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Numerical simulation of a class of space fractional bistable systems based on the Fourier spectral method:

Numerical simulation of a class of space fractional bistable systems based on the Fourier... Nonlinear vibration arises everywhere in a bistable system. The bistable system has been widely applied in physics, biology, and chemistry. In this article, in order to numerically simulate a class of space fractional-order bistable system, we introduce a numerical approach based on the modified Fourier spectral method and fourth-order Runge-Kutta method. The fourth- order Runge-Kutta method is used in time, and the Fourier spectrum is used in space to approximate the solution of the space fractional-order bistable system. Numerical experiments are given to illustrate the effectiveness of this method. Keywords Space fractional bistable system, Fourier spectral method, fractional Laplacian, numerical simulation Introduction Fractional partial differential equations are becoming widely used as a suitable modeling approach for many fields in 1–6 science and engineering. We mainly study models arising in the application areas of theoretical biology, physics, and chemistry, modeled by some space fractional bistable systems ∂u α ¼ d Δ u þ a u þ a v þ f ðu, vÞ, 1 11 12 ∂t (1) ∂v α ¼ d Δ v þ a u þ a v þ gðu, vÞ 1 21 22 ∂t where u = u (x, y, t) and v = v (x, y, t) are unknown functions. ðx, yÞ2V ¼½a, b × ½c, d, t > 0, the smooth boundary is ∂V. The parameters d ,i = 1 and 2 2 R are diffusion coefficients, and f (u , v ) and g (u , v ) are the reaction terms. The i i i i i system (1) is subjected to some initial condition u (x, y,0)= u (x, y), v (x, y,0)= v (x, y), and the homogeneous Neumann 0 0 ∂u ∂v boundary condition, namely,  ∂V ¼ ∂V ¼ 0. The functions u (x, y, t) and v (x, y, t) are assumed to be a causal function ∂n ∂n of time, that is, vanishing for t< 0. The general response expression contains parameters describing the order of the fractional derivatives that can be varied to obtain various responses. In this article, we use the fractional Laplacian operator by the Riesz fractional derivatives as follows Tianjin Xin Hua Vocational College, Tianjin, P. R. China School of Science, Inner Mongolia University of Technology, Hohhot, P. R. China School of Computer and Information, Inner Mongolia Medical University, Hohhot, P. R. China Institute of Economics and Management, Jining Normal University, Jining, P. R. China Corresponding authors: Wang Yulan, School of Science, Inner Mongolia University of Technology, Inner Mongolia, Hohhot 010059, P. R. China. Email: wylnei@163.com Li Cao, School of Computer and Information, Inner Mongolia Medical University, Inner Mongolia, Hohhot, P. R. China. Email: caolidd@sina.com.cn Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/ en-us/nam/open-access-at-sage). Liu et al. 605 Figure 1. Numerical solutions at different α =2, d =2 ×10 , F = 0.045, k = 0.0625. (a) t = 2100. (b) t = 4400. (c) t = 6700. (d) t = 10,000. (e) t = 30,000. Figure 2. Numerical solutions at different α = 1.9, d =2×10 , F = 0.045, k = 0.0625. (a) t = 2500. (b) t = 4600. (c) t = 6500. (d) t = 10,000. (e) t = 20,000. α α α ∂ u ∂ u 1 α α α α Δ u ¼ þ ¼ D u þ D u þ D u þ D u (2) α α x R x R y L y R πα ∂jxj ∂jyj 2 cos α α α α with