A Meshless Simulations for 2D Nonlinear Reaction-diffusion Brusselator System

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1 Copright 3 Tech Science Press CMES, vol.95, no.4, pp.59-8, 3 A Meshless Simulations for D Nonlinear Reaction-diffusion Brusselator Sstem Ahmad Shirzadi, Vladimir Sladek, Jan Sladek 3 Abstract: This paper is concerned with the development of a numerical approach based on the Meshless Local Petrov-Galerkin (MLPG method for the approimate solutions of the two dimensional nonlinear reaction-diffusion Brusselator sstems. The method uses finite differences for discretizing the time variable and the moving least squares (MLS approimation for field variables. The application of the weak formulation with the Heaviside tpe test functions supported on local subdomains (around the nodes used in MLS approimation to semi-discretized partial differential equations ields the finite-volume local weak formulation. A predictor-corrector scheme is used to handle the nonlinearit of the problem within each time step. Numerical test problems are given to verif the accurac of the proposed method. Under particular conditions, this sstem ehibits Turing instabilit which results in a pattern forming instabilit. This concept is studied and a test problem is given. Kewords: Meshless Local Petrov-Galerkin method; Moving least squares; Finite differences; Nonlinear problems; Brusselator equations; Turing instabilit; Pattern formation. Introduction In 95 Turing showed [Turing (95] that a simple mathematical model describing self-diffusing and reacting chemicals could give rise to stationar spatial concentration patterns of fied characteristic length from a random initial configuration and proposed that reaction-diffusion models might have relevance in describing morphogenesis, the growth of biological form. In the late 96s, IIa Department of Mathematics, Persian Gulf Universit, Bushehr, Iran. shirzadi@pgu.ac.ir, shirzadi.a@gmail.com Institute of Construction and Architecture, Slovak Academ of Sciences, 8453 Bratislava, Slovakia. vladimir.sladek@savba.sk 3 Institute of Construction and Architecture, Slovak Academ of Sciences, 8453 Bratislava, Slovakia. jan.sladek@savba.sk

2 6 Copright 3 Tech Science Press CMES, vol.95, no.4, pp.59-8, 3 Prigogine among others at the Universit of Brussels developed the Brusselator model [Prigogine and Lefever (968;Nicolis and Prigogine (997] which is one of the simplest reaction-diffusion models ehibiting Turing instabilit. From the mathematical point of view, the Brusselator is a sstem of two coupled nonlinear partial differential equations. The linear stabilit analsis can be used for predicting the parameter values that result in the Turing instabilit in a particular reaction-diffusion sstem. Based on the linear approimation one can also effectivel predict the characteristic length of the resulting pattern. Linear analsis cannot, however, predict the spatial characteristics of the resulting patterns, since the pattern selection is governed b comple nonlinear dnamics. Thus some nonlinear analsis is needed. Since analtical solutions are unavailable, finding efficient numerical methods for solving these kind of problems has been an active area of research. Meshless methods as a tool for numerical simulations of linear or nonlinear problems, especiall partial differential equations have become popular in recent ears. This paper aims to present a new meshless method for the numerical simulations of the Brusselator equations. The proposed method is based on the MLPG formulation [Atluri and Zhu (998a;Atluri and Zhu (998b;Atluri, Kim, and Cho (999;Atluri (4;Atluri and Shen (] which uses the local weak form equations and the MLS approimation to transform the model equations to the final sstem of algebraic equations. An application of the MLPG method for numericall solving coupled pair of nonlinear diffusion equations can be found in [Abbasband, Sladek, Shirzadi, and Sladek (]. For a review of applications of the MLPG method in engineering and sciences the readers are referred to [Sladek, Stanak, Han, Sladek, and Atluri (3]. Analsis of the convergence of the MLPG method is well studied in [Shirzadi and Ling (3]. The non-linear reaction-diffusion Brusselator sstem is given b: u t v t ( = A + u u v (B + u + D u + u ( = Bu u v v + D v + v, in the two-dimensional region Ω = [a,b], with initial condition, ( u(,, = f (,, ( v(,, = g(,, and Dirichlet boundar condition: u(,t = q (,t, Ω t, t >, v(,t = q (,t,

