Abstract
The existence of positive solutions to the system of ordinary differential equations related to the Belousov-Zhabotinsky reaction is established. The key idea is to use a new successive approximation of solutions, ensuring its positivity. To obtain the positivity and invariant region for numerical solutions, the system is discretized as difference equations of explicit form, employing operator splitting methods with linear stability conditions. Algorithm to solve the alternate solution is given.
Highlights
The existence of positive solutions to the system of ordinary differential equations related to the Belousov-Zhabotinsky reaction is established
To obtain the positivity and invariant region for numerical solutions, the system is discretized as difference equations of explicit form, employing operator splitting methods with linear stability conditions
We introduce a new difference scheme which produces positive numerical solutions for arbitrary large ∆t; for the partial difference equations, we impose the linear stability condition
Summary
The construction of time-local positive solutions to the system of first order ordinary differential equations (ODE) is discussed. For general matrix-valued functions F , one may construct ul+1 for each l ∈ N by perturbation theory, at least time-locally. Assume that ∥ul(t)∥ ≤ 2∥a∥ and uli (t) ≥ 0 hold for t ∈ [0, T0] and i = 1, . We can see that uli+1 ≥ 0 for i by the same contradiction argument above This means that ∥ul(t)∥ ≤ 2∥a∥ and uli ≥ 0 hold for all l ∈ N and i = 1, . In Theorem 1, it is not needed to use neither the existence of stable solutions to (P), comparison principle, nor a priori estimates by Lyapunov functions
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