D , D u and D u, D u being the RiemannLiouville fractional operators. x L x R y L y R Many approaches are used to solve the reaction-diffusion system. These methods include the finite difference method, 8–17 18–19 reproducing the kernel method (RKM), variational iteration method (VIM), homotopy perturbation methods 20–22 23–27 (HPM), etc. There are few numerical methods for higher order space fractional reaction-diffusion system. In this article, in order to numerically simulate space fractional-order reaction-diffusion system, we introduce a novel 28–30 numerical approach based on the modified Fourier spectral method and fourth-order Runge-Kutta method. Some pattern formations are shown by using this new approach, and the results have good agreement with theoretical results. Simulation results show the effectiveness of the method. Bifurcation analysis of system In this section, we give the Turing bifurcation conditions of system (1). We assume the equilibrium point of the non- diffusive system (1) is E =(u , v ), so 0 0 a u þ a v þ f ðu ,v Þ¼ 0, 11 0 12 0 0 0 (3) a u þ a v þ gðu ,v Þ¼ 0 11 0 12 0 0 0 ∗ ∗ Now, we do linear stability analysis of the equilibrium point E . We evaluate the Jacobian matrix A of the system at E as g t  b u ¼ (4) The characteristic roots for A are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ ð0Þ¼ trðAÞ ± ½trðAÞ   4detðAÞ (5) 1;2 0 0 0 2 606 Journal of Low Frequency Noise, Vibration and Active Control 41(2) Figure 3. Numerical solutions at different α =2, d =2×10 , F = 0.045, k = 0.0625. (a) t = 600. (b) t = 2500. (c) t = 3600. (d) t = 10,000. (e) t = 30,000. where ∂f ∂g trðAÞ ¼ a þ a þ ðu , v Þþ ðu , v Þ 11 22 0 0 0 0 ∂u ∂v and ∂f ∂g ∂g ∂f detðAÞ ¼ a þ ðu , v Þ × a þ ðu , v Þ  a þ ðu , v Þ × a þ ðu , v Þ 11 0 0 22 0 0 21 0 0 12 0 0 ∂u ∂v ∂u ∂v In general, if real part of eigenvalues λ (0) is negative, then the non-diffusive system (1) is stable. 1,2 Next, we derive cross-diffusion driven instability conditions and show that these are a generalization of the classical diffusion-driven instability conditions in the absence of cross-diffusion. b b d d 11 12 11 12 A ¼  (6) b b d d 21 22 21 22 ∂f ∂f ∂g ∂g where b ¼ a11 þ ðu , v Þ,b ¼ a þ ðu , v Þ,b ¼ a þ ðu , v Þ,b ¼ a þ ðu , v Þ. 11 0 0 12 12 0 0 21 21 0 0 22 22 0 0 ∂u ∂v ∂u ∂v The characteristic roots for A are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ ðαÞ¼ trðA Þ ± ½trðA Þ  4detðA Þ (7) 1;2 α α α where trðA Þ¼ b þ b  αðd þ d Þ α 11 22 11 22 and detðA Þ¼ðd d  d þ d Þα ðb d þ b d  b d Þα þðb d  b d Þ α 11 22 12 21 11 22 22 11 12 21 11 22 12 21 Turing bifurcation occurs when the equilibrium state is stable in absence of non-diffusion but it becomes unstable in presence of cross-diffusion. Thus, the only way for the E to become an unstable point of the cross-diffusion system (1) is when the real part of eigenvalues λ (k) is positive; then, the cross-diffusion system (1) is unstable. 