System of differential equations

The processes that are developing in time can be described by a system of differential equations. 

We should first of all ascertain which parameters determine a process under consideration. The validated approach to the problem implies that all parameters should have the well-defined physical meaning. 

Next step is the development of a physical model of the process. Then we convert our physical model to a mathematical language. The mathematical formulation follows a physical model that has been worked in detail. 

Let us suppose that N parameters yi, y2. yN affect the process under study. In a general case every parameter depends upon time and on all other parameters, that is. 

Interatomic Bonding in Solids Fundamentals,Simulation,andApplicaUons, First Edition. Valim Levitin. 

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At first we tried to explain the phenomenon on the base of the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary [12]. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. We worked out the mathematical description of both gas-vapor diffusion and evaporation-condensation processes in cone s channel. Solving the system of differential equations for evaporation-condensation processes, we ve derived the formula for the dependence of top s (or inner) liquid column growth on time. But the calculated curves for the kinetics of inner column s length are 1-2 orders of magnitude smaller than the experimental ones [12]. 

As an example we take again the Lindemaim mechanism of imimolecular reactions. The system of differential equations is given by equation (A3.4.127T equation (A3.4.128 ) and equation (A3.4.129T The rate coefficient matrix is... 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

M. Hochbruck, Ch. Lubich, and H. Selhofer Exponential integrators for large systems of differential equations. SIAM J. Sci. Comp. (1998) (to appear)... 

In this section we consider the classical equations of motion of particles in cases where the highest-frequency oscillations are nearly harmonic The positions y t) = j/i (t) evolve according to the second-order system of differential equations... 

These equations form a fourth-order system of differential equations which cannot be solved analytically in almost all interesting nonseparable cases. Further, according to these equations, the particle slides from the hump of the upside-down potential — V(Q) (see fig. 24), and, unless the initial conditions are specially chosen, it exercises an infinite aperiodic motion. In other words, the instanton trajectory with the required periodic boundary conditions,... 

Initial conditions for the system of differential equations shown before are given by the values of state variables known at the inlet of the reactor ... 

Some systems may show stiff properties, especially those for oxidations. Here the system of differential equations to be integrated are not stiff . Even at calculated runaway temperature, ordinary integration methods can be used. The reason is that equilibrium seems to moderate the extent of the runaway temperature for the reversible reaction. 

One of the possibilities is to study experimentally the coupled system as a whole, at a time when all the reactions concerned are taking place. On the basis of the data obtained it is possible to solve the system of differential equations (1) simultaneously and to determine numerical values of all the parameters unknown (constants). This approach can be refined in that the equations for the stoichiometrically simple reactions can be specified in view of the presumed mechanism and the elementary steps so that one obtains a very complex set of different reaction paths with many unidentifiable intermediates. A number of procedures have been suggested to solve such complicated systems. Some of them start from the assumption of steady-state rates of the individual steps and they were worked out also for stoichiometrically not simple reactions [see, e.g. (8, 9, 5a)]. A concise treatment of the properties of the systems of consecutive processes has been written by Noyes (10). The simplification of the treatment of some complex systems can be achieved by using isotopically labeled compounds (8, 11, 12, 12a, 12b). Even very complicated systems which involve non-... 

A) Definition of Stability According to Liapounov.—Given a system of differential equations of an autonomous system... 

Stability on the Basis of Abridged Equations.—The argument used in deriving the characteristic equation (3.8) was to neglect P2(x,y) and Qz(x,y), and to proceed on the basis of linear terms. It is possible to obtain more precise information regarding the validity of this assumption. Consider the system of differential equations... 

The error in Runge-Kutta calculations depends on h, the step size. In systems of differential equations that are said to be stiff, the value of h must be quite small to attain acceptable accuracy. This slows the calculation intolerably. Stiffness in a set of differential equations arises in general when the time constants vary widely in magnitude for different steps. The complications of stiffness for problems in chemical kinetics were first recognized by Curtiss and Hirschfelder.27... 

This system of differential equations can be written in matrix form as... 

The Cauchy problem for a system of differential equations of first order. Stability condition for Euler s scheme. We illustrate those ideas with concern of the Cauchy problem for the system of differential equations of first order... 

Equations of gas dynamics with heat conductivity. We are now interested in a complex problem in which the gas flow is moving under the heat conduction condition. In conformity with (l)-(7), the system of differential equations for the ideal gas in Lagrangian variables acquires the form... 

Example With this aim, it seems worthwhile giving the following system of differential equations ... 

