Numerical simulations for a variable order fractional Schnakenberg model Z. Hammouch, T. Mekkaoui, and F. B. M. Belgacem Citation: AIP Conference Proceedings 1637, 1450 (2014); doi: 10.1063/1.4907312 View online: http://dx.doi.org/10.1063/1.4907312 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1637?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Modeling and numerical simulation of a harpsichord J. Acoust. Soc. Am. 136, 2202 (2014); 10.1121/1.4899986 Modeling and numerical simulation of a piano. J. Acoust. Soc. Am. 129, 2542 (2011); 10.1121/1.3588451 Sample path properties of fractional Riesz–Bessel field of variable order J. Math. Phys. 49, 013509 (2008); 10.1063/1.2830431 Numerical simulation of interface waves by highorder spectral modeling techniques J. Acoust. Soc. Am. 95, 681 (1994); 10.1121/1.408428 Numerical simulation of interface waves by highorder spectral modeling techniques J. Acoust. Soc. Am. 92, 2456 (1992); 10.1121/1.404528
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Corrigendum: Numerical simulations for a variable order fractional Schnakenberg model, Z. Hammouch, T. Mekkaoui, and F. B. M. Belgacem, AIP Conf. Proc. 1637, 1450 (2014) In the original article PDF file, as supplied to AIP Publishing, the name and affiliation of author F. B. M. Belgacem was missing due to a Latex compiling error. This article was updated on 29 January 2015 to correct that error.
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Numerical simulations for a Variable Order Fractional Schnakenberg model Z.Hammouch∗ , T.Mekkaoui∗ and F.B.M.Belgacem† ∗
E3MI, Département de Mathématiques, FST Errachidia, Université Moulay Ismail BP.509 Boutalamine 52000 Errachidia, Morocco † E3MI, Corresponding author: F.B.M.Belgacem, email:
[email protected], Department of Mathematics, Faculty of Basic Education, PAAET, Al-Aardhia, Kuwait. Abstract. This paper is concerned with the numerical solutions of a variable-order space-time fractional reaction-diffusion model. The space-time fractional derivative is considered in the sense of Riesz-Feller, the system is defined by replacing the second order space derivatives with the variable Riesz-Feller derivatives. The problem is solved by an explicit finite difference method. Finally, simulation results to this problem are presented and discussed. Keywords: Explicit finite difference, Variable-order Riesz-feller derivative, Schnakenberg model. PACS: 87.10.Ed, 87.10.Ed, 87.15.Vv.
1. INTRODUCTION Fractional calculus is an old mathematical topic from 17th century. However it can be considered as a new topic of research. Actually, in the last few years Fractional Order Partial Differential Equations (FOPDEs) have played an important role in applied mathematics, physics, engineering, economics and other science [1]-[8]. The dynamic models of a large number of phenomena can be modeled by FOPDEs which are characterized by fractional space and/or time derivatives. Because the fractional derivative of a function depends on the values of the function over the entire interval, it is suitable for describing the memory effect and modeling of the systems with long range interactions both in space and time. Besides, systems of fractional reaction-diffusion (SFRD) have gained considerable attention to study nonlinear phenomena arising in the disciplines of science and engineering [9]-[13]. Of these particular interests are patterns formation [2], since it is best described by the fractional-order models because the fractional derivatives take into consideration the whole memory of the system [14]. On the other hand, numerical methods yield approximate solutions to the governing equations through the discretization of space and time [15]-[21]. The finite difference method is the most classic method for fractional differential equations. It has been extensively considered in the literature for constant order time or space fractional diffusion equations; Roop [22] investigated the numerical approximation of the variational solution to the fractional advection-dispersion equation, Meerschaert [23] examined finite-difference approximations for fractional advection-dispersion flow equations, Zhuang and Li [24] analyzed an implicit finite difference approximation for the time-fractional diffusion equation. However, variable-order fractional partial differential equations is relatively new topic, there is a growing number of papers on numerical approximation of these equations, Coimbra [25] proposed a first-order accurate approximation for the solution of variable-order differential equations. Lin et al. [26] proposed a new explicit finite-difference method for solving variable-order fractional partial differential equations with Riesz-Feller derivative. The main purpose of this paper is to consider a variable-order fractional version of the well-known Schnakenberg model. since the behaviour of chemical species may change in time and space. Recently, it was found that the variational order derivative is very useful to describe efficiently such situation. We then modify the Schnakenberg model by replacing the partial derivative with respect to space by the variable-order derivative. The resulting problem is very difficult to handle analytically. We solved it numerically via an explicit finite difference scheme. The structure of the remainder of this paper is as follows: In Section 2, some basic mathematical definitions from fractional calculus are given. In Section 3, the detailed description and formulation of the problem is presented. In Section 4, we present the numerical simulations of the solutions behaviour. Finally, we conclude our work in Section 5.