1,2 Fourier spectral method 2πLj 2πLj For any integer N> 0, consider x ¼ jΔx ¼ , y ¼ jΔy ¼ ,L ¼ b  a, j ¼ 0; 1,/,N  1. uðx,y,tÞ is transformed into j j N N the discrete Fourier space as N1 N1 XX 1 N N ik xjik yj x y b u k , k , t ¼ FðuÞ¼ u x , y , t e  ≤ k ,k ≤  1 (8) x y j j x y N 2 2 j¼0 j¼0 Liu et al. 607 Figure 4. Numerical solutions at different α = 1.7, d =3×10 , F = 0.015, k = 0.055. (a) t = 900. (b) t = 2300. (c) t = 3800. (d) t = 10000. (e) t = 30000. Figure 5. Numerical solutions at different α = 1.8, d =3×10 , F = 0.015, k = 0.055. (a) t = 900. (b) t = 2300. (c) t = 3800. (d) t = 10,000. (e) t = 30,000 and the inverse formula is N2 N2 X X 1 ik xjik yj x y u x , y , t ¼ F b u ¼ b u k , k , t e ,0 ≤ j ≤ N  1 (9) j j x y N N k ¼ ky¼ 2 2 It is easy to know u (x, y, t)= F fF½uðx, y, tÞg. 2 2 For system (1) with f ðu, vÞ¼ rða  u þ u vÞ and gðu, vÞ¼ rðb  u vÞ, using Fourier transform, we can get n h io n h io h i h i h i ∂u α 1 1 1 1 1 ¼ d ðik Þ þ ik b u þ a F F b u þ a F F bv þ r a  F F b u þ F b u F bv 1 x y 12 12 ∂t (10) n h io n h io h i h i ∂bv 1 1 1 1 ¼ d ðik Þ þ ik b u þ a F F b u þ a F F bv þ r b  F F b u F bv 2 x y 22 22 ∂t We use the fourth-order Runge-Kutta method to solve the ordinary differential equation (10) which is as follows k ¼ g t  b u , 1 n τ τk k ¼ g t þ ,b u þ , 2 n n 2 2 τ τk (11) k ¼ g t þ , u þ , 3 n n 2 2 k ¼ g t þ τ,b u þ τk , 4 n n 3 b u ¼ b u þ ðk þ 2k þ 2k þ k Þ nþ1 n 1 2 3 4 6 608 Journal of Low Frequency Noise, Vibration and Active Control 41(2) Figure 6. Numerical solutions at different α = 1.9, d =3 × 10 , F = 0.015, k = 0.055. (a) t = 900. (b) t = 2300. (c) t = 3100. (d) t = 3800. (e) t = 30,000. ∂b u ∂b v where τ is step-size and gðt  uÞ¼ = . For convenience of expression, we denote ∂t ∂t U ¼ b u ðtÞ,bu ,/, b u ðtÞ , Gðt,UÞ¼ g t,b uðtÞ , g t,b uðtÞ , n ¼ 1,/ (12) 0 1 N1 Equation (10) is reduced to ∂U ¼ Gðt, UÞ ∂t Next, we can obtain the standard fourth-order Runge-Kutta formula K ¼ Gðt  U Þ, 1 n n τ τK K ¼ G t þ , U þ , 2 n n 2 2 τ τK (14) K ¼ G t þ , U þ , 3 n n 2 2 K ¼ Gðt þ τ, U þ τK Þ, 4 n n 3 U ¼ U þ ðK þ 2K þ 2K þ K Þ nþ1 n 1 2 3 4 Then, we can derive that by solving the following formula k ¼ g t ,b u ,b u ,/,b u , j1 j n 0,n 1,n N1,n τ τk τk ðN1Þ1 k ¼ g t þ ,b u þ ,/,b u þ , j2 j n 0,n N1,n 2 2 2 τ τk τk 02 ðN1Þ2 (15) k ¼ g t þ ,b u þ ,/,b u þ , j3 j n 0,n N1,n 2 2 2 k ¼ g t þ τ,b u þ τk ,/,b u þ τk , j4 j n 0,n 03 N1,n ðN1Þ3 b u ¼ b u þ k þ 2k þ 2k þ k j, nþ1 j,n j1 j2 j3 4 Finally, we find the numerical solution using the inverse discrete Fourier transform. Numerical simulation In this section, numerical simulation of the fractional Gray-Scott (GS) model with a perturbation to the spatially ho- mogeneous steady-state equation is obtained. The domain of interest is taken to be V =[1, 1] , discretized using N =128 points in each spatial coordinate. On account of the evolution of the numerical solution, u is similar with that of the v, but we will not present it in this article. Liu et al. 609 Numerical experiment Consider the following fractional Gray-Scott (GS) model: ∂u ¼ d Δ u  uv þ Fð1  uÞ, ∂t (16) ∂v α ¼ d Δ v  uv þðF þ kÞ v, ðx, y, tÞ2V × ½0, T ∂t 5 5 We take d =2× 10 , d =1× 10 , F =0.045, k =0.0625, u = ones (N), v = zeros (N), 1 2 vðN =2, N =2: NÞ¼ 0:5, vðN =2  1, 1 : N =2Þ¼ 1, the numerical results are shown in Figures 1 and 2. We take d =2× 5 5 10 , d =1 × 10 , F =0.045, k =0.0625, u = ones (N), v = zeros (N), vðN =2, N =2: NÞ¼ 0:5, vðN =2  2, 1 : N =2Þ¼ 1, 5 5 the numerical results are shown in Figure 3. We take d = 3×10 , d = 1×10 , F = 0.015, k = 0.055, u = ones (N), v = zeros 1 2 (N), vðN =2, N =2: NÞ¼ 0:5, vðN =2  2, 1 : N =2Þ¼ 1; the numerical results are shown in Figures 4–6. Conclusion and remarks In the article, a