Ordinary differential equations standard

Solving Newton s equation of motion requires a numerical procedure for integrating the differential equation. A standard method for solving ordinary differential equations, such as Newton s equation of motion, is the finite-difference approach. In this approach, the molecular coordinates and velocities at a time it + Ait are obtained (to a sufficient degree of accuracy) from the molecular coordinates and velocities at an earlier time t. The equations are solved on a step-by-step basis. The choice of time interval Ait depends on the properties of the molecular system simulated, and Ait must be significantly smaller than the characteristic time of the motion studied (Section V.B). 

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]. 

For fast reactions Da becomes large. Based on that assumption and standard correlations for mass transfer inside the micro channels, both the model for the micro-channel reactor and the model for the fixed bed can be reformulated in terms of pseudo-homogeneous reaction kinetics. Finally, the concentration profile along the axial direction can be obtained as the solution of an ordinary differential equation. 

This set of ordinary differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. When Euler s method is used to integrate in time, the equations become... 

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,... 

Under stationary conditions dc /dt = 0, and an ordinary differential equation results with Eq. (10.5) as boundary conditions, which can be solved explicitly by standard techniques. The resulting expression for the current density is ... 

Here I describe a simple numerical method for solving Maxwell s equation in the frequency domain. As the structure to be analysed is onedimensional, Maxwell s equations turn into a system of two coupled ordinary differential equations that can be solved with standard numerical routines. 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

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

Once again, we can use the standard solution technique for ordinary differential equations (Kreyszig, 1982), resulting in a solution. 

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... 

Equation (6.2.21) is an ordinary differential equation having large parameter h r). Its solution under certain standard conditions could be obtained by means of the steepest descent (called also VKB) method. 

The set of four ordinary differential equations (7.64) to (7.67) for the dynamical system are quite sensitive numerically. Extreme care should be exercised in order to obtain reliable results. We advise our students to experiment with the standard IVP integrators ode... in MATLAB as we have done previously in the book. In particular, the stiff integrator odel5s should be tried if ode45 turns out to converge too slowly and the system is thus found to be stiff by numerical experimentation. 

If the one-dimensional approximation is adequate, the problem is reduced to a routine integration of a set of ordinary differential equations. Procedures for integrating such sets of equations are given in standard works on numerical analysis (see, for example, H5, M2, and M3). The working equations for simple forward-difference equations are given in the next paragraph. 

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 ref 146 the authors present a non-standard (nonlinear) two-step explicit P-stable method of fourth algebraic order and 12th phase-lag order for solving second-order linear periodic initial value problems of ordinary differential equations. The proposed method can be extended to be vector-applicable for multi-dimensional problem based on a special vector arithmetic with respect to an analytic function. 

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... 

In the second, a solution is obtained for the sinusoidal steady state. Generally, through transformations of the type discussed in Example 1.8, this too requires solutions of ordinary differential equations. While in some cases numerical solution is required, analytic solutions are possible for a large number of problems. Analytic solutions to some typical equations are reviewed in this chapter. For more details, see standard textbooks on engineering math. ... 

---