10th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences AIP Conf. Proc. 1637, 1450-1455 (2014); doi: 10.1063/1.4907312 © 2014 AIP Publishing LLC 978-0-7354-1276-7/$30.00
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2. VARIABLE-ORDER RIESZ FRACTIONAL DERIVATIVE There are some preferences dealt with in the definition of fractional derivatives such as Riemann-Liouville, Caputo, Riesz, Weyl, Grunwald-Letnikov, Coimbra etc. It is noted that each type of fractional derivative has different properties and applications. For the readers that are not familiar with the variable-order derivative Riesz-Feller, we recall in this section its basic definition. Assume first that: 1 ≤ α(x,t) < 2, Xa ≤ x ≤ Xb , 0 ≤ t ≤ T and u(x,t) = 0 when x ≤ Xa or x > xb . According to [26] the generalized Riesz fractional derivative is defined by Z Z − sec(α(x,t) π2 ) d 2 θ u(ξ ,t)dξ d 2 xb u(ξ ,t)dξ α(x,t) ρ + σ , (1) R u(x,t) = x Γ(2 − α(x,t)) dθ 2 xa (θ − ξ )α(x,t)−1 dθ 2 θ (ξ − θ )α(x,t)−1 θ =x where ρ ≥ 0, σ ≥ 0 and ρ + σ = 1. Remark 2.1 When ρ = σ = 0.5, the above equation defines the variable-order Riesz fractional derivative and when ρ = 1 and σ = 0 we recover the variable-order Riemann-Liouville derivative multiplied by the coefficient cos(α(x,t) π2 ).
3. THE VOF-SCHNAKENBERG MODEL Reaction-diffusion equations are useful in many areas of science and engineering. In applications to population biology, the reaction term models growth, and the diffusion term accounts for migration [27]. Many of these mathematical models are a generalized reaction-diffusion system of two chemical species which can be written in terms of nondimensional variables as ∂U = D∆U + kF(U), in Ω(t), (2) ∂t where Ω(t) is a time-dependent domain, and u f (u, v) du 0 U= , F= , D= , v g(u, v) 0 dv where u, v are the two chemical species concentrations, f and g are reaction kinetics, k is a reaction parameter, is the diffusion matrix. The Schnakenberg model is one of the simplest reaction-diffusion models, it describes the behavior of a chemical activator u in the presence of a chemical inhibitor v. It has been used in stability analysis applications and pattern formation, allowing the prediction of the interaction mechanisms between molecular chemical systems and morphogenic constructions such as bone formation and growth [27]. It is related to Meinhardt-Gierer’s substrate depletion model, but with a simpler nonlinearity. The 1D model is described by the following equations [27]: ∂ u ∂ 2u 2 ∂t = ∂ x2 + γ(a − u + u v), (3) 2v ∂ v ∂ 2 = d 2 + γ(b − u v). ∂t ∂x In Eq. (3), the movement of both the activator and the inhibitor substances is described as a diffusive term with an equivalent diffusion constant given by the coefficient d. The reaction term for the activator substance Eq. (3)1 is described as a constant production given by the parameter a, a linear consumption, and a nonlinear kinetic reaction representing the production of the activator u in presence of the inhibitor v. On the other hand, the reaction term for the inhibitor substance Eq. (3)2 is given as a constant production with coefficient b and a nonlinear kinetic reaction for the consumption of the inhibitor substance in the presence of the activator, and γ is a positive dimensionless constant [27]. In this work we consider a more general version of (3): ∂ u = x Rα(x,t) u(x,t) + γ(a − u + u2 v) ∂t (4) ∂ v α(x,t) 2 = dxR v(x,t) + γ(b − u v). ∂t
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where (x,t) ∈ Ω = [−L, L] × [0, T ] and x Rα(x,t) is a generalized Riesz fractional derivative of time-space dependent order α(x,t) (1 < α(x,t) ≤ 2). First, we consider the linear stability of (4). The spatially homogeneous steady state solutions of (4) satisfying a − u + u2 v = 0, (5) b − u2 v = 0. are
(u, v) = a + b,
b (a + b)2
.