numerical method that combines the Fourier spectral method with the Runge-Kutta method is proposed to study a class of space fractional bistable system. This approach has general meanings and thus can be used to solve same types of nonlinear space fractional partial differential equations with periodic boundary condition in science and engi- neering. Some pattern formations are shown by using this new approach, and the results have good agreement with theoretical results. Simulation results show the effectiveness of the method. All computations are performed by the MATLABR2017b software. Acknowledgments The authors would like to express their thanks to the unknown referees for their careful reading and helpful comments. Declaration of Conflicting Interests The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by Inner Mongolia Natural Science Foundation under grant numbers (2021MS01009 and 2019MS07008). ORCID iD Wang Yulan  https://orcid.org/0000-0001-5292-246X References 1. Jena RM, Chakraverty S, Jena SK, et al. Analysis of time-fractional fuzzy vibration equation of large membranes using double parametric based Residual power series method. ZAMM-J Appl Math Mech 2021; 101: e202000165. 2. Fikret A, Aliev NA, and Safarova YV. Mamedova solution of the problem of analytical construction of optimal regulators for a fractional order oscillatory system in the general case. J Appl Comput Mech 2021; 7: 970–976. 3. He JH and El-Dib YO. 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Chem Eng Sci 1983; 38(1): 29–43. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Journal of Low Frequency Noise, Vibration and Active Control" SAGE

Numerical simulation of a class of space fractional bistable systems based on the Fourier spectral method:

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Abstract

Nonlinear vibration arises everywhere in a bistable system. The bistable system has been widely applied in physics, biology, and chemistry. In this article, in order to numerically simulate a class of space fractional-order bistable system, we introduce a numerical approach based on the modified Fourier spectral method and fourth-order Runge-Kutta method. The fourth- order Runge-Kutta method is used in time, and the Fourier spectrum is used in space to approximate the solution of the space fractional-order bistable system. Numerical experiments are given to illustrate the effectiveness of this method. Keywords Space fractional bistable system, Fourier spectral method, fractional Laplacian, numerical simulation Introduction Fractional partial differential equations are becoming widely used as a suitable modeling approach for many fields in 1–6 science and engineering. We mainly study models arising in the application areas of theoretical biology, physics, and chemistry, modeled by some space fractional bistable systems ∂u α ¼ d Δ u þ a u þ a v þ f ðu, vÞ, 1 11 12 ∂t (1) ∂v α ¼ d Δ v þ a u þ a v þ gðu, vÞ 1 21 22 ∂t where u = u (x, y, t) and v = v (x, y, t) are unknown functions. ðx, yÞ2V ¼½a, b × ½c, d, t > 0, the smooth boundary is ∂V. The parameters d ,i = 1 and 2 2 R are diffusion coefficients, and f (u , v ) and g (u , v ) are the reaction terms. The i i i i i system (1) is subjected to some initial condition u (x, y,0)= u (x, y), v (x, y,0)= v (x, y), and the homogeneous Neumann 