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1、2Nonlinear Differential Equations: Application to Chemical KineticsProject SummaryLevel of difficulty:1Keywords:Nonlinear system of differential equations, stability, integration schemes, Euler explicit scheme, RungeKutta scheme, delayed differential equationApplication fields:Chemical kinetics, bio
2、logy2.1 Physical Problem and Mathematical ModelingThe laws governing chemical kinetics can be written as systems of ordinarydifferential equations. In the case of complex reactions with several different participating molecules, these equations are nonlinear and present interesting mathematical prop
3、erties (stability, periodicity, bifurcation, etc.). The numer-ical solution of this type of system is a domain of study in itself with a flour-ishing literature. Very efficient numerical methods to solve systems of ODEsare implemented in MATLAB, as in most such software. The first model of reaction
4、that we shall study in this chapter can be completely solved using such a standard package. We will therefore use the ode solvers provided by MATLAB , assuming that the user masters the underlying theory and the basic concepts such as convergence, stability, and precision (see Chap. 1). The other mo
5、del includes a delay term. We choose here not to use the delayequation solver dde23 and describe a specific numerical treatment. Both are examples of models presented in Hairer, Norsett, and Wanner (1987).We first study the so-called Brusselator model, which involves six reactants and products A,B,D
6、,E,X, and Y :342 Nonlinear Differential Equations: Application to Chemical Kinetics Av1 X, B + Xv2 Y + D,bimolecular reaction, 2X + Yv3 3X,trimolecular autocatalytic reaction, Xv4 E,(2.1)where viare the constant chemical reaction rates. The concentrations of the species as functions of time t are de
7、noted by A(t), B(t), D(t), E(t), X(t), and Y (t). Mass conservation in the chemical reactions leads to the followingdifferential equations: A?= v1A, B?= v2BX, D?= v2BX, E?= v4X, X?= v1A v2BX + v3X2Y v4X, Y?= v2BX v3X2Y.We start by eliminating the two equations governing the production of species D a
8、nd E, since they are independent of the four others: A?= v1A, B?= v2BX, X?= v1A v2BX + v3X2Y v4X, Y?= v2BX v3X2Y.The system can be furthermore simplified by assuming that A and B are kept constant and by taking all reactions rates equal to 1. The resulting system of two equations with two unknowns c
9、an be written as the initial value problem ?U?(t) = F(U(t), U(0) = U0= (X0,Y0)T,(2.2)where U(t) = (X(t),Y (t)Tis the vector modeling the variations of concen- tration of substances X and Y , andF(U) =?A (B + 1)X + X2Y BX X2Y? .2.2 Stability of the SystemThe stability of the system is its propensity
10、to evolve toward a constant or steady solution. This steady solution U(t) = Uc, if it exists, satisfies U?(t) = 0,and can therefore be calculated by solving F(Uc) = 0. The solution Ucis called a critical point. In the above example it is easy to compute: Uc= (A,B/A)T.The stability of the system can
11、also be regarded as its ability to relax in finite2.2 Stability of the System35time to the steady state when a perturbation (t) = U(t) Ucis appliedto the solution. In order to study the influence of variations (t), the right- hand side of the system is linearized around the critical point using a Ta
12、ylor expansion:U?(t) = F(U) = F(Uc) + FU=Uc(U Uc) + O(|U Uc|2),whereF =F1 XF1 YF2 XF2 Y =?2XY (B + 1)X2 B 2XYX2? .Assuming small variations (t), the term O(|U Uc|2) can be neglected,leading to the linear differential system ?(t) = J(t), (0) = 0,(2.3)where the Jacobian matrix J = FU=Ucis in this case
13、J =?B 1A2 BA2? .In the case that J is diagonalizable, it can be decomposed as J = MDM1, where Dij= iijand its integer powers are Jn= MDnM1for n 0. Werecall the definition of the exponential of a matrix J (see for instance Allaire and Kaber (2006):eJ=?n=01 n!Jn=?n=01 n!MDnM1= M?n=01 n!Dn?M1= MeDM1,wh
14、ere eDis the diagonal matrix (eD)ij= ijei, formed out of the exponentialof the eigenvalues of the matrix J. With this definition, the differential system (2.3) can be directly integrated:(t) = etJ0.(2.4)The long-time behavior is obtained by making t + in the exact solution (2.4) of the linearized sy
15、stem (2.3). If all eigenvalues of J have negative real part, then et 0 as t +. Therefore the matrix eJt= MeDtM1 0 as t + and the solution (t) goes to 0. The Taylor expansion around the critical point is valid in this case, and the solution of the nonlinear system (2.2) tends toward the critical poin
16、t. In this very simple example, the eigenvalues of the matrix J can be explic- itly calculated as the roots of the characteristic polynomial. The reader can easily verify that they are362 Nonlinear Differential Equations: Application to Chemical Kinetics=B A2 1 2,with = (A2 B + 1)2 4A2,and that their real part is negative if B 1.5, the system
