×

Realization of the method of continuation on a parameter for a system of two equations. (Russian. English summary) Zbl 1025.65029

The authors develop and generalize their previous results obtained in [Vychisl. Tekhnol. 6, 9-15 (2001; Zbl 0992.34007)].
The aim is, first, to construct functions \(x = x(\alpha)\) and \(y = y(\alpha)\) which solve the following system of nonlinear equations with a parameter \(\alpha\): \[ f(x,y,\alpha) = 0, \quad g(x,y,\alpha) = 0. \] The algorithm starts with studying an initial value problem to a system of two ordinary differential equations. Subsequent numerical integration makes it possible to find the parametric curve \(x = x(\alpha)\), \(y = y(\alpha)\). Here the authors use the \((m,k)\)-method of solving stiff systems suggested by E. A. Novikov in [Explicit methods for stiff systems, Nauka, Novosibirsk (1997; Zbl 0934.65067)].
Then, the method is illustrated by finding parametric relations of steady states for the Slin’ko-Chumakov model from the theory of oscillating catalytic reactions and for the Volterra–Sal’nikov model arising in the theory of polymerization.

MSC:

65H10 Numerical computation of solutions to systems of equations
65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
80A32 Chemically reacting flows
82D60 Statistical mechanics of polymers
PDFBibTeX XMLCite