A Meshless Method for Solving the 2D Brusselator Reaction-Diffusion System

Transcription

1 Copright Tech Science Press CMES, ol., no., pp.-8, A Meshless Method for Soling the D Brusselator Reaction-Diffusion Sstem M. Mohammadi, R. Mokhtari, and R. Schaback Abstract: In this paper, the two-dimensional (D) Brusselator reaction-diffusion sstem is simulated numericall b the method of lines. The proposed method is implemented as a meshless method based on spatial trial functions in the reproducing kernel Hilbert spaces. For efficienc and stabilit reasons, we use the Newton basis introduced recentl b Müller and Schaback. The method is shown to work in all interesting situations described b Hopf bifurcations and Turing patterns. Kewords: D Brusselator reaction-diffusion sstem, method of lines, meshless method, Newton basis functions, reproducing kernel Hilbert space. Introduction Reaction-diffusion equations frequentl arise in the stud of chemical and biological sstems. The so-called Brussels school [Herschkowitz-Kaufman and Nicolis (97); Laenda, Nicolis, and Herschkowitz-Kaufman (97); Lefeer (968); Lefeer and Nicolis (97); Nicolis and Prigogine (977); Prigogine and Lefeer (968)] deeloped and analzed the behaiour of a non-linear oscillator [Lefeer (968); Prigogine and Lefeer (968)] associated with the chemical sstem δ U, ρ +U V + D, U +V U, U E () Facult of Mathematical Sciences and Computer, Kharazmi Uniersit, Taleghani Ae., Tehran 686, Iran. Department of Mathematical Sciences, Isfahan Uniersit of Technolog, Isfahan 86-8, Iran. Corresponding author. Institut für Numerische und Angewandte Mathematik, Uniersität Göttingen, Lotzestraße 6-8, D-77 Göttingen, German.

2 Copright Tech Science Press CMES, ol., no., pp.-8, where δ and ρ are input chemicals, D and E are output chemicals and U and V are intermediates. Let u(,t) and (,t) be the concentrations of U and V, and assume that the concentrations of the input compounds δ and ρ are held constant during the reaction process. Then using the law of mass action, the kinetic equations associated with () are gien b the following sstem of reaction-diffusion equations, known as the Brusselator sstem [Prigogine and Lefeer (968)]: u t (,t) = δ + u (ρ + )u + µ u, t (,t) = ρu u + µ, () where u and represent dimensionless concentrations of two reactants, δ, ρ, and diffusion coefficients µ and µ are positie constants. The parameter ρ is often chosen as a parameter for studing bifurcation. The Brusselator sstem occurs in a large number of phsical problems such as the formation of ozone from atomic ogen, in enzmatic reactions, and arises in laser and plasma phsics from multiple coupling between modes. No analtical solution of the sstem is known so far, and numerical solutions hae to be used. Moreoer, there is er little literature on the numerical solution of the sstem. Known techniques are the Adomian decomposition method [Adomian (99); Lin, Liu, and Li ()], second order finite difference method [Gumel, Langford, Twizel, and Wu (); Twizell, Gumel, and Cao (999)], modified Adomian decomposition method [Wazwaz ()], dualreciprocit boundar element method [Ang ()], differential quadrature method [Mittal and Jiwari ()], modified cubic B-spline differential quadrature method [Jiwari and Yuan ()], Multistage homotop perturbation [Lee, Park, and Jang ()], and radial basis functions collocation method [ul Islam, Haq, and Ali ()]. Unlike traditional numerical methods in soling partial differential equations (PDEs), meshless methods [Dong, Alotaibi, Mohiuddine, and Atluri (); Han and Atluri ()] need no mesh generation. Collocation methods are trul meshless and simple to program, and the allow arious approaches for soling PDEs. Taking translates of kernels as trial functions, meshless collocation in unsmmetric and smmetric form dates back to [Franke and Schaback (998b); Franke and Schaback (998a); Kansa (986)] and has proen to be highl successful, because the arising linear sstems are eas to generate and allow good accurac at low computational cost. In addition, it was proen recentl [Schaback ()] that smmetric collocation [Fasshauer (997); Franke and Schaback (998b); Franke and Schaback (998a)] using kernels is optimal along all linear PDE solers using the same input data. This motiates the use of kernels for soling PDEs. An oeriew of kernel methods before 6 is gien in [Schaback and Wendland (6)], while recent ariations of the theme are in [Hon and Schaback (8);