3 Meshless Simulations for Brusselator Sstem 6 or subject to Neumann s boundar conditions on the boundar Ω. Recall that A denotes the value of the source term and B is the bifurcation parameter setting the distance to the onset of instabilit. The chemical u is the activator concentration and v is the inhibitor. D u and D v are diffusion coefficients, f, g, q and q are known functions. Some eisting papers on the stud of this problem were concerned with theoretical concepts such as smmetr-breaking and bifurcations [Nicolis and Prigogine (997;Walgraef, Dewel, and Borckmans (98]. The theor b Turing [Turing (95] predicts that a spatiall uniform stationar state that is stable against perturbations in the absence of diffusion might become unstable against perturbations in the presence of diffusion resulting in a pattern forming instabilit. Some papers on numerical studies of this model emploed the computational approach and addressed the problem of pattern selection as a function of sstem parameters in two-dimensional and three-dimensional sstems [Verdasca, Wit, Dewel, and Borckmans (99;Borckmans, Wit, and Dewel (99;Leppänen (4]. For papers which deal with the numerical solution of the reactiondiffusion Brusselator sstem see Twizell et al. [Twizell, Gumel, and Cao (999] for a second order finite difference method, Adomian [Adomian (995] for decomposition method, Whe-Teong [Ang (3] for the dual-reciprocit boundar element method, S-u. Islam et al. [Siraj-ul, Arshed, and Sirajul (] for a numerical scheme based on RBF collocation method, Mittal and Jiwari [Mittal and Jiwari (] for polnomial based differential quadrature method, Kamranian et al. [Kamranian, Tatari, and Dehghan (] for the finite point method and so on. The remaining content of the paper is organized as follows. Section describes the basics of the MLS approimation used in this stud. The time discretization approach is presented in Section 3. Construction of local weak equations are illustrated in Section 4 and the obtained local weak equations are discretized in Section 5. The predictor-corrector scheme which is proposed to handle the nonlinear terms is also illustrated in Section 5. To simplif the evaluation of local integrals, certain techniques are described in Section 6. Two test problems are given in Section 7 to verif the accurac of the method. To address the problem of pattern selection as a function of sstem parameters, another test problem is given at the end of Section 7. Our conclusions are summarized in the last section. The MLS approimation Consider a sub-domain Ω, the neighborhood of a point which is located in the problem domain Ω. To approimate the distribution of function u in Ω over a number of randoml located nodes i,i =,,...n, the MLS approimant u h ( of u, Ω, is defined b u h ( = p T ( a( Ω, (3

4 6 Copright 3 Tech Science Press CMES, vol.95, no.4, pp.59-8, 3 where p T ( = [p (, p (,..., p m (] is a complete monomial basis of order m, and a( is a vector containing coefficients a j (, j =,,...m which are functions of the space coordinates. For eample, for a -D problem, p T ( = [,,] and p T ( = [,,,,, ], for linear basis (m = 3 and quadratic basis (m = 6, respectivel. The coefficient vector a( is determined b minimizing a weighted discrete L norm, defined as J( = n i= w i ([p T ( i a( û i ] (4 = [P.a( û] T.W.[P.a( û], where w i ( is the weight function associated with the node i, with w i ( > for all in the support of w i (, i denotes the value of at node i, n is the number of nodes in Ω for which the weight functions w i ( >, the matrices P and W are defined as p T ( P = p T (... p T ( n n m w (..., W = w n ( and û T = [û,û,...,û n ] where û i, i =,,...,n in (4 are the fictitious nodal values, and not the nodal values of the unknown trial function u h ( in general. In this work the Gaussian weight function is used: w i ( = ep[ ( d i c i ] ep[ ( r i ep[ ( r i c i ] c i ], d i r i,, d i r i, where d i = i, c i is a constant controlling the shape of the weight function w i and r i is the size of the support domain. The stationarit of J in (4 with respect to a( leads to the following linear relation between a( and û, A(a( = B(û, (5 where the matrices A( and B( are defined b A( = P T WP = B(P = n i= w i (p( i p T ( i, B( = P T W = [w (p(,w (p(,...,w n (p( n ].

