Ordinary Differential Equations Systems

I presented a group of subroutines—CORE, CHECKSTEP, STEPPER, SLOPER, GAUSS, and SWAPPER—that can be used to solve diverse theoretical problems in Earth system science. Together these subroutines can solve systems of coupled ordinary differential equations, systems that arise in the mathematical description of the history of environmental properties. The systems to be solved are described by subroutines EQUATIONS and SPECS. The systems need not be linear, as linearization is handled automatically by subroutine SLOPER. Subroutine CHECKSTEP ensures that the time steps are small enough to permit the linear approximation. Subroutine PRINTER simply preserves during the calculation whatever values will be needed for subsequent study. 

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. 

Although in this chapter we have chosen to linearize the mathematical system after reduction to a system of ordinary differential equations, the linearization can be performed prior to or after the reduction of the partial differential equations to ordinary differential equations. The numerical problem is identical in either case. For example, linearization of the nonlinear partial differential equations to linear partial differential equations followed by application of orthogonal collocation results in the same linear ordinary differential equation system as application of orthogonal collocation to the nonlinear partial differential equations followed by linearization of the resulting nonlinear ordinary differential equations. The two processes are shown ... 

The plug-flow problem may be formulated with a variable cross-sectional area and heterogeneous chemistry on the channel walls. Although the cross-sectional area varies, we make a quasi-one-dimensional assumption in which the flow can still be represented with only one velocity component u. It is implicitly assumed that the area variation is sufficiently small and smooth that the one-dimensional approximation is valid. Otherwise a two- or three-dimensional analysis is needed. Including the surface chemistry causes the system of equations to change from an ordinary-differential equation system to a differential-algebraic equation system. 

Hindmarsh, A. C. "GEAR Ordinary Differential Equation System Solver" Lawrence Livermore Laboratory, Report UCID-30001, Revision 3, December, 1974. 

Given the initial conditions (concentrations of the 22 chemical species at t = 0), the concentrations of the chemical species with time are found by numerically solving the set of the 22 stiff ordinary differential equations (ODE). An ordinary differential equation system solver, EPISODE (17) is used. The method chosen for the numerical solution of the system includes variable step size, variable-order backward differentiation, and a chord or semistationary Newton method with an internally computed finite difference approximation to the Jacobian equation. 

The system of differential equations is integrated using CVODE numerical integration package. CVODE is a solver for stiff and nonstiff ordinary differential equation systems [60]. The fraction of dose absorbed is calculated as the sum of all drug amounts crossing the apical membrane as a function of time, divided by the dose, or by the sum of all doses if multiple dosing is used. 

Manipulation of symbolic expressions and numerics (e.g., differentiation integration Taylor series Laplace transforms ordinary differential equations systems of linear equations, polynomials, and sets vectors matrices and tensors)... 

A. C. Hindmatsh, GEAR Ordinary Differential Equation System Solver, UCID-30001 Rev. 2. Lawrence Livermore Lab., Livermore, California, 1972. 

Ordinary differential equation systems are broached in Chapter 2. Conditioning, stability, and stiffness are described in detail by giving specific information on how to handle them whenever they arise. The BzzMath library also implements a wide set of algorithms to solve classical problems and chemical/process engineering problems. 

Chapter 3 reports a collection of literature and industrial problems based on ordinary differential equation systems. The basics of the physical problem are described and the model behind it is given together as the initial conditions. Implementation tricks, special functions of the classes, and suggestions to improve the solution s accuracy and efficiency are provided through various examples. 

An ordinary differential equations system is indicated with the acronym ODE. 

Ordinary differential equation systems with boundary conditions (see Chapter 6). 

Ordinary Differential Equations Systems Others tend to... 

---