Ordinary differential equation standard form

The ordinary differential equations for f and C now form a fifth-order system which can be solved using a standard NAG library routine. The results are shown in Fig. 10.73. This figure also shows the numerical results for concentration obtained using a full numerical approach, and there is reasonable agreement between the two. 

The treatment in this chapter has been theoretical. For a brief, dear, and very practical description of computational details for a number of standard problems, [10] is unsurpassed, and [12] can be recommended for programming techniques for automatic computers. For information on ordinary differential equations, the reader should consult [2], and for partial differential equations, [1]. For general methods of reduction to algebraic form as well as methods of solution, see [5], [7], and [8]. 

Since we can take each variable except msio2 1° be constant, Equation 26.3 has the form of an ordinary differential equation in time. We can use standard techniques to solve the equation for msio2(0- The solution corresponding to the initial condition msio2 = m0 at t =0 is,... 

The linearfirst-onkr ordinary differential equation has the standard form of... 

Since the orthogonal collocation or OCFE procedure reduces the original model to a first-order nonlinear ordinary differential equation system, linearization techniques can then be applied to obtain the linear form (72). Once the dynamic equations have been transformed to the standard state-space form and the model parameters estimated, various procedures can be used to design one or more multivariable control schemes. 

An important issue in the boundary-layer problem, and in differential-algebraic equations generally, is the specification of consistent initial conditions. We think first of the physical problem (not in Von Mises form), since the inlet profiles of u, v, and T, as well as pressure p, must be specified. However, all the initial conditions are not independent, as they would be for a system of standard-form ordinary differential equations. So assuming that the axial velocity u and temperature T profiles are specified, the radial velocity must be required to satisfy certain constraints. 

Develop two method-of-lines simulations to solve this problem. In the first, formulate the problem as standard-form ordinary differential equations, y7 = ff(f, y). In the second, formulate the problem in differential-algebraic (DAE) form, 0 = g(t, y, y ). Standard-form stiff, ordinary-differential-equation (ODE) solvers are readily avalaible. DAE solvers are less readily available, but Dassl is a good choice. The Fortran source code for Dassl is available at http //wwwjietlib.org. 

Consider a first-order ordinary-differential equation in the standard form... 

Many chemical-kinetics problems, such as the homogeneous mass-action kinetics problems discussed in Section 16.1, are easily posed as a system of standard-form ordinary differential equations (ODE),... 

Presume that a problem is described as an ordinary-differential-equation initial-value problem, such as the mass-action kinetics or plug-flow problems discussed earlier. In the standard form, such a problem might be written as... 

Taken together, the system of equations represents a set of stiff ordinary differential equations, which can be solved numerically. Because more than one dependent-variable derivative can appear in a single equation (e.g., the momentum equation has velocity and pressure derivatives), it is usually more convenient to use differential-algebraic equation (DAE) software (e.g., Dassl) for the solution rather than standard-form ODE software. 

Equation (140) is a second-order ordinary differential equation and standard methods [74] show that its solution must be of the form... 

We also use a restricted form of Equation 19 for the kinetics studies described previously. Smog chamber analyses uses just the first and last terms so that they depend on ordinary differential equations. These are solutions which describe the time-dependent behavior of a homogeneous gas mixture. We used standard Runge-Kutta techniques to solve them at the outset of the work, but as will be shown here, adaptations of Fade approximants have been used to improve computational efficiency. 

In Eq. (5.8.12) we already specified H in terms of T, Pm, o- The determination of G in the same variables is more involved we base our derivation on Eq. (1.13.19) adapted to the present situation. This is actually an ordinary differential equation of standard form since all variables save T are fixed. Invoking Eq. (1.3.27) as the solution to the first order differential equation (1.13.19) one obtains the expression... 

Adopting the Galerkin method as a particular form of weighted residuals, i.e., considering the weights W, to be the same as the trial functions N after standard transformations of integrals in the relation (11), the next system of the ordinary differential equations with respect to nodal concentrations Q(t) may be derived ... 

As mentioned above, the goal of MD is to compute the phase-space trajectories of a set of molecules. We shall just say a few words about numerical technicalities in MD simulations. One of the standard forms to solve these ordinary differential equations i.s by means of a finite difference approach and one typically uses a predictor-corrector algorithm of fourth order. The time step for integration must be below the vibrational frequency of the atoms, and therefore it is typically of the order of femtoseconds (fs). Consequently the simulation times achieved with MD are of the order of nanoseconds (ns). Processes related to collisions in solids are only of the order of a few picoseconds, and therefore ideal to be studied using this technique. 

In fact, the ordinary differential equations in each variable from such a separation have the standard form of the Lam6 equation [12] ... 

Solution of Eq. (3.5) can be obtained by any standard method for solving second order ordinary differential equation. It is also possible to carry out the reverse process, i.e., given a differential equation of Eq. (3.5), the corresponding integral form of Eq. (3.4) can be obtained by using calculus of variations. 

The new set now has N+l coupled ordinary differential equations (Eqs. 7.5 and 7.4). Thus, the standard form of Eq. 7.1 is recovered, and we are not constrained by the time appearing explicitly or implicitly. In this way, numerical algorithms are developed only to deal with autonomous systems. 

In the simulation, the first step is to develop mathematical modeling. The modeling, based on first principles, is done by applying a standard input/ output approach for time-dependent systems with one or multiple inputs x t) and one or multiple outputs y t). The mathematical descriptions of components and hardware are formulated in the form of ordinary differential equations with the time t as the independent variable. The system description is represented mathematically by a system of coupled, nonlinear, first-order differential (or integral) equations ... 

Ordinary Points of a Linear Differential Equation. We shall have occasion to discuss ordinary linear differential equations of the second order with variable coefficients whose solutions cannot he obtained in terms of Lhe elementary functions of mathematical analysis, la such cases one of the standard procedures is to derive n pair of linearly independent solutions in the form ofinfinite series and from these series to compute tables of standard solutions. With the aid of such tables the solution appropriate to any given initial conditions may then he readily found. The object of this note is to outline briefly the procedure to he followed in these instances for proofs of the theorems... 

This is now an ordinary second-order differential equation. The solution Y = const, corresponds to thermal equilibrium according to Eq. (11.21). There is, however, another steady-state solution to the equation, if it can be brought into the standard form... 

The yeast cell cycle has also been analyzed at this high level of chemical detail [17]. The molecular mechanism of the cycle in the form of a series of chemical equations was described by a set of ten nonlinear ordinary differential kinetic rate equations for the concentrations of the cyclins and associated proteins and the cell mass, derived using the standard principles of biochemical kinetics. Numerical solution of these equations 3uelded the concentrations of molecules such as the cyclin, Cln2, which is required to activate the cell cycle, or the inhibitor, Sid, which helps to retain the cell in the resting Gi phase. The rate constants and concentrations ( 50 parameters) were estimated from published measurements and adjusted so that the solutions of the equations yielded appropriate variations, i.e., similar to those experimentally measured, of the concentrations of the constituents of the system and the cell mass. The model also provides a rationalization of the behavior of cells with mutant forms of various system constituents. 

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