5 Meshless Simulations for Brusselator Sstem 63 The MLS approimation is well defined if and onl if the rank of P equals m. A necessar condition for a well-defined MLS approimation is that at least m weight functions are non-zero (i.e. n > m for each sample point Ω and that the nodes in Ω will not be arranged in a special pattern such as on a straight line. Solving for a( from (5 and substituting it into (3 gives a relation which ma be written as the form of an interpolation function similar to that used in FEM, as u h ( = Φ T (.û = n i= and essentiall u i û i, where Φ T ( = p T (A (B(, or φ i ( = m p j ([A (B(] ji. j= φ i (û i ; u h ( i u i ; Ω, (6 φ i ( is called the shape function of the MLS approimation corresponding to nodal point i. The support of the nodal point i is usuall taken to be a circle of radius r i, centered at i. The fact that φ i ( =, for not in the support of nodal point i preserves the local character of the Moving Least Squares approimation. The partial derivatives of φ i ( are obtained as φ i,k = m j= [p j,k (A B ji + p j (A B,k + A,k B ji], (7 in which A,k = (A,k represents the derivative of the inverse of A with respect to k, kth coordinate of, which is given b A,k = A A,k A, where (,i denotes ( / i. 3 The finite difference approimation The finite-difference approimation of the time derivatives in the θ method is given as follows θ u k+ + ( θ u k = u k+ u k, θ. (8 t Considering Eq. ( at the time instants k t and (k + t, one obtains, respectivel θ u k+ = θd u u k+ + θa + θ(u k+ v k+ θ(b + u k+ ( θ u k = ( θd u u k + ( θa + ( θ(u k v k ( θ(b + u k

6 64 Copright 3 Tech Science Press CMES, vol.95, no.4, pp.59-8, 3 Hence and from (8, we have u k+ u k t ( = D u u k + θd u u k+ u k + A + θ(u k+ v k+ (9 + ( θ(u k v k (B + (θu k+ + ( θu k Using Crank-Nicholson scheme (θ = Eq. (9 becomes: u k+ u k t = D u ( u k + u k+ + A + ((u k+ v k+ + (u k v k ( (B + u k+ + u k or [ + t (B + D t u ] u k+ t (u k+ v k+ [ = t (B + + D t u ] u k + t (u k v k + ta. Similarl, we have [ t D v ] v k+ t Bu k+ + t (u k+ v k+ [ t = + D v ] v k + t Bu k t ( (u k v k Therefore, the time dependent PDEs are transformed to the semi-discrete PDEs of the elliptic tpe for the field variables u k+ and v k+, assuming the fields u k and v k being known from the computation in the previous time step. To treat the nonlinearit, a predictor-corrector scheme is proposed in this paper. 4 The local weak form In this paper, we construct the weak form over local sub-domains such as, located entirel inside Ω which is a circle of radius r and centered at node i. The local weak form of the equations ( and ( for i = ( i, i can be written as [[ + t t ] u k+ t (u k+ v k+ ] u d = (B + D u [[ t (B + + D t u ] u k + t ] (u k v k + ta u d, ( (

7 Meshless Simulations for Brusselator Sstem 65 [[ t D v ] v k+ t Bu k+ + t ] (u k+ v k+ u d = [[ t + D v ] v k + t Bu k t ] (u k v k u d where u and v are trial functions and u is a test function. If the Heaviside step function,, u ( =, /, (3 is chosen as the test function in each sub-domain, then u = and the local weak forms ( and (3 are transformed into the following equations ( + t t + D u t (B + u k+ d D u t (u k+ v k+ d = u k n ds + t ( t u k+ n ds (B + u k d ( (u k v k + A d, (4 + t t v k+ d D v t v k+ n ds ((u k+ v k+ Bu k+ d = ((u k v k Bu k d v k d + D v t v k n ds (5 5 Discretized equations and the predictor-corrector scheme We consider N nodal points, some of them located on the boundaries and some in the domain. Let u k ( = u(,k t be approimated according to (6 with denoting the nodal unknowns as û k,i. In each time step, our aim is to compute û k+,i, i =,,...,N assuming û k,i, i =,,...,N being known from the previous time step. Since there are nonlinear terms in the equations, we propose a predictor-corrector scheme in each time step to obtain a desired accurac. So we assume that û (l k+ is an approimation of û k+ after l correction step and û k+ is identified with the value û ( f k+ obtained in the final correction. We will begin from û( k+ = û k in nonlinear term. For internal nodes, from (4, (5 and using the MLS approimation (6, we

