Asymptotic Analysis Of The Solution Of Schnakenberg Equation

The study of Chemical reaction kinetics attaches great important to chemistry and chemical engineering research,among them,the research of chemical reaction equilibrium is a very important topic.Due to the chemical reaction equations are generally non-linear,and it almost impossible to solve directly,so people more care about the research of approximate solution and the stability of the equilibrium state. The multiple scale method is a very reliable way to solve approximate solution for this type of equation.Until 1979,the Schnakenberg studied some chemical reaction process,and set up a series of chemical reaction kinetics model,and discussed some of the limit cycles in linear model. First part of the paper,we mainly considered the following forms of Schnakenberg equations.In the original model equations, u is respectively for the concentration of the X, and v is respectively for the concentration of the Y, u is respectively for the concentration change rate of the reactants X, and v is respectively for the concentration change rate of the reactants Y, t is the reaction time, ε> 0 is a small disturbance quantity.lt requires that t satisfied t0≤εt≤T.This paper discussed the disturbance of the chemical reaction near the zero equilibrium,and calculated the first-order asymptotic expressions of the solutions of the equations by the multiple scales.Then,in order to make the error estimation for the asymptotic solution that we have solved,we first turned the problem to the initial value problem of a differential equations satisfied the remainder term of the approximate solution and the corresponding integral equations, fully proved that the asymptotic solution we have solved is reasonable and uniformly valid by using the Nonlinear Gronwall inequality.We can also solve the high order approximate solutions and their error estimates by this method.According to the chemical reaction Aâ‰' X,Bâ†'Y,2X+Yâ†' 3XIf x is the concentration of X, y is the concentration of Y,then we can get the following forms of equations by the law of conservation of mass.The system has a unique equilibrium point, in the case of a=b,we studied the asymptotic behavior of solution of the problem on t-â†'∞, and proved that solution of initial value problems will converge to the equilibrium solution by index when the parameter satisfied certain conditions,and gave the convergence rate of the solution.At the end of the paper,we verified the stability of the equilibrium solution of a class of wider Schnakenberg equations by attractor theory.