Reaction-diffusion model. 

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Context 1

... Differential Equation (PDE) is a relation involving anunknown function of several independent variables and its partial derivatives with respect to those variables. System of partial differential equations have attracted much attention in a verity of applied science because of their wide applicability(Jost, 2002). A few nonlinear differential equations have known exact solutions, but many of which are important in applications do not (Pinney, 1955).The nonlinear models of real-life problems are still difficult to solve either numerically or theoretically. There has recently been much attention devoted to the search for better andmore efficient solution methods for determining a solution, approximate or exact, analytically or numerically, to nonlinear models (Batiha, 2009). A possible connection between patterns in biological systems and patterns that could from spontaneously in chemical reaction-diffusion systems was first proposed by Turing (1952). Turing’s analysis stimulated considerab le theoretical research on mathematical models of pattern formation, but Turing-type patterns were not observed in controlled laboratory experiments until 1990 (Turing, 1952 and Turing, 1990). The central idea behind the theory is that two homogeneously distributed substances within a certain space, one "locally activated" and the other capable of "long-range inhibition," can produce novel shapes and gradients. The results of these substance are shown in Figure 1 . What is special about such a model is that it can explain patternformation without a preformed pattern. That is, the reactiondiffusionmodel can explain how those initial patterns form in thefirst place. While fly development begins with a maternal injectionof bicoid into the oocyte, a reaction-diffusion system can theoreticallygive rise to a pattern without an initial a symmetry. Figure 2 shows how a reaction-diffusion system of two molecules,P and S, works. P stimulates the production of itself (autocatalysis) aswell as the production of its inhibitor S - S inhibits P. Furthermore, Pdiffuses slower than S. The initial conditions ( Figure2 B, Time 1) area random distribution of both substances no pre-pattern. Now whathappens? Since P stimulates production of itself and diffuses slowly,P concentrates into a peak ( Figure2 A, Time 1). Furthermore, sinceP is stimulating production of its inhibitor S which quickly diffuses,P concentrations fall as you move from the peak. This results in asingle peak ( Figure 2 A, Time 2). However, as S can only travel sofar, multiple peaks are capable of formation as shown in ( Figure 2 B, Time 3). The results are what are called "standing waves." Local activation(P) and long-range inhibition (S) form regular patterns(Notethat the rate of degradation was not considered in this model)(Turing, 1952). Reaction-diffusion systems are mathematical models which explainhow the concentration of one or more substances distributed inspace changes under the influence of two processes: local chemicalreactions in which the substances are transformed into each other,and diffusion which causes the substances to spread out over a surfacein space. The class of nonlinear reaction diffusion system is given in the following form: where ; diffusion coefficients , and the control parameters of the system are positive constants; are source terms; and are the two chemical concentrations under investigation, the nonlinear reaction term ( ) represents the type of cubic autocatalytic Chemical or Biochemical reaction, with homogeneous Dirichlet or Neumann boundary condition on a bounded, locally Lipschitz domain . The following four model equations are typical in this ...