(6)
According to the standard reaction-diffusion theory, the stability of the homogeneous steady state (6) can be determined γ(1 + 2uv) γu2 from its Jacobian matrix J = . At the homogeneous steady states the determinant and the trace −2γuv −γu2 of J are det(J) = γ 2 (a + b)2 , (7) ba Tr(J) = a+b − (a + b)2 . Hence the stability of the homogeneous steady state to homogeneous perturbations will occur when b − a < (a + b)3 .
4. NUMERICAL SIMULATIONS BY AN EXPILICIT FINITE DIFFERENCE SCHEME This section describes the numerical technique to solve the system (4) subject to periodic boundary conditions by an explicit finite difference as described by Lin et al [26]. The grid points in the space domain [−L, L] and the time domain [0, T ] are labeled : xi = ih for i = 0, ..., m and t j = jτ for j = 0, ..., n − 1, respectively. The Grunwald formula for the α(x,t)−fractional derivative approximation [26] gives 2 Z θ j j i+1 d u(ξ ,t)dξ 1 −αi+1 −αi+1 ≈ h ω u(xk ,t j ), ∑ i+1−k 2 Γ(2 − α(xi ,t j )) dθ xa (θ − ξ )α(x,t) − 1 θ =xi k=0 (8) 2 Z θ m j j 1 d u(ξ ,t)dξ −α i−1 u(xk ,t j ), ≈ h−αi−1 ∑ ω(k−i)−1 Γ(2 − α(x ,t )) dθ 2 xa (θ − ξ )α(x,t) − 1 θ =x i+1 j k=i−1 i+1 j
where
α ωk i
j αi = (−1) and αij = α(xi ,t j ). Substitution of (8) into (4) yields k k
! m j j j j i+1 2 αi−1 αi+1 j j j j j j j+1 −α −α ui = ui + ai τ ραi h i+1 ∑ ωi+1−k uk + σ αi h i−1 ∑ ω−i+1+k uk + γ(a − uij + uij vij ), k=0 k=i−1 j j j i+1 αi+1 j+1 j j j −αi+1 j j −αi−1 v = v + a τd ρα h ω ∑ i i i i+1−k vk + σ αi h i k=0
m
∑
j
αi−1 ω−i+1+k vkj
!
(9)
j2
+ γ(b − ui vij ),
k=i−1
where aij = − sec(α(xi ,t j )). The stability and the accuracy of the method have been discussed in [26]. The numerical scheme (9) represents a system of algebraic equations with block diagonal matrix. The computer code for numerical scheme (4), was written in Matlab software. The time and spatial steps used in the simulation were variying from 0 to 0.8 and from −1 to 1 respectively. The initial conditions are u(x, 0) = 2 − 10−5 cos(2πx) + 10−2 cos(76πx) + 10−2 cos(78πx), (10) (−5) v(x, 0) = 1 + 10 cos(2πx),
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and the boundary conditions are u(0,t) = u(1,t) = 2.02,
(11)
v(−1,t) = v(1,t) = 1. The constants a, b, d and γ are taken as −2, 4, 1 and 5 respectively. Figures 1 shows the results of computer solution of (4) considered above, for α(x,t) = 2 (the standard Schnakenberg model). Figure 2 depicts the numerical solutions 1 3 1 , Figure 3 u(x,t) and v(x,t) for the constant fractional case (α(x,t) = 1.98) and for α(x,t) = + exp − 2 2 2 2 x t +1 and Figure 4 illustrate the transition of pattern formation in u(x,t) and v(x,t) from the previous cases. These spatiotemporal patterns behaviour is important and may have a physical meaning in reaction-diffusion phenomena.