0 0 ∂u ∂v boundary condition, namely,  ∂V ¼ ∂V ¼ 0. The functions u (x, y, t) and v (x, y, t) are assumed to be a causal function ∂n ∂n of time, that is, vanishing for t< 0. The general response expression contains parameters describing the order of the fractional derivatives that can be varied to obtain various responses. In this article, we use the fractional Laplacian operator by the Riesz fractional derivatives as follows Tianjin Xin Hua Vocational College, Tianjin, P. R. China School of Science, Inner Mongolia University of Technology, Hohhot, P. R. China School of Computer and Information, Inner Mongolia Medical University, Hohhot, P. R. China Institute of Economics and Management, Jining Normal University, Jining, P. R. China Corresponding authors: Wang Yulan, School of Science, Inner Mongolia University of Technology, Inner Mongolia, Hohhot 010059, P. R. China. Email: wylnei@163.com Li Cao, School of Computer and Information, Inner Mongolia Medical University, Inner Mongolia, Hohhot, P. R. China. Email: caolidd@sina.com.cn Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/ en-us/nam/open-access-at-sage). Liu et al. 605 Figure 1. Numerical solutions at different α =2, d =2 ×10 , F = 0.045, k = 0.0625. (a) t = 2100. (b) t = 4400. (c) t = 6700. (d) t = 10,000. (e) t = 30,000. Figure 2. Numerical solutions at different α = 1.9, d =2×10 , F = 0.045, k = 0.0625. (a) t = 2500. (b) t = 4600. (c) t = 6500. (d) t = 10,000. (e) t = 20,000. α α α ∂ u ∂ u 1 α α α α Δ u ¼ þ ¼ D u þ D u þ D u þ D u (2) α α x R x R y L y R πα ∂jxj ∂jyj 2 cos α α α α with D , D u and D u, D u being the RiemannLiouville fractional operators. x L x R y L y R Many approaches are used to solve the reaction-diffusion system. These methods include the finite difference method, 8–17 18–19 reproducing the kernel method (RKM), variational iteration method (VIM), homotopy perturbation methods 20–22 23–27 (HPM), etc. There are few numerical methods for higher order space fractional reaction-diffusion system. In this article, in order to numerically simulate space fractional-order reaction-diffusion system, we introduce a novel 28–30 numerical approach based on the modified Fourier spectral method and fourth-order Runge-Kutta method. Some pattern formations are shown by using this new approach, and the results have good agreement with theoretical results. Simulation results show the effectiveness of the method. Bifurcation analysis of system In this section, we give the Turing bifurcation conditions of system (1). We assume the equilibrium point of the non- diffusive system (1) is E =(u , v ), so 0 0 a u þ a v þ f ðu ,v Þ¼ 0, 11 0 12 0 0 0 (3) a u þ a v þ gðu ,v Þ¼ 0 11 0 12 0 0 0 ∗ ∗ Now, we do linear stability analysis of the equilibrium point E . We evaluate the Jacobian matrix A of the system at E as g t  b u ¼ (4) The characteristic roots for A are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ ð0Þ¼ trðAÞ ± ½trðAÞ   4detðAÞ (5) 1;2 0 0 0 2 606 Journal of Low Frequency Noise, Vibration and Active Control 41(2) Figure 3. Numerical solutions at different α =2, d =2×10 , F = 0.045, k = 0.0625. (a) t = 600. (b) t = 2500. (c) t = 3600. (d) t = 10,000. (e) t = 30,000. where ∂f ∂g trðAÞ ¼ a þ a þ ðu , v Þþ ðu , v Þ 11 22 0 0 0 0 ∂u ∂v and ∂f ∂g ∂g ∂f detðAÞ ¼ a þ ðu , v Þ × a þ ðu , v Þ  a þ ðu , v Þ × a þ ðu , v Þ 11 0 0 22 0 0 21 0 0 12 0 0 ∂u ∂v ∂u ∂v In general, if real part of eigenvalues λ (0) is negative, then the non-diffusive system (1) is stable. 