Ordinary differential equation solvers

Hindmarsh, A., "Gear Ordinary Differential Equation Solver", UDID-3001, Rev. 1, August 20, 1972, Computer Center Library, University of California, Berkeley, CA. 

Equation (4.78) is a set of nonlinear algebraic equation and may be solved using various techniques [64], Often the nonlinear differential Eq. (4.77) are solved to the steady-state condition in place of the algebraic equations using the stiff ordinary differential equation solvers described in Chapter 2 [65], See Appendix I for more information on available numerical codes. 

CONP Kee, R. J., Rupley, F. and Miller, J. A. Sandia National Laboratories, Livermore, CA. A Fortran program (conp.f) that solves the time-dependent kinetics of a homogeneous, constant pressure, adiabatic system. The program runs in conjunction with CHEMKIN and a stiff ordinary differential equation solver such as LSODE (lsode.f, Hindmarsh, A. C. LSODE and LSODI, Two Initial Value Differential Equation Solvers, ACM SIGNUM Newsletter, 15, 4, (1980)). The simplicity of the code is particularly valuable for those not familiar with CHEMKIN. 

The user specified subroutines allow for connections to various other programs such as process simulators and ordinary differential equation solvers. Currently, MINOPT is connected to the DASOLV (Jarvis and Pantelides, 1992) integrator, and can solve MINLP models with differential and algebraic constraints. 

Ordinary Differential Equation Solvers in MATLAB For Initial Value Problems ... 

Hindmarsh, A. C. Gear Ordinary Differential Equation Solver. Tech. Report No. UCM-30001, Rev. 2. Lawrence Livermore Lab. 1972... 

A commercial stiff ordinary differential equation solver subroutine, DVOGER, is available in the IMSL Library (3). This subroutine uses Gear s method for the solution of stiff ODE s with analytic or numerical Jacobians. The pyrolysis model was solved using DVOGER and the analytical Jacobians of Eqs. (14) and (15). For a residence time of 0.0511 in dimensionless time, defined as t/t where 9... 

One particular class of ordinary differential equation solvers (ODE-solvers) handles stiff ODEs and these are widely known as stiff solvers. In our context, a system of ODEs sometimes becomes stiff if it comprises very fast and also very slow steps or relatively high and low concentrations. A typical example would be an oscillating reaction. Here, a highly sophisticated step-size control is required to achieve a reasonable compromise between accuracy and computation time. It is well outside the scope of this chapter to expand on the intricacies of modem numerical... 

A large number of ordinary differential equation solver software packages (i.e., ODE solvers) which are extremely user friendly have become available. We shall use POLYMATH S to solve the examples in the printed text. However, the CD-ROM contains an example that uses ASPEN, as well as all the MATLAB and POLYMATH solution programs to the example programs. With POLYMATH one simply enters Equations (E4-7.3) and (E4-7.4) and the corresponding parameter value into the computer Table E4-7.1) with the initial (rather, boundary) conditions and they are solved and displayed as shown in Figure E4-7.1,... 

For the ordinary differential equation solver (ODE solver), contact ... 

P. N. Brown, A. C. Hindmarsh, G. D. Byrne, Variable coefficient ordinary differential equation solver, procedure available on http //www.netlib.org (1989). 

ODE (Ordinary Differential Equation) solver option config.ODEMethod = 23 ... 

A large number of ordinary differential equation solver software packages (i.e., ODE solvers), which are extremely user friendly, have become available. We... 

The design Equation 14-2 and the rate law r(X) are now combined and solved using an ordinary differential equation solver ODE. The POLYMATH-program and the result is shown as follows ... 

The third term of heat balance (Equation 5.54) is the heat added to the system because of the eventual temperature difference between the injected flow V2 and the main flow denoted as — T). The above equations can be solved using a simple ordinary differential equation solver for each interval between two injection points. In this case, the boundary conditions of the /th interval have to be adapted considering the reaction mass injected at point J and its temperature as well as the temperature and concentrations at the end of the interval j - 1. 

Hindmarsh, A.C., 1980, LSODE and LSODI, two initial value ordinary differential equations solvers, ACM-SIGNUM Newslett., 15, 10-15. 

Click on d(x)+ on the left side [Ordinary differential Equations Solver] (see Figure A.3). 

Table 5.1 Ordinary differential equation solvers in MATLAB...

Using either the IMSL subroutine IVPAG or an ordinary differential equation solver in MATLAB, solve these equations for th and T22 as a function of time and calculate as a function of time. Take = 0.2, e = 0.01, and the rest of the parameters as given in Problem 3C.1. Compare the predicted values with the experimental ones for LDPE given in Appendix A.l, Table A.4. 

C.5 NumericalSolutionof Example 3.4. Solve the system of equations given in Example 3.4 using either one of the ordinary differential equation solvers in MATLAB or IVPAG in the IMSL subroutines. Assume initially that the rods are randomly oriented. 

The other question of concern in this problem was spherulite size. Because differential equation for determining R. We would then use the IMSL ordinary differential equation solver, IVPAG, or a MATLAB solver such as odel5s to find the radius, R, as a function of time. 

The numerical solution of the primary circuit is carried out in two stages. In the first stage, IHX flow, pump flows, pump speeds and the sodium levels are calculated utilizing a standard ordinary differential equation solver based on the Hamming s predictor-corrector method. 

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