(a)
(b)
FIGURE 1. Numerical solutions of (4) for α(x,t) = 2: (a) u(x,t), (b) v(x,t).
(a)
(b)
FIGURE 2. Numerical solutions of (4) for α(x,t) = 1.98: (a) u(x,t), (b) v(x,t).
5. CONCLUSION In this paper, a variable-order time-space fractional system was numerically studied on finite spatial and time domains. The temporal patterns behaviour in the steady-state solutions of this system when the fractional derivative function
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(a)
(b)
3 1 1 FIGURE 3. Numerical solutions of (4) for α(x,t) = + exp − 2 2 : (a) u(x,t), (b) v(x,t). 2 2 x t +1
(a)
(b)
3 1 1 FIGURE 4. Approximate solution of (4) for α(x,t) = + exp − 2 2 variable, at different times: (a) u(x, :), (b) v(x, :). 2 2 x t +1
1 3 1 index α(x,t) = + exp − 2 2 varies from 1.98 to 2. The system of fractional reaction-diffusion equations 2 2 x t +1 has proven useful in understanding the dynamics of nonlinear phenomena. The results derived from the variable-order fractional system are of a more general nature.
REFERENCES 1. R. Hifler, Applications of fractional calculus in physics, World Scientific, Singapore, 2000. 2. I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999. 3. F. Mainardi, A. Mura, G. Pagnini, and R. Gorenflo, In K. Tas et al. (Eds.), Mathematical Methods in Engineering (pp. 23-55), Springer-Verlag, Dordrecht, 2007. 4. R. Metzler, and J. Klafter, Phys. Rep., 339 (2000), 1-77.
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5. T. A. M. Langlands, and B. I. Henry, J. Comput. Phys., 205 (2005), 719-736 6. V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreire-Mejias, and H. R. Hicks, J. Comput. Phys., 192 (2003), 406-421. 7. M. Garg, and P. Manohar, Fract. Calc. Appl. Anal., 13 (2010), 191-207. 8. K. B. Oldham, and S. Spanier, Academic Press, San Diego, 1974. 9. J. Buceta, and K. Lindenberg, Physica A, 325 (2003), 230-242. 10. D. A. Garzon-Alvarado, and A. M. Ramirez-Martinez, Theor. Biol. Med. Model., 8 (2011), 24. 11. V. Gafiychuk, B. Datsko, V. Meleshko, and D. Blackmore, Chaos, Soliton and Fractals, 41 (2009), 1095-1104. 12. B. I. Henry, and S. L. Wearne, Physica A, 276 (2000), 448-455. 13. Y. Nikolova, and L. Boyadjiev, Fract. Calc. Appl. Anal., 13 (2010), 57-67. 14. K.K. Akil, S.V. Muniady and E. Lim, Malaysian J. of Fundamental and Applied Sciences, 8 (2012), 122-126. 15. FBM. Belgacem„ Elliptic boundary value problems with indefinite weights: Variational formulations of the principal eigenvalue and applications, Addison-Wesley-Longman, Pitman Research Notes Series Vol. New York, USA. 16. B. Baeumer, M. Kovacs, and M. M. Meerschaert, Comput. Math. Appl., 55 (2008), 2212-2226. 17. M. M. Khader, Commun. Nonlinear Sci., 16 (2011), 2535-2542. 18. FBM. Belgacem, Problems of Nonlinear Analysis in Engineering Systems, 2, (1999) 51-58. 19. FBM. Belgacem, Cambridge Sci. Pub., UK. Chap. 6, (2007) 51-60. 20. S. G. Samko and B. Ross, Integr. Transf. Spec. Funct. 1 (1993) 277-300. 21. L. Ramirez and C. Coimbra, Ann. Phys. 16 (2007), 543-552. 22. J.P. Roop, J. Comp. Appl and Math., 193 (2006) 243-268. 23. M. M. Meerschaert, and C. Tadjeran, Appl. Numer. Math., 56 (2006), 80-90. 24. P. Zhuang and F. Liu, J.Applied Math. and Comp, 22 (2006) 87-99. 25. C. Coimbra, Ann. Phys. (8) 12 (2003), 692-703. 26. R. Lin, F. Liu, V. Anh and I. Turner, Applied Mathematics and computation, 212, (2009) 435-445. 27. A.A.M. Arafa, S.S. Rida and H. Mohamed, Applied Mathematia Modelling, (2012).