Context 2

... Differential Equation (PDE) is a relation involving anunknown function of several independent variables and its partial derivatives with respect to those variables. System of partial differential equations have attracted much attention in a verity of applied science because of their wide applicability(Jost, 2002). A few nonlinear differential equations have known exact solutions, but many of which are important in applications do not (Pinney, 1955).The nonlinear models of real-life problems are still difficult to solve either numerically or theoretically. There has recently been much attention devoted to the search for better andmore efficient solution methods for determining a solution, approximate or exact, analytically or numerically, to nonlinear models (Batiha, 2009). A possible connection between patterns in biological systems and patterns that could from spontaneously in chemical reaction-diffusion systems was first proposed by Turing (1952). Turing’s analysis stimulated considerab le theoretical research on mathematical models of pattern formation, but Turing-type patterns were not observed in controlled laboratory experiments until 1990 (Turing, 1952 and Turing, 1990). The central idea behind the theory is that two homogeneously distributed substances within a certain space, one "locally activated" and the other capable of "long-range inhibition," can produce novel shapes and gradients. The results of these substance are shown in Figure 1 . What is special about such a model is that it can explain patternformation without a preformed pattern. That is, the reactiondiffusionmodel can explain how those initial patterns form in thefirst place. While fly development begins with a maternal injectionof bicoid into the oocyte, a reaction-diffusion system can theoreticallygive rise to a pattern without an initial a symmetry. Figure 2 shows how a reaction-diffusion system of two molecules,P and S, works. P stimulates the production of itself (autocatalysis) aswell as the production of its inhibitor S - S inhibits P. Furthermore, Pdiffuses slower than S. The initial conditions ( Figure2 B, Time 1) area random distribution of both substances no pre-pattern. Now whathappens? Since P stimulates production of itself and diffuses slowly,P concentrates into a peak ( Figure2 A, Time 1). Furthermore, sinceP is stimulating production of its inhibitor S which quickly diffuses,P concentrations fall as you move from the peak. This results in asingle peak ( Figure 2 A, Time 2). However, as S can only travel sofar, multiple peaks are capable of formation as shown in ( Figure 2 B, Time 3). The results are what are called "standing waves." Local activation(P) and long-range inhibition (S) form regular patterns(Notethat the rate of degradation was not considered in this model)(Turing, 1952). Reaction-diffusion systems are mathematical models which explainhow the concentration of one or more substances distributed inspace changes under the influence of two processes: local chemicalreactions in which the substances are transformed into each other,and diffusion which causes the substances to spread out over a surfacein space. The class of nonlinear reaction diffusion system is given in the following form: where ; diffusion coefficients , and the control parameters of the system are positive constants; are source terms; and are the two chemical concentrations under investigation, the nonlinear reaction term ( ) represents the type of cubic autocatalytic Chemical or Biochemical reaction, with homogeneous Dirichlet or Neumann boundary condition on a bounded, locally Lipschitz domain . The following four model equations are typical in this ...

Context 3

... Differential Equation (PDE) is a relation involving anunknown function of several independent variables and its partial derivatives with respect to those variables. System of partial differential equations have attracted much attention in a verity of applied science because of their wide applicability(Jost, 2002). A few nonlinear differential equations have known exact solutions, but many of which are important in applications do not (Pinney, 1955).The nonlinear models of real-life problems are still difficult to solve either numerically or theoretically. There has recently been much attention devoted to the search for better andmore efficient solution methods for determining a solution, approximate or exact, analytically or numerically, to nonlinear models (Batiha, 2009). A possible connection between patterns in biological systems and patterns that could from spontaneously in chemical reaction-diffusion systems was first proposed by Turing (1952). Turing’s analysis stimulated considerab le theoretical research on mathematical models of pattern formation, but Turing-type patterns were not observed in controlled laboratory experiments until 1990 (Turing, 1952 and Turing, 1990). The central idea behind the theory is that two homogeneously distributed substances within a certain space, one "locally activated" and the other capable of "long-range inhibition," can produce novel shapes and gradients. The results of these substance are shown in Figure 1 . What is special about such a model is that it can explain patternformation without a preformed pattern. That is, the reactiondiffusionmodel can explain how those initial patterns form in thefirst place. While fly development begins with a maternal injectionof bicoid into the oocyte, a reaction-diffusion system can theoreticallygive rise to a pattern without an initial a symmetry. Figure 2 shows how a reaction-diffusion system of two molecules,P and S, works. P stimulates the production of itself (autocatalysis) aswell as the production of its inhibitor S - S inhibits P. Furthermore, Pdiffuses slower than S. The initial conditions ( Figure2 B, Time 1) area random distribution of both substances no pre-pattern. Now whathappens? Since P stimulates production of itself and diffuses slowly,P concentrates into a peak ( Figure2 A, Time 1). Furthermore, sinceP is stimulating production of its inhibitor S which quickly diffuses,P concentrations fall as you move from the peak. This results in asingle peak ( Figure 2 A, Time 2). However, as S can only travel sofar, multiple peaks are capable of formation as shown in ( Figure 2 B, Time 3). The results are what are called "standing waves." Local activation(P) and long-range inhibition (S) form regular patterns(Notethat the rate of degradation was not considered in this model)(Turing, 1952). Reaction-diffusion systems are mathematical models which explainhow the concentration of one or more substances distributed inspace changes under the influence of two processes: local chemicalreactions in which the substances are transformed into each other,and diffusion which causes the substances to spread out over a surfacein space. The class of nonlinear reaction diffusion system is given in the following form: where ; diffusion coefficients , and the control parameters of the system are positive constants; are source terms; and are the two chemical concentrations under investigation, the nonlinear reaction term ( ) represents the type of cubic autocatalytic Chemical or Biochemical reaction, with homogeneous Dirichlet or Neumann boundary condition on a bounded, locally Lipschitz domain . The following four model equations are typical in this ...