Nonstiff system

Integrate the following nonstiff system (Lapidus and Seinfeld, 1971) ... 

ODEPACK A collection of codes for solving stiff and nonstiff systems of initial value problems. 

Petzold, L.R., 1983. Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations. SIAM J. Sci. Stat. Comput. 4, 136-148. 

While both of these methods work adequately with nonstiff systems, delay times that are long compared with characteristic times in the system, such as the period of oscillation, can lead to serious numerical difficulties in systems like the Oregonator that are very stiff in the absence of delay. 

To solve an ODE without a solve block, use an ODE solver. Mathcad offers ODE solvers for stiff and nonstiff systems of first-order ODEs, as outlined in Table 5.3. 

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. 

For reasons that will be explained in due course, a substitution iterative method is adopted when the system to be solved is nonstiff and, in this case, the algorithm adopted belongs to the Adams-Moulton family. Conversely, in stiff problems, the nonlinear system is solved using the Newton method and the algorithm belongs to the Gear family. 

A bit of confusion may arise between stiff and nonstiff problems and well- and ill-conditioned systems. For example, the equation... 

The rigidity of the system of differential equations depends on the equations, their initial conditions, and the nnmerical method. Nonstiff methods can be employed to solve stiff problems, but these require much more computational time. One example is the propagation of flame fronts. The stiffness of the equations has to do with complex chemical process and differences in time scales. 

The eigenvalues of J are —k cu and —k2- When the activated species is very reactive, h k CM and the system is stiff Let us examine what happens to the performances of ode45 and odel 5s. QSSA ex.m uses cputime to compare the CPU times required to solve the ODE-IVP with the two solvers when k cu= I and kz is increased from a value of 1 (Table 4.2). As the system becomes stiff, ode45 requires more CPU time to simulate the response, due to a need to use very small time steps to preserve numerical stability, odel 5s performs much better when the system is stiff, showing little change in performance. The concentrations of A, A, and B are plotted for the nonstiff case 2 = 1 in Figure 4.8. As expected, ca initially grows as it is produced by the first reaction, and then decreases later as it is consumed by the second reaction. 

The two main routines for solving ODE-IVPs are ode45 and odelBs. For nonstiff problems, the exphcit ode45 method is recommended. For stilF systems, the imphcit ode15s is preferred. A hst of available ODE solvers is returned by help funfun. For... 

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