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Numerical simulations for a Variable Order Fractional Schnakenberg model Z.Hammouch and T.Mekkaoui E3MI, Département de Mathématiques, FST Errachidia, Université Moulay Ismail BP.509 Boutalamine 52000 Errachidia, Morocco Abstract. This paper is concerned with the numerical solutions of a variable-order space-time fractional reaction-diffusion model. The space-time fractional derivative is considered in the sense of Riesz-Feller, the system is defined by replacing the second order space derivatives with the variable Riesz-Feller derivatives. The problem is solved by an explicit finite difference method. Finally, simulation results to this problem are presented and discussed. Keywords: Explicit finite difference, Variable-order Riesz-feller derivative, Schnakenberg model. PACS: 87.10.Ed, 87.10.Ed, 87.15.Vv.
1. INTRODUCTION Fractional calculus is an old mathematical topic from 17th century. However it can be considered as a new topic of research. Actually, in the last few years Fractional Order Partial Differential Equations (FOPDEs) have played an important role in applied mathematics, physics, engineering, economics and other science [1]-[8]. The dynamic models of a large number of phenomena can be modeled by FOPDEs which are characterized by fractional space and/or time derivatives. Because the fractional derivative of a function depends on the values of the function over the entire interval, it is suitable for describing the memory effect and modeling of the systems with long range interactions both in space and time. Besides, systems of fractional reaction-diffusion (SFRD) have gained considerable attention to study nonlinear phenomena arising in the disciplines of science and engineering [9]-[13]. Of these particular interests are patterns formation [2], since it is best described by the fractional-order models because the fractional derivatives take into consideration the whole memory of the system [14]. On the other hand, numerical methods yield approximate solutions to the governing equations through the discretization of space and time [15]-[21]. The finite difference method is the most classic method for fractional differential equations. It has been extensively considered in the literature for constant order time or space fractional diffusion equations; Roop [22] investigated the numerical approximation of the variational solution to the fractional advection-dispersion equation, Meerschaert [23] examined finite-difference approximations for fractional advection-dispersion flow equations, Zhuang and Li [24] analyzed an implicit finite difference approximation for the time-fractional diffusion equation. However, variable-order fractional partial differential equations is relatively new topic, there is a growing number of papers on numerical approximation of these equations, Coimbra [25] proposed a first-order accurate approximation for the solution of variable-order differential equations. Lin et al. [26] proposed a new explicit finite-difference method for solving variable-order fractional partial differential equations with Riesz-Feller derivative. The main purpose of this paper is to consider a variable-order fractional version of the well-known Schnakenberg model. since the behaviour of chemical species may change in time and space. Recently, it was found that the variational order derivative is very useful to describe efficiently such situation. We then modify the Schnakenberg model by replacing the partial derivative with respect to space by the variable-order derivative. The resulting problem is very difficult to handle analytically. We solved it numerically via an explicit finite difference scheme. The structure of the remainder of this paper is as follows: In Section 2, some basic mathematical definitions from fractional calculus are given. In Section 3, the detailed description and formulation of the problem is presented. In Section 4, we present the numerical simulations of the solutions behaviour. Finally, we conclude our work in Section 5.