8 66 Copright 3 Tech Science Press CMES, vol.95, no.4, pp.59-8, 3 have the following linear equations: ( + t t + D u t (B + N φ j d û j=( (l+ k+, j D u t (ũ (l k+ ( ṽ (l k+ (d = ( t (B + N ( φ j j= n ds û (l+ k+, j N j=( φ j d û k, j N ( φ ( j j= n ds û k, j + t (ũ k( ṽ k ( + A d, N ( φ j d ˆv (l+ k+, j D v t j= + t N ( = j= t where ( (ũ (l k+ ( ṽ (l k+ ( d B t φ j d ˆv k, j + D v t ( (ũ k ( ṽ k ( Bũ k ( N ( φ j j= n ds ˆv (l+ k+, j N ( φ j j= d, N ( φ j (d û (l+ k+, j j= n ds ˆv k, j (6 (7 ũ k ( = N φ j (û k, j = j= N φ j (û ( f k, j, ṽ k( = j= N φ j ( ˆv k, j = j= N φ j ( ˆv ( f k, j, j= ũ (l N k+ = φ j (û (l k+, j, j= N ṽ(l k+ = φ j ( ˆv (l k+, j. j= The Dirichlet boundar conditions are imposed directl using the equations of the boundar conditions and the MLS approimation (6. To impose the Neumann s boundar conditions, we adopt a method proposed in Abbasband and Shirzadi (. 6 Computational techniques The MLS approimation with Gaussian weight function and quadratic basis (m = 6 are used in all test problems. We have to deal with two kinds of integrals in the discretized local integral equations (6 and (7. Basicall, both the integrations over the circular sub-domain and/or its boundar can be performed analticallsladek and Sladek (; Sladek, Sladek, and Zhang ( with saving the

9 Meshless Simulations for Brusselator Sstem 67 computational time substantiall because of eliminating the evaluation of the shape functions and/or its derivatives at the integration points. Then, we could write g(d = πr g( i + π 8 r4 g,kk ( i + O(r 6, g(ds = πr g( i + π r3 g,kk ( i + O(r 5, g n (ds = πr g,kk ( i + O(r 4. In order to avoid the inaccurac of the second and higher-order derivatives of the shape functions in meshless approimations, we prefer a numerical integration over. Since the number of the integration points in the domain integrals is much higher than in the contour integrals, the analtical integration in case of domain integrals ields more substantial savings in the computational time. Therefore we appl the analtical integrations over with omitting the terms O(r 4 involving the derivatives of the shape functions. Thus summarizing, the required integrals over are computed b using the regular Gauss-Legendre quadrature rule with seven integration points as π = φ j (ds φ j ( i + r cos(θ, i + r sin(θr dθ = πr φ j ( i + r cos(πθ + π, i + r sin(πθ + πdθ πr 7 p=w p φ j ( i + r cos(πθ p + π, i + r sin(πθ p + π where w p and θ p are the Gauss quadrature integration rule weights and points on [-, ]. The same technique can be used for evaluation of contour integrals of the normal derivatives of the shape functions n k (φ i,k ( with utilizing Eq. (7 for the gradients of the shape functions. Finall, the required integrals over sub-domains are approimated as follows: φ j (d πφ j ( i r.

