤ 2. The Riesz fractional derivative of order α is defined by [22] α ∂ u α 1 α α (x, t)=–(–) 2 u(x, t)=– D u(x, t)+ D ∞u(x, t) , ∂|x|α 2 cos(πα/2) –∞ x x + α α where –∞Dx u(x, t)andxD+∞u(x, t) are the left- and right-side Riemann–Liouville frac- tional derivatives, respectively. ∂u Multiplying (1.2)by ∂t and then integrating with respect to x,weobtain 2 d 1 ∂u α 2 + (–) 4 u +2P(u) dx =0, dt R 2 ∂t that is, the conservation of energy 2 1 ∂u 1 α 2 4 + (– ) u L2 + P(u) dx = const, (1.5) 2 ∂t L2 2 R where P(u) is consistent with the aforementioned. To the best of our knowledge, there are very few works developing conservative numer- ical methods for fractional SGEs. The main objective of this paper is to propose a conser- vative numerical method for space-fractional SGEs and try to reduce the computational complexity. This paper is arranged as follows. In the next section, we propose a conservative differ- ence scheme for space-fractional SGEs. Subsequently, we prove that the difference scheme preserves the energy conservation law. The boundedness, solvability, and convergence of the difference scheme are rigorously established. In Sect. 4,weintroducearevisedNewton method for implementation of the difference scheme. In Sect. 5, we present some numer- ical results to demonstrate the effectiveness of the difference scheme. Finally, we give a simple conclusion. Xing and Wen Advances in Difference Equations (2018)2018:238 Page 3 of 22 2 A conservative difference scheme for fractional SGEs 2.1 Notation To develop a finite difference scheme for problem (1.2)–(1.4), we assume that its solution is negligibly small outside of the interval =(xR, xL), that is, u|x∈R\ =0.wechoosethe T xR–xL time step τ = N and mesh size h = M with two positive integers N and M.Denote h = {xi|xi = xL + ih,1≤ i ≤ M –1}, τ = {tn|tn = nτ,0≤ n ≤ N} and n ≈ n ui u(xi, tn), Ui = u(xi, tn). T Let νh = {w|w =(w1, w2,...,wM–1) } be the space of grid functions. For a given grid func- { n| ∈ × } tion w = wi (xi, tn) h τ , we define the finite difference operators wn+1 – wn wn – wn–1 wn = i i , wn = i i , i t τ i t¯ τ n+1 n–1 n+1 n–1 n wi – wi n wi + wi w ˆ = , w¯ = . i t 2τ i 2 Foranytwogridfunctionsun and vn,wedefine M–1 n n n n n2 n n n | | u , v = h ui vi , u = u , u , u ∞ = sup ui . (2.1) lh ∈Z i=1 i ≤ ≤ ∈ 2 Let 0 σ 1 be given. For any u lh, the fractional Sobolev norm u Hσ and seminorm |u|Hσ can be defined as π π 2 –2σ | |2σ 2 | |2 –2σ | |2σ 2 u Hσ = h 1+h k u(k) dk, u Hσ = h h k u(k) dk. –π –π 2.2 A conservative implicit difference scheme Lemma 2.1 ([23]) Suppose that u ∈ L2+α(R). Then α –αα 2 2 –h h f (x)=–(– ) f (x)+O h , (2.2) where ∞ (–1)k (α +1) αf (x)= g(α)f (x – kh), g(α) = . h k k (α/2 – k +1) (α/2 + k +1) k=–∞ If we define ⎧ ⎨ ∈ ∗ u(x, t)ifx [xL, xR], u (x, t)=⎩ 0ifx ∈ (–∞, xL) ∪ (xR, ∞), Xing and Wen Advances in Difference Equations (2018)2018:238 Page 4 of 22 then, for 1 < α ≤ 2, we can get M–1 α (α) 2 n –α n 2 –(– ) ui =–h gi–l ul + O h . (2.3) l=1 Denote M–1 α n –α (α) n δh ui = h gi–l ul . l=1 Considering equation (1.2)atpoints(xi, tn), we derive ∂2u ∂αu (x , t )– (x , t )=–sin u(x , t ), 1 ≤ i ≤ M –1,1≤ n ≤ N – 1. (2.4) ∂t2 i n ∂|x|α i n i n Combining the Taylor expansion and equation (2.3), we get ∂2u (x , t )= Un + O τ 2 , ∂t2 i n i tt ∂αu (x , t )=δαU¯ n + O h2 + τ 2 , (2.5) ∂|x|α i n h i ¯ n 2 – sin u(xi, tn)=ζ Ui + O τ , n+1 n–1 ¯ n cos(Ui )–cos(Ui ) where ζ (Ui )= n+1 n–1 . Ui –Ui Substituting (2.5)into(2.4), we get n α ¯ n ¯ n n Ui tt¯ + δh Ui = ζ Ui + ri , (2.6) n 2 2 where ri = O(h + τ ). In addition, from conditions (1.3)and(1.4)wehave n n ≤ ≤ U0 =0, UM =0, 0 n N, (2.7) ∂U U0 = ϕ(x ), (x ,0)=ψ(x ), 1 ≤ i ≤ M – 1. (2.8) i i ∂t i i n n n Omitting the small term ri in (2.6) and using the numerical solution ui to replace Ui ,we obtain the following difference scheme for solving problem (1.2)–(1.4): n α ¯ n ¯ n ≤ ≤ ≤ ≤ ui tt¯ + δh ui = ζ ui ,1i M –1,1 n N – 1, (2.9) n n ≤ ≤ u0 =0, uM =0, 0 n N, (2.10) τ 2 u0 = ϕ(x ), u1 = u0 + τψ(x )– δαu0 + sin u0 ,1≤ i ≤ M – 1. (2.11) i i i i i 2 h i i Xing and Wen Advances in Difference Equations (2018)2018:238 Page 5 of 22 3 Numerical analysis 3.1 Some useful lemmas Lemma 3.1 ([13]) If u, v are any two grid functions in νh, then there exists a linear operator α 1 α – 2 2 h = h G such that α –α α α δh u, v = h Gu, v = h u, h v , where ⎛ ⎞ g(α) g(α) ...g(α) ⎜ 0 –1 –M+2⎟ ⎜ (α) (α) (α) ⎟ ⎜ g1 g0 ...g–M+3⎟ G = ⎜ . ⎟ ; ⎝ . .. ⎠ (α) (α) ··· (α) gM–2 gM–3 g0 1 1 G 2 is the unique positive definite square root of G, that is,(G 2 )2 = G.
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