10th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences AIP Conf. Proc. 1637, 1450-1455 (2014); doi: 10.1063/1.4907312 © 2014 AIP Publishing LLC 978-0-7354-1276-7/$30.00
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2. VARIABLE-ORDER RIESZ FRACTIONAL DERIVATIVE There are some preferences dealt with in the definition of fractional derivatives such as Riemann-Liouville, Caputo, Riesz, Weyl, Grunwald-Letnikov, Coimbra etc. It is noted that each type of fractional derivative has different properties and applications. For the readers that are not familiar with the variable-order derivative Riesz-Feller, we recall in this section its basic definition. Assume first that: 1 ≤ α(x,t) < 2, Xa ≤ x ≤ Xb , 0 ≤ t ≤ T and u(x,t) = 0 when x ≤ Xa or x > xb . According to [26] the generalized Riesz fractional derivative is defined by Z Z − sec(α(x,t) π2 ) d 2 θ u(ξ ,t)dξ d 2 xb u(ξ ,t)dξ α(x,t) ρ + σ , (1) R u(x,t) = x Γ(2 − α(x,t)) dθ 2 xa (θ − ξ )α(x,t)−1 dθ 2 θ (ξ − θ )α(x,t)−1 θ =x where ρ ≥ 0, σ ≥ 0 and ρ + σ = 1. Remark 2.1 When ρ = σ = 0.5, the above equation defines the variable-order Riesz fractional derivative and when ρ = 1 and σ = 0 we recover the variable-order Riemann-Liouville derivative multiplied by the coefficient cos(α(x,t) π2 ).
3. THE VOF-SCHNAKENBERG MODEL Reaction-diffusion equations are useful in many areas of science and engineering. In applications to population biology, the reaction term models growth, and the diffusion term accounts for migration [27]. Many of these mathematical models are a generalized reaction-diffusion system of two chemical species which can be written in terms of nondimensional variables as ∂U = D∆U + kF(U), in Ω(t), (2) ∂t where Ω(t) is a time-dependent domain, and u f (u, v) du 0 U= , F= , D= , v g(u, v) 0 dv where u, v are the two chemical species concentrations, f and g are reaction kinetics, k is a reaction parameter, is the diffusion matrix. The Schnakenberg model is one of the simplest reaction-diffusion models, it describes the behavior of a chemical activator u in the presence of a chemical inhibitor v. It has been used in stability analysis applications and pattern formation, allowing the prediction of the interaction mechanisms between molecular chemical systems and morphogenic constructions such as bone formation and growth [27]. It is related to Meinhardt-Gierer’s substrate depletion model, but with a simpler nonlinearity. The 1D model is described by the following equations [27]: ∂ u ∂ 2u 2 ∂t = ∂ x2 + γ(a − u + u v), (3) 2v ∂ v ∂ 2 = d 2 + γ(b − u v). ∂t ∂x In Eq. (3), the movement of both the activator and the inhibitor substances is described as a diffusive term with an equivalent diffusion constant given by the coefficient d. The reaction term for the activator substance Eq. (3)1 is described as a constant production given by the parameter a, a linear consumption, and a nonlinear kinetic reaction representing the production of the activator u in presence of the inhibitor v. On the other hand, the reaction term for the inhibitor substance Eq. (3)2 is given as a constant production with coefficient b and a nonlinear kinetic reaction for the consumption of the inhibitor substance in the presence of the activator, and γ is a positive dimensionless constant [27]. In this work we consider a more general version of (3): ∂ u = x Rα(x,t) u(x,t) + γ(a − u + u2 v) ∂t (4) ∂ v α(x,t) 2 = dxR v(x,t) + γ(b − u v). ∂t
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where (x,t) ∈ Ω = [−L, L] × [0, T ] and x Rα(x,t) is a generalized Riesz fractional derivative of time-space dependent order α(x,t) (1 < α(x,t) ≤ 2). First, we consider the linear stability of (4). The spatially homogeneous steady state solutions of (4) satisfying a − u + u2 v = 0, (5) b − u2 v = 0. are
(u, v) = a + b,
b (a + b)2
.
(6)
According to the standard reaction-diffusion theory, the stability of the homogeneous steady state (6) can be determined γ(1 + 2uv) γu2 from its Jacobian matrix J = . At the homogeneous steady states the determinant and the trace −2γuv −γu2 of J are det(J) = γ 2 (a + b)2 , (7) ba Tr(J) = a+b − (a + b)2 . Hence the stability of the homogeneous steady state to homogeneous perturbations will occur when b − a < (a + b)3 .