3 A Meshless Method for Soling the D Brusselator Reaction-Diffusion Sstem Lee, Ling, and Schaback (9); Mohammadi and Mokhtari (); Mohammadi and Mokhtari (); Mohammadi and Mokhtari (); Mohammadi, Mokhtari, and Panahipour (), Mohammadi, Mokhtari, and Panahipour (), Mohammadi, Mokhtari, and Toutian Isfahani (); Mokhtari and Mohammadi (); Mokhtari and Mohseni (); Mokhtari, Toutian Isfahani, and Mohammadi ()] and the references therein. It is well known that representations of kernel-based approimants in terms of the standard basis of translated kernels are notoriousl unstable. The Newton basis [Müller and Schaback (9)] with a recursiel computable set of functions which anish at increasingl man data points, turns out to be more stable. It is orthonormal in the natie Hilbert space and complete, if infinitel man data locations are reasonabl chosen. Recentl, an adaptie calculation of Newton basis arising from a pioted Cholesk factorization which is computationall cheap, has been introduced [Pazouki and Schaback ()]. For time-dependent partial differential equations, meshless kernel-based methods are based on a fied spatial interpolation, but since the coefficients are time-dependent, one obtains a sstem of ordinar differential equations. This is the well-known method of lines, and it turned out to be accurate in seeral problems [Dereli and Schaback ()]. In this stud, a method of lines, implemented as a meshless method based on spatial trial spaces spanned b the Newton basis functions in the natie Hilbert space of the reproducing kernel is deeloped for the numerical simulation of the twodimensional Brusselator reaction-diffusion sstem. The rest of the paper is organized as follows. In Section, we describe the behaiour of the Brusselator sstem. In Section, we gie the goerning equations. Kernel-based trial functions, and particularl the Newton basis functions, are summarized in Section. In Section, we turn to Newton basis functions satisfing the Brusselator sstem and proide a method of lines which leads to an ODE sstem. The implementation of the method is gien in Section 6. Some numerical eamples are presented in Section 7. The last section is deoted to a brief conclusion. Analsis of the Brusselator sstem We first describe the behaiour of the Brusselator sstem [Ma and Wang (); Roussel ()]. As one is interested in the stabilit analsis of a nonlinear reactiondiffusion sstem, one tpicall first determines the stationar state of the model in the absence of diffusion. This is done b soling the sstem () with conditions u t = t = and µ = µ =. So the onl equilibrium point of the ordinar differential equation (ODE) sstem is (u, ) = ( δ, ρ δ ). The Jacobian at the equilibrium

4 6 Copright Tech Science Press CMES, ol., no., pp.-8, point is gien b [ ρ δ J = ρ δ ] and its eigenalues satisf the characteristic equation λ + ( ρ + δ )λ + δ =. So the eigenalues of J clearl depend on ρ + δ and the quantit ( ρ + δ ) δ. When the real parts of the eigenalues are negatie, the equilibrium point is a stable focus and when the cross zero and become positie the equilibrium point becomes an unstable focus, with orbits spiralling out. The stabilit properties and the eistence of a limit ccle in the diffusion-free Brusselator sstem are summarized in Tab. in relation to the four regions of Fig.. Table : Nature of the critical point and eistence of the limit ccle. Region ρ + δ Eigenalues Tpe of critical point Limit ccle eists < Positie real Unstable node Yes < < Positie real parts Unstable focus Yes = < Imaginar Stable fine focus No > < Negatie real parts Stable focus No > Negatie real Stable node No The appearance or the disappearance of a periodic orbit through a local change in the stabilit properties of an equilibrium point is known as the Hopf bifurcation. In a differential equation a Hopf bifurcation tpicall occurs when a comple conjugate pair of eigenalues of the linearized flow at an equilibrium point becomes purel imaginar. So the equilibrium point ( δ, ρ δ ) undergoes a Hopf bifurcation at ρ = ρ H = + δ, with oscillations being obsered for ρ > ρ H. Now we want to stud the stabilit analsis of the Brusselator reaction-diffusion sstem (). Note that the stead state of the diffusion-free sstem is also a stead state of the reaction-diffusion one. If (u,) = (u, ), then the spatial deriaties of this constant function are zero, as are the reaction terms, so the time deriaties must be zero too. This solution is called the homogeneous stead state. To stud

5 A Meshless Method for Soling the D Brusselator Reaction-Diffusion Sstem 7. ρ+δ = = ρ.... δ Figure : Stabilit regions of the diffusion-free Brusselator sstem. the behaiour of the sstem, we will linearize () around the equilibrium point, i.e., we substitute u = U + δ, = V + ρ δ, where U and V are displacements from equilibrium, which now depend both on time and space, and neglect high order terms in u and. Then U t = (ρ )U + δ V + µ U, V t = ρu δ V + µ V, and so [ ] U V t [ U = J V ] [ U + D V ], where [ ρ δ J = ρ δ ] [ µ, D = µ ].