10 68 Copright 3 Tech Science Press CMES, vol.95, no.4, pp.59-8, 3 and similarl, the domain integrals involving the nonlinear terms are approimated as: (ũ (l k+ ṽ (l k+d π(ũ(l k+ (i ṽ (l k+ (i r, ( (ũ k ṽ k + A d π( (ũ k( i ṽ k ( i + A r, ( (ũ (l k+ ( ṽ (l ( π (ũ (l k+ (i ṽ (l N ( k+ d ( B j= k+ (i B ((ũ k ṽ k Bũ k d π N j= φ j ( i û (l+ k+, j φ j (d û (l+ k+, j r ( (ũ k ( i ṽ k ( i Bũ k ( i r. Recall that the assumed approimations for the domain integrals are meaningful provided that the radius of the sub-domains is sufficientl small. 7 Test problems The domain and boundar integrals are evaluated as demonstrated in the previous section. Because of the computational techniques described in the previous section, the radius of each local sub-domains, r, should be small enough. A ver small r also causes much cancelation error and therefore, it is chosen as.5h r.h. The shape parameter of the Gaussian weight function is chosen as c i.6h, and the radius of the support of the weight function corresponding to node i, r i, is chosen as r i 7h where h is the minimum distance between nodes. For the predictor corrector scheme, 3 or 4 corrections are calculated in each time step. 7. Eample. For the first test problem consider the reaction diffusion Brusselator Sstem ( with D u = D v = 4, B = and A = in the region Ω = {(, :, } and the boundar condition etracted from the eact solution given b: u(,,t = ep( t, v(,,t = ep( t + +. In this eample, the relative error which will be reported is defined as: N i= e u R = (u i ū i N i= (u, i

11 Meshless Simulations for Brusselator Sstem 69 and the infinit-norm are defined as: e u = Ma{ u i ū i, i =,,...,N}, u i and ū i are the eact and approimate value of u at point i, respectivel, and N is the number of nodes. In order to show the convergence of the proposed method with respect to the spatial variables, Table presents the results obtained with different number of nodal points and fied time step t =. at time instant t =. B going through each column of Table, one can see increasing accurac with increasing the number of nodal points. Table : The results obtained with different number of nodal points, t =. at t =. N e u e v e u R e v R The results obtained at time instant t = 4 with using N = 44 nodal points and different size of time step t are presented in Table. B going through each column of Table, one can see increasing accurac with decreasing the size of the time step t. Table 3 presents the results obtained at different time instants with Table : Results obtained at t = 4 with using N = 44 nodal points and various time steps t t e u e v e u R e v R using N = 44 nodal points and the time step t =.. B going through each column of Table 3, we can see increasing accurac of u and decreasing accurac of v. B the finite differences applied to discretize the time variable, the behavior of v is natural. The reason of increasing the accurac of u b increasing time variable t is cancelation error; b increasing t, u and ver small values of u causes cancelation error.

12 7 Copright 3 Tech Science Press CMES, vol.95, no.4, pp.59-8, 3 Table 3: Results obtained with using time step t =., N = 44 nodal points at different time instants t e u e v e u R e v R Error profiles of u and v at t=4 obtained b using N = 44 number of nodal points and t =. are presented in Figures and. 5 Error of u Figure : Error profiles of u with N = 44 and t =. at t = Eample. For the second test problem, we consider the reaction diffusion Brusselator Sstem ( with D u = D v =., B = and A = in the region Ω = {(, :

13 Meshless Simulations for Brusselator Sstem 7 Error of v Figure : Error profiles of v withn = 44 and t =. at t = 4, } with the initial and Neumann s boundar conditions given b: u(,, = 3 3, v(,, = 3 3, u(,,t = u(,,t = u(,,t = u(,,t =, = = = = v(,,t = v(,,t = v(,,t = v(,,t =, = = = = for which the eact solution is unknown. For small values of the diffusion coefficients D u and D v, if B + A > then the stead state solution of the Brusselator sstem ( converges to equilibrium point (A, B/A. This is valid for Eample and is confirmed b the numerical solutions. Here, the relative error which will be reported is defined as: e u R = N i= (ūn i ū N i N, i= (ūn i

