Unrestricted Harmonic Balance III, Application to Running and Standing Chemical Waves
In former papers the method of “Unrestricted Harmonic Balance” (UHB) was developed for time-periodic phenomena in non-linear chemical reaction systems with mass action kinetics by Fourier expansion and solution of the non-linear algebraic system. This method is now applied to a chemical reaction-diffusion system characterized by the phenomenon of substrate inhibition that can form running and standing chemical waves depending on the choice of parameters. The system of parabolic partial differential equations with cyclic boundary conditions in time t and space s on a ring fiber of length L is transformed to a system of ODE by introduction of a constant profile wave variable w = 2π(s-ct)/L, where c is the wave velocity that is unknown in advance. The concentrations are found as a Fourier series of harmonics of w. Standing waves can be found as a limiting case of running waves or constructed directly by means of a special ansatz including only cosine-functions.
KeywordsFourier Series Standing Wave Substrate Inhibition Ring Fiber Parabolic Partial Differential Equation
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