6 8 Copright Tech Science Press CMES, ol., no., pp.-8, We again want to determine if the stead state is stable against small perturbations. But this time we want to introduce a spatial aspect. Suppose that perturbations are inhomogeneous in space. The conenient form is [ U V ] = [ U V ] e λt e ik, where = (, ), and k represents a ector of two waenumbers (k,k ). The question now will be whether or not conditions can be found under which the stead state is unstable (Re(λ) > ) when a wiggl disturbance is introduced. Since an disturbance oer a finite domain can be snthesized b adding up sine waes, this will answer the question of whether the stead state is stable against small, but otherwise arbitrar perturbations. If we substitute the perturbation into the linearized equation, after cancelling off the common factors of e λt e ik, we get λ [ U V ] = J [ U V ] k D [ U V ], or (λi + k D J) [ U V ] [ = ], where I is the identit matri. This is a homogeneous equation in [U,V ] which onl has nontriial solutions if λi + k D J = λ +λ ( k (µ +µ ) + ρ + δ ) + k µ µ + k ( µ ( ρ) + µ δ ) +δ =. Our task will be to determine whether this characteristic equation has solutions for which the real part of λ is positie, and if so, under what conditions. Because of the appearance of the unknown waenumber k in this equation, this equation has a different solution for eer k. This quadratic equation is of the form λ + qλ + p =. The solutions are λ = { q ± } q p. There will be at least one root with a positie real part proided one of the following two conditions are met:

7 A Meshless Method for Soling the D Brusselator Reaction-Diffusion Sstem 9. q <, which leads to ρ > + δ + k (µ + µ ) > + δ. So the stead state ma also go through a Hopf instabilit if ρ > +δ eoling then into a homogeneous limit ccle characterized b a critical frequenc ω = δ.. q > and p <, which leads to the following bifurcation conditions: ( ) µ + δ < ρ < + δ. µ Note that this bifurcation occurs onl when the stead state would be stable in the absence of diffusion. Thus this is a purel diffusie instabilit. Moreoer, it occurs for a finite range of waenumbers k. Therefore, this instabilit will form a spatial pattern of some sort since adding up a bunch of sine waes within a finite range of waelengths should produce a nontriial wae pattern. This is a Turing bifurcation. Technicall, a Turing bifurcation is the destabilization of an other stable stead state b diffusie terms, leading to pattern formation. Goerning equations We consider the D Brusselator sstem with the initial and Dirichlet or Neumann boundar conditions: u t (,t) = δ + u (ρ + )u + µ u Ω R, t (,T ], () t (,t) = ρu u + µ (u(,t),(,t)) = ( f D (,t),g D (,t) ) D Ω, t [,T ], ( ) u n (,t), n (,t) = ( f N (,t),g N (,t) ) N Ω, t [,T ], u(,) = u (), (,) = (), () Ω. () where u,, f D, g D, f N and g N are known functions, Ω R is the domain set, Ω = N D is the boundar of the domain set Ω, and is the Laplace operator.

8 Copright Tech Science Press CMES, ol., no., pp.-8, Kernel-based trial functions We take a smooth smmetric positie definite kernel K : Ω Ω R on the spatial domain Ω. Behind each such kernel there is a reproducing natie Hilbert space N K = span{k(, ) Ω}, of functions on Ω in the sense f,k(, ) NK = f () for all Ω, f K, and whose inner product is linked to the kernel itself ia K(, ),K(, ) NK = K(,) for all Ω. The most important eamples are the Whittle-Matern kernels r m d/ K m d/ (r), r =,, R d, reproducing in the Sobole space W m(rd ) for m > d/, where K ν is the modified Bessel function of the second kind [Schaback ()]. The following will be independent of the kernel chosen, but users should be aware that the kernel should be smooth enough to allow sufficientl man deriaties for the PDE and additional smoothness for fast conergence [Wendland ()]. For scattered nodes,..., n Ω, the translates K j () = K( j,) are the trial functions we want to start with. Since the kernel K is smooth and eplicitl aailable, we can take deriaties with respect to both arguments cheapl, and this implies that we get cheap deriaties of the K j. But the standard basis of translates leads to an ill-conditioned kernel matri A = (K( j, k )) j,k n, and hence the translates are notoriousl unstable. The Newton basis with a recursiel computable set of basis functions and anishing at increasingl man data points turns out to be more stable. It is orthonormal in the natie Hilbert space and complete, if infinitel man data locations are reasonabl chosen. The Newton basis functions {N k ()} n k= can be epressed b N k () = n K(, j )c jk, k n. (6) j= If N() = (N (),...,N n ()), and T () = (K(, ),...,K(, n )), from (6) we hae N() = T () C, where C = ( c jk is the coefficient matri. Hence the alue ) j,k n matri V = (N j ( i )) i, j n is of the form V = A C. It has been proed that the Cholesk decomposition A = L L T with a nonsingular lower triangular matri L