14 7 Copright 3 Tech Science Press CMES, vol.95, no.4, pp.59-8, 3 and the infinit-norm are defined as: e u = Ma{ ū N i ū N i, i =,,...,N }, where ū N i is the approimation of u at node i obtained with using N nodal points and { i, i =,,...,N } { i, i =,,...,N }. The numerical solution obtained at some points with using N = 44 nodal points and t =. at different time instants are presented in Table 4. This table confirms that (u, v (A, B/A when t. The results obtained at time instant t = 4 with using N = 44 and N = Table 4: The solution obtained at some points. (.,. (.4,.6 (.5,.5 (.8,.9 t u v u v u v u v nodal points and different time step t are presented in Table 5. From this table, it can be seen that b decreasing the size of the time step t, the accurac of the numerical solution increases. Table 5: Results obtained at t = 4, with different time step t, N = 44 and N = points. t e u e v e u R e v R Initial profiles of u and v are presented in Figures 3 and 4. Plots of the numerical solutions obtained for u and v at t = b using N = 44 nodal points and t =

15 Meshless Simulations for Brusselator Sstem 73. are presented in Figures 5 and 6. Figures 7 and 8 presents the numerical solution at point (.5,.5 versus time t. From these figures we can see that at ever point of the domain the solution (u,v tends to (A,B/A...5 u(,, Figure 3: Initial concentration profile of u...5 v(,, Figure 4: Initial concentration profile of v.

16 74 Copright 3 Tech Science Press CMES, vol.95, no.4, pp.59-8, 3 u(,, Figure 5: The solution u at t =, N = 44, t = v(,, Figure 6: The solution v solutions at t =, N = 44, t =..

17 Meshless Simulations for Brusselator Sstem u(.5,.5,t obtained b MLPG Figure 7: Plot of u(.5,.5, t versus time t with N = 44, t = v(.5,.5,t obtained b MLPG Figure 8: Plot of v(.5,.5, t versus time t with N = 44, t = Eample 3. Pattern formation in Brusselator sstem Consider the Brusselator equation ( with no flu boundar conditions. The threshold for the Turing instabilit is B T c = [ + A D u /D v ]. The wave number cor-

18 76 Copright 3 Tech Science Press CMES, vol.95, no.4, pp.59-8, 3 u, t= v, t= Figure 9: u and v at t= responding to the most unstable Turing mode is given b kc = A/ D u D v. The corresponding characteristic length of the pattern is defined b λ c = π k c. This test problem considers ( with A = 4.5 and D v D u = 8 resulting in B T c 6.7. The pa-

19 Meshless Simulations for Brusselator Sstem 77 u, t= v, t= Figure : u and v at t=5 rameter B is chosen as B = 6.75 resulting in spot patterns as shown in Leppänen (4. The domain of the problem is chosen as [,] and the diffusion coefficients are fied with D v =. D u =.5 resulting in λ c.79. The initial

20 78 Copright 3 Tech Science Press CMES, vol.95, no.4, pp.59-8, 3 u, t= v, t= Figure : u and v at t= configuration was random perturbations around the uniform stationar state. Figures 9- presents the process of the pattern formation for the considered model. In all contour graphs, coloration is determined b a constant threshold value, u s for

21 Meshless Simulations for Brusselator Sstem 79 u, t= v, t= Figure : u and v at t=4 u and v s for v, such that in the regions with white color, u < u s and v < v s while the regions which eperience the concentration u > u s and v > v s are colored with green. The results confirm that the profiles of the function v are alwas 8 out of phase to those of u.

22 8 Copright 3 Tech Science Press CMES, vol.95, no.4, pp.59-8, 3 8 Conclusions A numerical method for solving the two dimensional nonlinear reaction-diffusion Brusselator sstem was presented. The time variable was discretized b using onestep θ method. Then the resultant elliptic tpe PDEs were considered in weak forms on the local subdomains and the MLS approimation was used to approimate the field variables. The Heaviside step function was used as test function in each local subdomain. The nonlinear terms were treated b using the predictor corrector scheme. Test problems reveal that the method is of high accurac and stabilit. A model sstem ehibiting Turing instabilit was given as a test problem and the resulting patterns were presented. Acknowledgements The financial supports b APVV-3- (Slovak Research and Development Agenc is greatl acknowledged. References Abbasband, S.; Shirzadi, A. (: MLPG method for two-dimensional diffusion equation with Neumann s and non-classical boundar conditions. Appl. Numer. Math., vol. 6, pp Abbasband, S.; Sladek, V.; Shirzadi, A.; Sladek, J. (: Numerical simulations for coupled pair of diffusion equations b MLPG method. CMES Compt. Model. Eng. Sci., vol. 7, no., pp Adomian, G. (995: The diffusion-brusselator equation. Computers and Mathematics with Applications, vol. 9, no. 5, pp. 3. Ang, W.-T. (3: The two-dimensional reaction-diffusion brusselator sstem: a dual-reciprocit boundar element solution. Eng. Anal. Bound. Elem., vol. 7, no. 9, pp Atluri, S. (4: The meshless method (MLPG for domain and BIE discretizations. Tech Science Press. Atluri, S.; Kim, H.-G.; Cho, J. Y. (999: A critical assessment of the trul meshless local Petrov-Galerkin (MLPG and local boundar integral equation (LBIE methods. Comput. Mech., vol. 4, pp Atluri, S.; Shen, S. (: Method. Tech Science Press. The Meshless Local Petrov-Galerkin (MLPG Atluri, S.; Zhu, T. (998: A new meshless local Petrov-Galerkin (MLPG approach in computational mechanics. Comput. Mech., vol., no., pp. 7 7.