1,2 Next, we derive cross-diffusion driven instability conditions and show that these are a generalization of the classical diffusion-driven instability conditions in the absence of cross-diffusion. b b d d 11 12 11 12 A ¼  (6) b b d d 21 22 21 22 ∂f ∂f ∂g ∂g where b ¼ a11 þ ðu , v Þ,b ¼ a þ ðu , v Þ,b ¼ a þ ðu , v Þ,b ¼ a þ ðu , v Þ. 11 0 0 12 12 0 0 21 21 0 0 22 22 0 0 ∂u ∂v ∂u ∂v The characteristic roots for A are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ ðαÞ¼ trðA Þ ± ½trðA Þ  4detðA Þ (7) 1;2 α α α where trðA Þ¼ b þ b  αðd þ d Þ α 11 22 11 22 and detðA Þ¼ðd d  d þ d Þα ðb d þ b d  b d Þα þðb d  b d Þ α 11 22 12 21 11 22 22 11 12 21 11 22 12 21 Turing bifurcation occurs when the equilibrium state is stable in absence of non-diffusion but it becomes unstable in presence of cross-diffusion. Thus, the only way for the E to become an unstable point of the cross-diffusion system (1) is when the real part of eigenvalues λ (k) is positive; then, the cross-diffusion system (1) is unstable. 1,2 Fourier spectral method 2πLj 2πLj For any integer N> 0, consider x ¼ jΔx ¼ , y ¼ jΔy ¼ ,L ¼ b  a, j ¼ 0; 1,/,N  1. uðx,y,tÞ is transformed into j j N N the discrete Fourier space as N1 N1 XX 1 N N ik xjik yj x y b u k , k , t ¼ FðuÞ¼ u x , y , t e  ≤ k ,k ≤  1 (8) x y j j x y N 2 2 j¼0 j¼0 Liu et al. 607 Figure 4. Numerical solutions at different α = 1.7, d =3×10 , F = 0.015, k = 0.055. (a) t = 900. (b) t = 2300. (c) t = 3800. (d) t = 10000. (e) t = 30000. Figure 5. Numerical solutions at different α = 1.8, d =3×10 , F = 0.015, k = 0.055. (a) t = 900. (b) t = 2300. (c) t = 3800. (d) t = 10,000. (e) t = 30,000 and the inverse formula is N2 N2 X X 1 ik xjik yj x y u x , y , t ¼ F b u ¼ b u k , k , t e ,0 ≤ j ≤ N  1 (9) j j x y N N k ¼ ky¼ 2 2 It is easy to know u (x, y, t)= F fF½uðx, y, tÞg. 2 2 For system (1) with f ðu, vÞ¼ rða  u þ u vÞ and gðu, vÞ¼ rðb  u vÞ, using Fourier transform, we can get n h io n h io h i h i h i ∂u α 1 1 1 1 1 ¼ d ðik Þ þ ik b u þ a F F b u þ a F F bv þ r a  F F b u þ F b u F bv 1 x y 12 12 ∂t (10) n h io n h io h i h i ∂bv 1 1 1 1 ¼ d ðik Þ þ ik b u þ a F F b u þ a F F bv þ r b  F F b u F bv 2 x y 22 22 ∂t We use the fourth-order Runge-Kutta method to solve the ordinary differential equation (10) which is as follows k ¼ g t  b u , 1 n τ τk k ¼ g t þ ,b u þ , 2 n n 2 2 τ τk (11) k ¼ g t þ , u þ , 3 n n 2 2 k ¼ g t þ τ,b u þ τk , 4 n n 3 b u ¼ b u þ ðk þ 2k þ 2k þ k Þ nþ1 n 1 2 3 4 6 608 Journal of Low Frequency Noise, Vibration and Active Control 41(2) Figure 6. Numerical solutions at different α = 1.9, d =3 × 10 , F = 0.015, k = 0.055. (a) t = 900. (b) t = 2300. (c) t = 3100. (d) t = 3800. (e) t = 30,000. ∂b u ∂b v where τ is step-size and gðt  uÞ¼ = . For convenience of expression, we denote ∂t ∂t U ¼ b u ðtÞ,bu ,/, b u ðtÞ , Gðt,UÞ¼ g t,b uðtÞ , g t,b uðtÞ , n ¼ 1,/ (12) 0 1 N1 Equation (10) is reduced to ∂U ¼ Gðt, UÞ ∂t Next, we can obtain the standard fourth-order Runge-Kutta formula K ¼ Gðt  U Þ, 1 n n τ τK K ¼ G t þ , U þ , 2 n n 2 2 τ τK (14) K ¼ G t þ , U þ , 3 n n 2 2 K ¼ Gðt þ τ, U þ τK Þ, 4 n n 3 U ¼ U þ ðK þ 2K þ 2K þ K Þ nþ1 n 1 2 3 4 Then, we can derive that by solving the following formula k ¼ g t ,b u ,b u ,/,b u , j1 j n 0,n 1,n N1,n τ τk τk ðN1Þ1 k ¼ g t þ ,b u þ ,/,b u þ , j2 j n 0,n N1,n 2 2 2 τ τk τk 02 ðN1Þ2 (15) k ¼ g t þ ,b u þ ,/,b u þ , j3 j n 0,n N1,n 2 2 2 k ¼ g t þ τ,b u þ τk ,/,b u þ τk , j4 j n 0,n 03 N1,n ðN1Þ3 b u ¼ b u þ k þ 2k þ 2k þ k j, nþ1 j,n j1 j2 j3 4 Finally, we find the numerical solution using the inverse discrete Fourier transform. Numerical simulation In this section, numerical simulation of the fractional Gray-Scott (GS) model with a perturbation to the spatially ho- mogeneous steady-state equation is obtained. The domain of interest is taken to be V =[1, 1] , discretized using N =128 points in each spatial coordinate. On account of the evolution of the numerical solution, u is similar with that of the v, but we will not present it in this article. Liu et al. 609 Numerical experiment Consider the following fractional Gray-Scott (GS) model: ∂u ¼ d Δ u  uv þ Fð1  uÞ, ∂t (16) ∂v α ¼ d Δ v  uv þðF þ kÞ v, ðx, y, tÞ2V × ½0, T ∂t 5 5 We take d =2× 10 , d =1× 10 , F =0.045, k =0.0625, u = ones (N), v = zeros (N), 1 2 vðN =2, N =2: NÞ¼ 0:5, vðN =2  1, 1 : N =2Þ¼ 1, the numerical results are shown in Figures 1 and 2. We take d =2× 5 5 10 , d =1 × 10 , F =0.045, k =0.0625, u = ones (N), v = zeros (N), vðN =2, N =2: NÞ¼ 0:5, vðN =2  2, 1 : N =2Þ¼ 1, 5 5 the numerical results are shown in Figure 3. We take d = 3×10 , d = 1×10 , F = 0.015, k = 0.055, u = ones (N), v = zeros 1 2 (N), vðN =2, N =2: NÞ¼ 0:5, vðN =2  2, 1 : N =2Þ¼ 1; the numerical results are shown in Figures 4–6. Conclusion and remarks In the article, a numerical method that combines the Fourier spectral method with the Runge-Kutta method is proposed to study a class of space fractional bistable system. This approach has general meanings and thus can be used to solve same types of nonlinear space fractional partial differential equations with periodic boundary condition in science and engi- neering. Some pattern formations are shown by using this new approach, and the results have good agreement with theoretical results. Simulation results show the effectiveness of the method. All computations are performed by the MATLABR2017b software. Acknowledgments The authors would like to express their thanks to the unknown referees for their careful reading and helpful comments. Declaration of Conflicting Interests The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by Inner Mongolia Natural Science Foundation under grant numbers (2021MS01009 and 2019MS07008). ORCID iD Wang Yulan  https://orcid.org/0000-0001-5292-246X References 1. Jena RM, Chakraverty S, Jena SK, et al. Analysis of time-fractional fuzzy vibration equation of large membranes using double parametric based Residual power series method. ZAMM-J Appl Math Mech 2021; 101: e202000165. 2. Fikret A, Aliev NA, and Safarova YV. Mamedova solution of the problem of analytical construction of optimal regulators for a fractional order oscillatory system in the general case. J Appl Comput Mech 2021; 7: 970–976. 3. He JH and El-Dib YO. 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Journal

"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE

Published: Dec 16, 2021

Keywords: Space fractional bistable system; Fourier spectral method; fractional Laplacian; numerical simulation

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