9 A Meshless Method for Soling the D Brusselator Reaction-Diffusion Sstem leads to the Newton basis N with N() = T () (L T ), and V = L. Hence the condition number of the collocation matri corresponding to Newton basis functions is smaller than the one corresponding to translated RBFs. Consequentl, using Newton basis functions for collocation will lead to more stable methods than using the basis of translates. The Newton basis functions can be recursiel calculated and hae the propert N j ( k ) =, k j n. If the alues of the Newton basis and linear maps L like deriaties are needed to be calculated at other points, we get the linear sstems V N T () = T () T, and V L (N T ( )) = L (T ( ) T ), respectiel. Method of lines We aim at the method of lines (MOL), which leads to a sstem of ordinar differential equations, and this implies that there will be neither time discretization at all nor artificial linearization of the differential equation. The problem of correct time-stepping will be automaticall soled b the ODE soler we inoke. The discretization is at points i, i n for the PDE, j, j m for the Dirichlet and z k, k l for the Neumann boundar conditions. We reorder these sequentiall into points w i, i +l, the j first and the i second, and form the Newton basis N,...,N +l for these points. Then N m+,...,n +l anish on the Dirichlet points, and N +,...,N +l also anish on the PDE points. We write our trial space functions as +l ũ(,t) = j= +l ṽ(,t) = j= α j (t)n j () β j (t)n j () (7) and care for the Dirichlet boundar conditions b soling ũ(w i,t) = f D (w i,t) = ṽ(w i,t) = g D (w i,t) = m α j (t)n j (w i ), i m, j= m β j (t)n j (w i ), i m, j= for the unknown ectors a (t) = (α (t),...,α m (t)) T, and b (t) = (β (t),...,β m (t)) T. This is just the Newton interpolant to the data fi D and g D i on the Dirichlet points. We will also need f D (w i,t) = m α j(t)n j (w i ), i m, j=

10 Copright Tech Science Press CMES, ol., no., pp.-8, g D (w i,t) = m β j(t)n j (w i ), i m, j= for the formulation of the MOL, where the prime denotes the deriatie with respect to t. Our unknowns in the trial space are onl the ectors a (t) = (α m+ (t),...,α (t)) T, b (t) = (β m+ (t),...,β (t)) T, a (t) = (α + (t),...,α +l (t)) T, b (t) = (β + (t),...,β +l (t)) T. Now we implement the Neumann boundar conditions at a point w i, m + n + i m + n + l as follows: f N (w i ) = m α j (t) N j j= n (w i) + α j (t) N j j=m+ n (w i) + +l α j (t) N j j=+ n (w i), g N (w i,t) = m β j (t) N j j= n (w i) + β j (t) N j j=m+ n (w i) + +l β j (t) N j j=+ n (w i). Thus the unknown ectors a (t) and b (t) can be written in terms of the unknown ectors a (t) and b (t) b soling the following equations: +l α j (t) N j j=+ n (w i,t) = f N (w i,t) m j= α j (t) N j n (w i,t) j=m+ α j (t) N j n (w i,t), +l β j (t) N j j=+ n (w i,t) = g N (w i,t) m j= β j (t) N j n (w i,t) j=m+ β j (t) N j n (w i,t),

