Ordinary differential equations stiffness conditions

The solution of nonlinear equations is, therefore, of significant interest not only as an independent problem but also in relation to the solution of DAE (differential algebraic equation) and ODE (ordinary differential equation) stiff problems with both initial and boundary conditions (Buzzi-Ferraris and Manenti, 2015). 

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. 

Simultaneous to the graph creation, kinetic properties in each vRxn are used to create the appropriate reaction rate equations (ordinary differential equations, ODE). These properties include rate constants (e.g., Michaelis constant, Km, and maximum velocity, Vmax, for enzyme-catalyzed reactions, and k for nonenzymatic reactions), inhibitor constants, A) and modes of inhibition or allosterism. The total set of rate equations and specified initial conditions forms an initial value problem that is solved by a stiff ODE equation solver for the concentrations of all species as a function of time. The constituent transforms for the each virtual enzyme are compiled by carefully culling the literature for data on enzymes known to act on the chemicals and chemical metabolites of interest. 

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. 

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. 

The mathematical model is shown in Table DC. It consists of a system of c+1 ordinary differential equations with initial conditions, these ODE s are stiff, even in the isothermd case, and of course stiffer in the adiabatic case, because of the exponential term due to ARRHENIUS law. 

The state trajectory u t) is computed by the implicit integrator DDASSL (Petzold 1982 Brenan, Campbell, and Petzold 1989). updated here to handle the initial condition of Eq. (B.1-2). The DDASSL integrator is especially designed to handle stiff, coupled systems of ordinary differential and algebraic equations. It employs a variable-order, variable-step predictor-corrector approach initiated by Gear (1971). The derivative vector applicable at t +i. is approximated in the corrector stage by a... 

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