Runge-Kutta procedure

Complex reactions require the solution of simultaneous differential equations, and the Runge-Kutta procedure is applicable to these problems. To illustrate the method, Scheme XIV will be used. The rate equations are, in incremental form. 

For each system considered, an appropriate computer program was written for solving the equations involved in the modeling presented in Section 4.1. The first-order nonlinear differential equations were solved by numerical integration using the Runge-Kutta procedure [141]. 

Experience shows that the relatively slow Runge - Kutta procedure is quite robust and hence it is a good choice for a first try. 

M is a singular Matrix. Zero entries on the main diagonal of this matrix identify the algebraic equations, and all other entries which have the value 1 represent the differential equation. The vector x describes the state of the system. As numeric tools for the solution of the DAE system, MATLAB with the solver odel5s was used. In this solver, a Runge Kutta procedure is coupled with a BDF procedure (Backward Difference Formula). An implicit numeric scheme is used by the solver. 

A simple computer program, SIMPLATE, written in FORTRAN that implements the above procedures for obtaining the similarity profiles for velocity and temperature is available as discussed in the Preface. In this program, the solution to the set of equations defining the velocity profile function is obtained using the basic Runge-Kutta procedure. 

A mathematical model of the kinetics using a RUNGE-KUTTA procedure was established and is in good agreement with the observations. Concentration profiles of all different copper ions are obtained and the redox-potential of the Cu+/Cu++-equilibrium in the electrolyte can be estimated. 

The resulting set of differential equations is numerically integrated by standart Runge Kutta procedure. The different parameters of the model are determined to obtain the best agreement between the theory and the experimental results. 

The calculation of the reactor now proceeds as follows. The gas-phase equations are integrated by means of a Runge-Kutta procedure. To do this, pm , Pcoi and T, have to be calculated first. This cannot be done directly from Eq. (j) since this equation contains and t)m which are functions of (pm ),- Assume first a value of (Pm )i- From the equilibrium relation (Eq. (h)), which replaces Eq. (f), pcoj is calculated, using the relation between Pm , Pcoj. and the other partial pressures. 

For the investigation of stationary waves in steady flow, all partial derivatives with respect to time are removed from equations (1) - (4) and (7) - (9). With the help of equations (1) and (5), the continuity, momentum and energy equations (2) - (4) may then be recast as a set of three simultaneous equations for dug/dx, dTg/dx and dp/dx. Equations (1) and (7) - (9) furnish expressions for duf/dx, dn/dx, dTf/dx and dm/dx. The resulting set of seven simultaneous first order differential equations can then be integrated numerically using a fourth order Runge-Kutta procedure. 

The numerical integration of (9.74) was carried out using the fourth-order Runge-Kutta procedure with adaptive step-size control as described in detail by Press et al. (1992). The parameters in the continuum description of the energy were chosen to be = 0.03 J/m, /3i = 15 J/m, tq = IJ/m, Cm = —0.01 and (Ss = 2.86 J/m. It is assumed that the elastic modulus and the Poisson s ratio are lO N/m and 0.3, respectively, for both the film and substrate materials. For the compressively strained film, the surface energy of the film (sketched schematically in Figure 8.26) attains a minimum when 6 = 0.12, which implies that the sidewalls of the stepped mounds would evolve naturally toward this angle. 

Hartree-Fock level of accuracy, to obtain the stagnation graph of Dnh Cnh) compounds. The third-order hnear autonomous system for the flow was integrated using Runge-Kutta procedures [108]. 

In this way, after the duration of the applied field is over, field free multistate nonadiabatic dynamics can be further carried out. The Eqs. 17.17 or 17.18 are solved numerically using e.g. the fourth order Runge-Kutta procedure. 

Equations (9.1), (9.2), and (9.3) are ordinary differential equations in which distance is the independent variable. The technique of integration is to start from a perturbed full equilibrium condition at the hot boundary of the ffame and integrate backwards across the ffame by an explicit method. Dixon-Lewis et al. (1979a,b 1981) used a fourth-order Runge-Kutta procedure with variable step size for this purpose. We continue here by reviewing brieffy the application of the method with both partial equilibrium and quasi-steady-state assumptions. 

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