11 A Meshless Method for Soling the D Brusselator Reaction-Diffusion Sstem for m + n + i m + n + l. We now write the PDE () at a point w i, m + i m + n as follows: m j= =δ+ α j(t)n j (w i ) + ( m j= (ρ + ) α j (t)n j (w i )+ α j(t)n j (w i ) j=m+ j=m+ ( m α j (t)n j (w i ) + j= + µ ( m j=α j (t) N j (w i ) + ) ( m α j (t)n j (w i ) j= ) α j (t)n j (w i ) j=m+ α j (t) N j (w i ) + j=m+ β j (t)n j (w i )+ j=m+ β j (t)n j (w i ) ) +l α j (t) N j (w i ), j=+ ) =ρ m j= β j(t)n j (w i ) + ( m α j (t)n j (w i ) + j= ( m j= β j(t)n j (w i ) j=m+ α j (t)n j (w i ) j=m+ α j (t)n j (w i )+ α j (t)n j (w i ) j=m+ + µ ( m j=β j (t) N j (w i ) + ) ) ( m j= β j (t) N j (w i ) + j=m+ ) β j (t)n j (w i )+ β j (t)n j (w i ) j=m+ ) +l β j (t) N j (w i ) j=+. (8) Thus we get an implicit sstem of first-order ordinar differential equations. The initial conditions also proide ũ (w i ) = m α j ()N j (w i ) + α j ()N j (w i ), m + i m + n, j= j=m+ ṽ (w i ) = m β j ()N j (w i ) + β j ()N j (w i ), m + i m + n. j= j=m+

12 Copright Tech Science Press CMES, ol., no., pp.-8, 6 Implementation If we introduce suitable column ectors and matrices into the sstem (8), we hae to satisf [ N N ][ a (t) b (t) ] [ R (a =,b ) R (a,b ) ], (9) with the initial conditions a () = (N ) ( (u (w i ), m + i m + n) T N a () ), where b () = (N ) ( ( (w i ), m + i m + n) T N b () ), R (a,b ) = δ + ((N a + N a ). ). (N b + N b ) (ρ + )(N a + N a ) + µ (D a + D a + D a ) N a (t), R (a,b ) = ρ (N a + N a ) ((N a + N a ). ). (N b + N b ) + µ (D b + D b + D b ) N b (t), in MATLAB notation for the pointwise product. and power. between two matrices or ectors of the same shape. The necessar matrices and ectors are N = (N j (w i )) i m, j m, N = (N j (w i )) m+ i, j m, N = (N j (w i )) m+ i,m+ j, a (t) = (N ) F D (t), b (t) = (N ) G D (t), a (t) = (N ) F D (t), b (t) = (N ) G D N(t), F D (t) = ( f D (w i,t), i m) T, G D (t) := (g D (w i,t), i m) T, F D (t) = ( f D (w i,t), i m) T, G D (t) := (g D (w i,t), i m) T, ( ) ( N a (t) = F N (t) N n n a (t) N ) n a (t) ( ) ( N b (t) = G N (t) N n n b (t) N ) n b (t) F N (t) = ( f N (w i,t), m + n + i m + n + l) T, G N (t) := (g N (w i,t), m + n + i m + n + l) T,

13 A Meshless Method for Soling the D Brusselator Reaction-Diffusion Sstem N n = N n = N n = ( ) Nj (w i ) n ( ) Nj (w i ) n ( ) Nj (w i ) n, + i +l, j m D = ( N j (w i )) m+ i, j m,, + i +l,m+ j, + i +l,+ j +l D = ( N j (w i )) m+ i,m+ j, D = ( N j (w i )) m+ i,+ j +l, where j is the column inde and i is the row inde. The sstem (9) is the ODE sstem generated b the MOL and one can inoke an ODE integrator to sole it. The method of lines thus reduces the PDE problem to a sstem of ODEs and relies numericall on the ODE solers inoked to sole these ODEs. The discretization of the PDE does not inole an time stepping at all, and therefore the time integration plas onl a role within the ODEs. But since ODEs are not in the focus of this paper, we ignore time stepping here. The matri of the left-hand side is time-independent, and in the case of the inertibilit of it, the approimate solutions u(,t) and (,t) will satisf the differential equations at all points w,...,w +l and all times, the latter within the accurac limit of the ODE integrator. Note that the nonlinearit of the PDE is presered, and a good ODE soler will automaticall use a reasonable time-stepping and detect stiffness of the ODE sstem. The matrices N, N, and N are the nonzero blocks of the alue matri of the Newton basis, which is upper triangular and nonsingular. Thus the square matrices N and N are upper triangular and nonsingular. Howeer, there is no guarantee that the matrices of normal deriaties are nonsingular. 7 Numerical results In this section we present the results of our scheme for the numerical solution of the Brusselator reaction-diffusion sstem ()-(). In all test problems, we take the Matern kernel with RBF parameter ν = = m d/ and RBF scale c =, i.e. we work with the kernel ( ) ( ) K(,) = K. We also assume that Ω = [,] [,], such that we work in the Hilbert space W (R ). We take uniforml distributed discretization points in the region Ω as shown in Fig.. We also take grid points along each ais for plotting of figures.

