Polynomial differential equation

Physical modeling is not as accurate as mathematical modeling. This should be attributed to the fact that in dimensionless equations, the dependent number is expressed as a monomial product of the determining numbers, whereas the corresponding phenomena are described by polynomial differential equations. Moreover, errors in the experimental determination of the several constants and powers of the dimensionless equations can also lead to inaccuracies. We should also keep in mind that the dimensionless-number equations are only valid for the limits within which the determining parameters are varied in the investigations of the physical models. 

The clearing up of this question is important from the point of view of practical chemistry as well as of mathematics. In the first place, if a polynomial differential equation has been fitted to experimental data, then it is a question, whether this equation can be considered as a model of reactions. In the second place, utilising the special structure of the kinetic differential equations, surprisingly strong theorems exist for the qualitative properties of the solutions. More precisely, certain systems of differential equations can be studied more efficiently if they can be models of chemical reactions. 

Section 4.7 treats an inverse problem whose solution can be formulated. This is the problem of the characterisation of kinetic differential equations among the polynomial differential equations. In other words how can one decide if a... 

Another kind of approach has been initiated in the recent investigations by Toth and Hars (1986a). They studied linear transforms of the Lorenz equation (1968) and of the Rossler model (1976) in order to obtain kinetic models. The failure of their efforts underline the importance of negative cross-effects. Based upon these results the following conjecture can be formulated if a nonkinetic polynomial differential equation shows chaotic behaviour then it cannot be transformed into a kinetic one. 

Polynomial differential equations, kinetic differential equations, kinetic initial value problems... 

Polynomial differential equations without negative cross-effects will usually be called kinetic differential equations from now on. 

Is the lack of negative cross-effects not too strong a restriction in the sense that a randomly selected polynomial differential equation is usually nonkinetic. How dense is the set of kinetic differential equations within the set of polynomial ones ... 

According to several different real situations different definitions have been given of the random event that a polynomial differential equation is kinetic (Toth, 1981b, pp. 44-8). The results can be summarised as follows. If one selects a polynomial differential equation with fixed coefficients and the random selection only concerns the exponents than the probability of getting a kinetic differential equation is 1. If the exponents are fixed and the coefficients are randomly chosen then the probability of getting a kinetic differential equation is 0. Finally, as a consequence of the statements above, if both the coefficients and the exponents are randomly selected then the probability of getting a kinetic differential equation is again 0. 

Verify that a polynomial differential equation without negative crosseffect can be induced by a reaction (called canonic reaction) that is constructed in the following way ... 

Try to formulate and give a (constructive) proof of the uniqueness theorem on the inducing reaction to a given polynomial differential equation mentioned in Subsection 4.7.1.4. 

Every change of time scale (including every change of time unit) preserves the character of polynomial differential equations in the sense above. 

Both deterministic and stochastic models can be defined to describe the kinetics of chemical reactions macroscopically. (Microscopic models are out of the scope of this book.) The usual deterministic model is a subclass of systems of polynomial differential equations. Qualitative dynamic behaviour of the model can be analysed knowing the structure of the reaction network. Exotic phenomena such as oscillatory, multistationary and chaotic behaviour in chemical systems have been studied very extensively in the last fifteen years. These studies certainly have modified the attitude of chemists, and exotic begins to become common . Stochastic models describe both internal and external fluctuations. In general, they are a subclass of Markovian jump processes. Two main areas are particularly emphasised, which prove the importance of stochastic aspects. First, kinetic information may be extracted from noise measurements based upon the fluctuation-dissipation theorem of chemical kinetics second, noise may change the qualitative behaviour of systems, particularly in the vicinity of instability points. 

The 0 part of the differential in equation 11.46 does have a known solution. The solution is a set of functions known as associated Legendre polynomials. (As with the Hermite polynomials, differential equations of the form in equation 11.46 had been previously studied, by the French mathematician Adrien Legendre, but for different reasons.) These polynomials, listed in Table 11.3, are functions of 0 only, but have two indices labeling the functions. One of the indices, an integer denoted , indicates the maximum power, or order, of 0 terms. (It also indicates the total order of the combination of cos 0 and sin 0 terms.) The second index, m, specifies which... 