This means that the system of differential equations (55)-(56) generates an approximation of order 1 in a summarized sense to the Cauchy problem (51) under the extra restrictions on the existence and boundedness of the derivative A t)cPu/dt in some suitable norm. 

Since the system of differential equations (56) approximates equation (50) in a summarized sense in compliance with approximating equation (85) by equation (86) with the number a, the additive scheme (86)-(88) generates an approximation of 0 t + /ip) ... 

As was pointed out, the set (1.31) is a rather complicated system of differential equations, and it is natural to attempt to replace them by much simpler equations. This is the first motivation for introducing the potential of the attraction field. Taking into account the equality... 

On integration of the above system of differential equations until B has been completely consumed U, W, and Z values are obtained. Monoazo dyes concentrations are then calculated using Ekjns. (5.4-175) and (5.4-176). The concentration of 1-naphthol is calculated knowing its surplus for the reaction and the concentration of bisazo dye S can be determined from any of the mass balances ... 

The analytic solution of the system of differential equations in eq. (39.14) can be written as follows ... 

Attree RW, Cabell MJ, Cushing RL, Pieroni JJ (1962) A calorimetric determination of the half-life of thorium-230 and a consequent revision to its neutron capture cross section. Can J Phys 40 194-201 Bateman H (1910) Solution of a system of differential equations occurring in the theory of radioactive transformations. Proc Cambridge Phil Soc 15 423-427 Beattie PD (1993) The generation of uranium series disequilibria by partial melting of spinel peridotite ... 

We strongly suggest the use of the reduced sensitivity whenever we are dealing with differential equation models. Even if the system of differential equations is non-stiff at the optimum (when k=k ), when the parameters are far from their optimal values, the equations may become stiff temporarily for a few iterations of the Gauss-Newton method. Furthermore, since this transformation also results in better conditioning of the normal equations, we propose its use at all times. This transformation has been implemented in the program for ODE systems provided with this book. 

The reciprocals of the time constants, x, and x2, are the rate constants k, and k2. The weights of the exponentials (ii and w2) are complicated functions of the transition rates in Eq. (6.25). Flowever, the rate constants are eigenvalues found by solving the system of differential equations that describe the above mechanism. A, and k2 are the two solutions of the quadratic equation ... 

Crosslinking of many polymers occurs through a complex combination of consecutive and parallel reactions. For those cases in which the chemistry is well understood it is possible to define the general reaction scheme and thus derive the appropriate differential equations describing the cure kinetics. Analytical solutions have been found for some of these systems of differential equations permitting accurate experimental determination of the individual rate constants. 

In Eq. (2.30), F is the Fock operator and Hcore is the Hamiltonian describing the motion of an electron in the field of the spatially fixed atomic nuclei. The operators and K. are operators that introduce the effects of electrons in the other occupied MOs. Hence, when i = j, J( (= K.) is the potential from the other electron that occupies the same MO, i ff IC is termed the exchange potential and does not have a simple functional form as it describes the effect of wavefunction asymmetry on the correlation of electrons with identical spin. Although simple in form, Eq. (2.29) (which is obtained after relatively complex mathematical analysis) represents a system of differential equations that are impractical to solve for systems of any interest to biochemists. Furthermore, the orbital solutions do not allow a simple association of molecular properties with individual atoms, which is the model most useful to experimental chemists and biochemists. A series of soluble linear equations, however, can be derived by assuming that the MOs can be expressed as a linear combination of atomic orbitals (LCAO)44 ... 

This can be inserted into the current expressions (30) and (32), whose divergences enter into the time development equations for the densities. (Note that the denominator of (76) simplifies greatly in all cases except where the silicon is nearly intrinsic. Here again, if the charge states are equilibrated, n+ and n can be eliminated in favor of n0, via (3), (4), and (75). Whether AH and /or nDH must be retained as distinct variables in the system of differential equations, or whether they, too, can be eliminated in favor of 0, will depend on whether or not they are able to come quickly into local equilibrium with the monatomic species. 

The system of differential equations, which describes the process on the basis of all stages presented above, agree well with experiment and reproduces the oscillation regime of the process [223],... 

Solutions are presented in the form of equations, tables, and graphs—most often the last. Serious numerical results generally have to be obtained with computers or powerful calculators. The introductory chapter describes the numerical procedures that are required. Inexpensive software has been used here for integration, differentiation, nonlinear equations, simultaneous equations, systems of differential equations, data regression, curve fitting, and graphing. 

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