14 6 Copright Tech Science Press CMES, ol., no., pp.-8,.9 PDE Boundar Figure : Points distribution in the region Ω. 7. Test problem Consider the Brusselator sstem together with the Dirichlet boundar conditions with ρ =, δ =, and µ = µ =.. The initial and boundar conditions are etracted from the eact solutions { u(,,t) = ep(.t), (,,t) = ep( + +.t). The concentrations profiles of u and at different time leels T =, T =, T =, and T = are shown in Figs. -6, respectiel. Absolute and relatie error distributions at time T = are shown in Figs. 7 and 8, respectiel. The results are in agreement with the results of [Jiwari and Yuan ()]. 7. Test problem In the second eperiment, we choose parameters ρ =, δ =, and µ = µ =., and start from zero initial conditions and fied boundar conditions taken as the homogeneous stead state (u,) = (δ, ρ δ ). Since the eact solutions are not known, we plot the error between the left and right hand sides of the equations of sstem () for the grid points of the region in Fig. 9. The plots of the alues of u and at the collocation point (.,.) ersus time shown in Fig., indicate that the solutions conerge toward the stationar ones (δ, ρ δ ) as t increases, wheneer ρ +δ >. In the net two test problems, we inestigate the behaiour of the sstem when the sign of ρ + δ changes and the Hopf bifurcation occurs.

15 A Meshless Method for Soling the D Brusselator Reaction-Diffusion Sstem u Figure : Plots of u and at T =, ρ =, δ =. (Test problem ).. u Figure : Plots of u and at T =, ρ =, δ =. (Test problem ) u Figure : Plots of u and at T =, ρ =, δ =. (Test problem )

16 8 Copright Tech Science Press CMES, ol., no., pp.-8, u Figure 6: Plots of u and at T =, ρ =, δ =. (Test problem ) 6 6 u error error Figure 7: Absolute error graph at time T =. (Test problem ).. u error. error Figure 8: Relatie error graph at time T =. (Test problem )

17 A Meshless Method for Soling the D Brusselator Reaction-Diffusion Sstem error error (a) (b) Figure 9: Error graph of PDE at time T =. (a) Equation ; (b) Equation. (Test problem ) u(.,.) (.,.) t t (a) (b) Figure : Plots of u(.,.) and (.,.) ersus time. (Test problem ) 7. Test problem Consider the Brusselator sstem with the following initial and Neumann boundar conditions: { u(,,) = +., (,,) = +.8, { u(,,t) (,,t) = = u(,,t) = = (,,t) = = u(,,t) = = u(,,t) = = (,,t) = =, = = (,,t) = =. Computations are carried out with the parameters ρ =, δ =, and µ = µ =.. The algorithm is tested up to time T =. The concentration profiles of u and at time from T = to T = 8 are shown in Fig.. It can be noted from Fig. that (u,) (δ, ρ δ ) as t increases, wheneer ρ + δ >. The results show an agreement with the results of [Jiwari and Yuan ()]. The plots of the alues

18 Copright Tech Science Press CMES, ol., no., pp.-8, of u and at the collocation point (.,.) ersus time are shown in Fig.. It can be noted from Fig., that (u(.,.),(.,.)) (,.) as t. The results show an agreement with the results of [Twizell, Gumel, and Cao (999)] and [ul Islam, Haq, and Ali ()]. 7. Test problem The algorithm is repeated with ρ =., δ = up to time T =. The concentrations profiles of u and at T = are shown in Fig.. The plots of the alues of u and at the collocation point (.,.) ersus time are shown in Fig.. It can be noted from Figs. and that the solutions are stable but oscillator and the numerical method is seen not to conerge to an fied concentration. The results show an agreement with the results of [Twizell, Gumel, and Cao (999)] and [ul Islam, Haq, and Ali ()]. 7. Test problem Consider the Brusselator sstem with the following initial and Neumann boundar conditions: { u(,,) =. +, (,,) = +, { u(,,t) (,,t) = = u(,,t) = = (,,t) = = u(,,t) = = u(,,t) = = (,,t) = =, = = (,,t) = =. The plots of the alues of u and at the collocation point (.,.) ersus time are shown in Fig. with the parameters ρ =., δ =, and µ = µ =.. It can be noted from Fig., that (u(.,.),(.,.)) (,.) as t. The concentration profiles of u and at time from T = to T = are shown in Fig. 6 with the parameters ρ =., δ =, and µ = µ =.. The results are similar to those obtained in [Jiwari and Yuan ()]. In the net test problem, we show the Turing pattern occuring in the Brusselator sstem b our scheme. 7.6 Test problem 6 Consider the Brusselator sstem with the parameters ρ = 8.7, δ =., µ =, and µ = 8. Fig. 7 shows the Turing pattern obtained at time T = starting from the initial conditions which are random perturbations around the stationar state (δ, ρ δ ), with no flu boundar conditions. This spott pattern erifies the fact that in two dimensions, reaction-diffusion sstems tpicall ehibit either stripes or spots [Leppänen ()].