Recently, Nicolini et al. (2013a, b) suggested a new approach for the calculation of low-dimensional slow manifolds in chemical kinetic systems. They transformed the original system of polynomial differential equations, which describes the chemical evolution, into a universal quadratic format. A region of attractiveness was found in the phase space, and a state-dependent rate function was defined that describes the evolution of the system. 

In the work of King, Dupuis, and Rys [15,16], the mabix elements of the Coulomb interaction term in Gaussian basis set were evaluated by solving the differential equations satisfied by these matrix elements. Thus, the Coulomb matrix elements are expressed in the form of the Rys polynomials. The potential problem of this method is that to obtain the mabix elements of the higher derivatives of Coulomb interactions, we need to solve more complicated differential equations numerically. Great effort has to be taken to ensure that the differential equation solver can solve such differential equations stably, and to... 

W. Gautschi. Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math., 3 381-397, 1961. 

The nonlinear element here is I => /(eg), eg being the grid voltage. If one introduces the usual approximation of the nonlinear characteristic by a polynomial, it can be shown that the differential equation takes the form... 

This section describes a number of finite difference approximations useful for solving second-order partial differential equations that is, equations containing terms such as d f jd d. The basic idea is to approximate f 2 z. polynomial in x and then to differentiate the polynomial to obtain estimates for derivatives such as df jdx and d f jdx -. The polynomial approximation is a local one that applies to some region of space centered about point x. When the point changes, the polynomial approximation will change as well. We begin by fitting a quadratic to the three points shown below. 

Let us notice that due to orthogonality of Legendre s polynomials many functions can be represented as a series, which is similar to Equation (1.162), and this fact is widely used in mathematical physics. Now, we will derive the differential equation, one of the solutions of which are Legendre s functions. 

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. 

The radial functions Sni p) and R i(r) may be expressed in terms of the associated Laguerre polynomials L p), whose definition and mathematical properties are discussed in Appendix F. One method for establishing the relationship between Sniip) and L p) is to relate Sni p) in equation (6.50) to the polynomial L p) in equation (F.15). That process, however, is long and tedious. Instead, we show that both quantities are solutions of the same differential equation. 

The differential equation satisfied by the associated Laguerre polynomials is given by equation (F.16) as... 

We next derive some recurrence relations for the Hermite polynomials. If we differentiate equation (D.l) with respect to s, we obtain... 

To find the differential equation that is satisfied by the Hermite polynomials, we first differentiate the second recurrence relation (D.6) and then substitute (D.6) with n replaeed by n — 1 to eliminate the first derivative of i ( )... 

To find the differential equation satisfied by the polynomials Pi p), we first multiply equation (E.5) by —p and add the result to equation (E.6) to give... 

Equation (F.5) can be used to find the differential equation satisfied by the polynomials Lkip). We note that the function /(p) defined as... 

When the polynomials L p) are introduced with equation (F.9), the differential equation is... 

In order to obtain well-behaved solutions for the differential equation (G.8), we need to terminate the infinite power series mi and U2 in (G.16) to a finite polynomial. If we let A equal an integer n (n = 0,1,2, 3,...), then we obtain well-behaved solutions... 

Since n and I are integers, equation (G.51) is identical to the associated Laguerre differential equation (F.16) with k = n + I and j = 21 + 1. Thus, the solutions u(p) are proportional to the associated Laguerre polynomials (p), whose properties are discussed in Appendix F... 

The inherent dynamic properties of a model are embedded in the characteristic polynomial of the differential equation. More specifically, the dynamics is related to the roots of the characteristic polynomial. In Eq. (2-27), the characteristic equation is xs + 1 = 0, and its root is -1/x. In a general sense, that is without specifying what C in is and without actually solving for C (t), we... 

We know that G(s) contains information on the dynamic behavior of a model as represented by the differential equation. We also know that the denominator of G(s) is the characteristic polynomial of the differential equation. The roots of the characteristic equation, P(s) = 0 pb p2,... pn, are the poles of G(s). When the poles are real and negative, we also use the time constant notation ... 

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