23 Meshless Simulations for Brusselator Sstem 8 Atluri, S.; Zhu, T. (998: A new meshless local Petrov-Galerkin (MLPG approach to nonlinear problems in computer modeling and simulation. Comput. Modeling Simulation in Engrg., vol. 3, pp Borckmans, P.; Wit, A. D.; Dewel, G. (99: Competition in ramped turing structures. Phsica A Statistical Mechanics and its Applications, vol. 88, no. -3, pp Kamranian, M.; Tatari, M.; Dehghan, M. (: The finite point method for reaction-diffusion sstems in developmental biolog. CMES Compt. Model. Eng. Sci., vol. 8, no., pp. 7. Leppänen, T. (4: Computational Studies of Pattern Formation in Turing Sstems. PhD thesis, Helsinki Universit of Technolog, Espoo, Finland, 4. Mittal, R.; Jiwari, R. (: Numerical solution of two-dimensional reactiondiffusion brusselator sstem. Appl. Math. Comp., vol. 7, no., pp Nicolis, G.; Prigogine, I. (997: Self-organization in nonequilibrium sstems: from dissipative structures to order through fluctuations. John Wile & Sons. Prigogine, I.; Lefever, R. (968: Smmetr breaking instabilities in dissipative sstems.ii. J. Chem. Phs., vol. 48, pp Shirzadi, A.; Ling, L. (3: Convergent overdetermined-rbf-mlpg for solving second order elliptic PDEs. Adv. Appl. Math. Mech., vol. 5, pp Siraj-ul, I.; Arshed, A.; Sirajul, H. (: A computational modeling of the behavior of the two- dimensional reaction-diffusion brusselator sstem. Appl. Math. Model., vol. 34, no., pp Sladek, J.; Stanak, P.; Han, Z. D.; Sladek, V.; Atluri, S. N. (3: Applications of the MLPG method in engineering & sciences: A review. CMES Compt. Model. Eng. Sci., vol. 9, no. 5, pp Sladek, V.; Sladek, J. (: Local integral equations implemented b MLSapproimation and analtical integrations. Eng. Anal. Bound. Elem., vol. 34, no., pp Sladek, V.; Sladek, J.; Zhang, C. (: On increasing computational efficienc of local integral equation method combined with meshless implementations. CMES Compt. Model. Eng. Sci., vol. 63, no. 3, pp Turing, A. (95: The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B, vol. 37, no. 64, pp Twizell, E. H.; Gumel, A. B.; Cao, Q. (999: A second-order scheme for the Brusselator reaction-diffusion sstem. J. Math. Chem., vol. 6, no. 4, pp (.

24 8 Copright 3 Tech Science Press CMES, vol.95, no.4, pp.59-8, 3 Verdasca, J.; Wit, A. D.; Dewel, G.; Borckmans, P. (99: Reentrant heagonal turing structures. Phsics Letters A, vol. 68, no. 3, pp Walgraef, D.; Dewel, G.; Borckmans, P. (98: Fluctuations near nonequilibrium phase transitions to nonuniform states. Phs. Rev. A, vol., pp