19 u u u u u A Meshless Method for Soling the D Brusselator Reaction-Diffusion Sstem Figure : Plots of u and at times T =,,,7,8, ρ =, δ =. (Test problem )

20 Copright Tech Science Press CMES, ol., no., pp.-8, u(.,.) (.,.) t t Figure : Plots of u(.,.) and (.,.) ersus time. (Test problem ) u Figure : Plots of u and at T =, ρ =., δ =. (Test problem ) u(.,.)..... (.,.) 6 t t Figure : Plots of u(.,.) and (.,.) ersus time. (Test problem )

21 u u u A Meshless Method for Soling the D Brusselator Reaction-Diffusion Sstem u(.,.) (.,.) t 6 8 t Figure : Plots of u(.,.) and (.,.) ersus time. (Test problem )

22 u u u Copright Tech Science Press CMES, ol., no., pp.-8, Figure 6: Plots of u and at times T =,,,7,,, ρ =., δ =. (Test problem ) Figure 7: Turing pattern with the parameters ρ = 8.7, δ =., µ =, µ = 8, and no flu boundar conditions. (Test problem 6 )

23 A Meshless Method for Soling the D Brusselator Reaction-Diffusion Sstem 8 Conclusion In this paper, the Newton basis functions were successfull used as spatial trial functions in the method of lines for the numerical solution of the D Brusselator reaction diffusion sstem. The method is shown to work in all interesting Hopf bifurcations and Turing patterns. References Adomian, G. (99): ol. 9, pp.. The diffusion Brusselator equation. Comput. Math. Appl., Ang, W.-T. (): The two-dimensional reaction-diffusion Brusselator sstem: a dual-reciprocit boundar element solution. Eng. Anal. Bound. Elem., ol. 7, pp Dereli, Y.; Schaback, R. (): The meshless kernel-based method of lines for soling the equal width equation. Preprint Göttingen,. Dong, L.; Alotaibi, A.; Mohiuddine, S.; Atluri, S. N. (): Computational methods in engineering: A ariet of primal & mied methods, with global & local interpolations, for well-posed or ill-posed bcs. CMES: Comput. Modeling Eng. Sci., ol. 99, pp. 8. Fasshauer, G. (997): Soling partial differential equations b collocation with radial basis functions. pp. 8. Vanderbilt Uniersit Press, Nashille, TN. Franke, C.; Schaback, R. (998): Conergence order estimates of meshless collocation methods using radial basis functions. Ad. Comput. Math., ol. 8, pp Franke, C.; Schaback, R. (998): Soling partial differential equations b collocation using radial basis functions. Appl. Math. Comp., ol. 9, pp Gumel, A.; Langford, W.; Twizel, E.; Wu, J. (): Numerical solutions for a coupled non-linear oscillator. J. Math. Chem., ol. 8, pp.. Han, Z. D.; Atluri, S. N. (): On the (Meshless local Petro-Galerkin) MLPG- Eshelb method in computational finite deformation solid mechanics - part II.. CMES: Comput. Modeling Eng. Sci., ol. 97, pp Herschkowitz-Kaufman, M.; Nicolis, N. (97): Localized spatial structures and non-linear chemical waes in dissipatie sstems. J. Chem. Phs., ol. 6, pp Hon, Y. C.; Schaback, R. (8): Solabilit of partial differential equations b meshless kernel methods. Ad. Comput. Math., ol. 8, pp