4. NUMERICAL SIMULATIONS BY AN EXPILICIT FINITE DIFFERENCE SCHEME This section describes the numerical technique to solve the system (4) subject to periodic boundary conditions by an explicit finite difference as described by Lin et al [26]. The grid points in the space domain [−L, L] and the time domain [0, T ] are labeled : xi = ih for i = 0, ..., m and t j = jτ for j = 0, ..., n − 1, respectively. The Grunwald formula for the α(x,t)−fractional derivative approximation [26] gives 2 Z θ j j i+1 d u(ξ ,t)dξ 1 −αi+1 −αi+1 ≈ h ω u(xk ,t j ), ∑ i+1−k 2 Γ(2 − α(xi ,t j )) dθ xa (θ − ξ )α(x,t) − 1 θ =xi k=0 (8) 2 Z θ m j j 1 d u(ξ ,t)dξ −α i−1 u(xk ,t j ), ≈ h−αi−1 ∑ ω(k−i)−1 Γ(2 − α(x ,t )) dθ 2 xa (θ − ξ )α(x,t) − 1 θ =x i+1 j k=i−1 i+1 j
where
α ωk i
j αi = (−1) and αij = α(xi ,t j ). Substitution of (8) into (4) yields k k
! m j j j j i+1 2 αi−1 αi+1 j j j j j j j+1 −α −α ui = ui + ai τ ραi h i+1 ∑ ωi+1−k uk + σ αi h i−1 ∑ ω−i+1+k uk + γ(a − uij + uij vij ), k=0 k=i−1 j j j i+1 αi+1 j+1 j j j −αi+1 j j −αi−1 v = v + a τd ρα h ω ∑ i i i i+1−k vk + σ αi h i k=0
m
∑
j
αi−1 ω−i+1+k vkj
!
(9)
j2
+ γ(b − ui vij ),
k=i−1
where aij = − sec(α(xi ,t j )). The stability and the accuracy of the method have been discussed in [26]. The numerical scheme (9) represents a system of algebraic equations with block diagonal matrix. The computer code for numerical scheme (4), was written in Matlab software. The time and spatial steps used in the simulation were variying from 0 to 0.8 and from −1 to 1 respectively. The initial conditions are u(x, 0) = 2 − 10−5 cos(2πx) + 10−2 cos(76πx) + 10−2 cos(78πx), (10) (−5) v(x, 0) = 1 + 10 cos(2πx),
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and the boundary conditions are u(0,t) = u(1,t) = 2.02,
(11)
v(−1,t) = v(1,t) = 1. The constants a, b, d and γ are taken as −2, 4, 1 and 5 respectively. Figures 1 shows the results of computer solution of (4) considered above, for α(x,t) = 2 (the standard Schnakenberg model). Figure 2 depicts the numerical solutions 1 3 1 , Figure 3 u(x,t) and v(x,t) for the constant fractional case (α(x,t) = 1.98) and for α(x,t) = + exp − 2 2 2 2 x t +1 and Figure 4 illustrate the transition of pattern formation in u(x,t) and v(x,t) from the previous cases. These spatiotemporal patterns behaviour is important and may have a physical meaning in reaction-diffusion phenomena.
(a)
(b)
FIGURE 1. Numerical solutions of (4) for α(x,t) = 2: (a) u(x,t), (b) v(x,t).
(a)
(b)
FIGURE 2. Numerical solutions of (4) for α(x,t) = 1.98: (a) u(x,t), (b) v(x,t).
5. CONCLUSION In this paper, a variable-order time-space fractional system was numerically studied on finite spatial and time domains. The temporal patterns behaviour in the steady-state solutions of this system when the fractional derivative function
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(a)
(b)
3 1 1 FIGURE 3. Numerical solutions of (4) for α(x,t) = + exp − 2 2 : (a) u(x,t), (b) v(x,t). 2 2 x t +1
(a)
(b)
3 1 1 FIGURE 4. Approximate solution of (4) for α(x,t) = + exp − 2 2 variable, at different times: (a) u(x, :), (b) v(x, :). 2 2 x t +1
1 3 1 index α(x,t) = + exp − 2 2 varies from 1.98 to 2. The system of fractional reaction-diffusion equations 2 2 x t +1 has proven useful in understanding the dynamics of nonlinear phenomena. The results derived from the variable-order fractional system are of a more general nature.
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