24 6 Copright Tech Science Press CMES, ol., no., pp.-8, Jiwari, R.; Yuan, J. (): A computational modeling of the behaior of the two-dimensional reaction-diffusion Brusselator sstem arising in chemical processes. J. Math. Chem. Kansa, E. J. (986): Application of Hard s multiquadric interpolation to hdrodnamics. In Proc. 986 Simul. Conf., Vol., pp. 7. Laenda, B.; Nicolis, G.; Herschkowitz-Kaufman, M. (97): Chemical instabilities and relaation oscillations. J. Theor. Biol., ol., pp Lee, C.-F.; Ling, L.; Schaback, R. (9): On conergent numerical algorithms for unsmmetric collocation. Ad. Comput. Math., ol., pp. 9. Lee, C. H.; Park, K.; Jang, B. (): Multistage homotop perturbation method for nonlinear reaction networks. J. Math. Chem., ol., pp Lefeer, R. (968): Dissipatie structures in chemical sstems. J. Chem. Phs., ol. 9, pp Lefeer, R.; Nicolis, G. (97): Chemical instabilities and sustained oscillations. J. Theor. Biol., ol., pp Leppänen, T. (): The theor of Turing pattern formation. aailable online at Lin, Y.; Liu, Y.; Li, Z. (): Smbolic computation of analtic approimate solutions for nonlinear differential equations with initial conditions. Comput. Phsics Commun., ol. 8, pp Ma, T.; Wang, S. (): Phs., ol., pp.. Phase transitions for the Brusselator model. J. Math. Mittal, R.; Jiwari, R. (): Numerical stud of two-dimensional reactiondiffusion Brusselator sstem b differential quadrature method. Int. J. Comput. Methods Eng. Sci. Mech., ol., pp.. Mohammadi, M.; Mokhtari, R. (): Soling the generalized regularized long wae equation on the basis of a reproducing kernel space. J. Comput. Appl. Math., ol., pp.. Mohammadi, M.; Mokhtari, R. (): A new algorithm for soling nonlinear Schrödinger equation in the reproducing kernel space. Iranian J. Sci. Tech. Section A. Sci., ol. 7, pp. 6. Mohammadi, M.; Mokhtari, R. (): A reproducing kernel method for soling a class of nonlinear sstems of PDEs. Math. Model. Anal., ol. 9, pp

25 A Meshless Method for Soling the D Brusselator Reaction-Diffusion Sstem 7 Mohammadi, M.; Mokhtari, R.; Panahipour, H. (): A Galerkinreproducing kernel method: Application to the d nonlinear coupled Burgers equations. Eng. Anal. Bound. Elem., ol. 7, pp Mohammadi, M.; Mokhtari, R.; Panahipour, H. (): Soling two parabolic inerse problems with a nonlocal boundar condition in the reproducing kernel space. Appl. Comput. Math., ol., pp Mohammadi, M.; Mokhtari, R.; Toutian Isfahani, F. (): Soling an inerse problem for a parabolic equation with a nonlocal boundar condition in the reproducing kernel space. Iranian J. Numer. Anal. Optimization, ol., pp Mokhtari, R.; Mohammadi, M. (): Numerical solution of GRLW equation using sinc-collocation method. Comput. Phsics Commun., ol. 8, pp Mokhtari, R.; Mohseni, M. (): A meshless method for soling mkd equation. Comput. Phsics Commun., ol. 8, pp Mokhtari, R.; Toutian Isfahani, F.; Mohammadi, M. (): Soling a class of nonlinear differential-difference equations in the reproducing kernel space. Abstr. Appl. Anal. Article ID. Müller, S.; Schaback, R. (9): Theor, ol. 6, pp Nicolis, G.; Prigogine, I. (977): Wile Interscience, New York. Pazouki, M.; Schaback, R. (): Appl. Math., ol. 6, pp A Newton basis for kernel spaces. J. Appro. Self-organization in non-equilibrium sstems. Bases for kernel-based spaces. J. Comput. Prigogine, I.; Lefeer, R. (968): Smmetries breaking instabilities in dissipatie sstems ii. J. Phs. Chem., ol. 8, pp Roussel, M. (): Reaction-diffusion equations. aailable online at roussel/nld/turing.pdf,. Schaback, R. (): Kernel based meshless methods. Lecture Note, Göttingen, Schaback, R. (): A computational tool for comparing all linear PDE solers. submitted, Schaback, R.; Wendland, H. (6): Kernel techniques: from machine learning to meshless methods. Acta Numerica, ol., pp. 69. Twizell, E.; Gumel, A.; Cao, Q. (999): A second-order scheme for the Brusselator reaction-diffusion sstem. J. Math. Chem., ol. 6, pp

26 8 Copright Tech Science Press CMES, ol., no., pp.-8, ul Islam, S.; Haq, S.; Ali, A. (): A computational modeling of the behaior of the two-dimensional reaction-diffusion Brusselator sstem. Appl. Math. Model., ol., pp Wazwaz, A.-M. (): The decomposition method applied to sstems of partial differential equations and to the reaction-diffusion Brusselator model. Appl. Math. Comput., ol., pp. 6. Wendland, H. (): Press. Scattered Data Approimation. Cambridge Uniersit