On approximate solutions to the wavefront speed problem (Q1609877)

The authors proposed an approximate method to obtain the speed \(c\) of wavefronts for the typical reaction-diffusion equation \[ \partial_{t}\rho=\partial_{xx}\rho+f(\rho) \] and then applied this method to a variety of source terms \(f\) which are of practical interest. They derived some analytical results that were previously known from other methods, and showed that the approximate method makes an exact solution \( \rho(z)\) \((z=x-ct)\) unnecessary. Several problems (including Forest Fire research, Chemical Kinetics) for which analytical results are not known, are also tackled by means of the comparison of numerical simulation, the approximate method and the previous estimations (the upper and lower bounds of \(c\)). It concludes that the approximate method is more accurate than the previous ones. In the case of time-delayed equations \[ a\partial_{tt}\rho+\partial_{t}\rho=\partial_{xx}\rho+f(\rho)+af'(\rho) \partial_{t}\rho, \] where \(a<1\) is the dimensionless delay time, the renormalization group (RG) approach has been applied for the first time, and its results are compared to those obtained by the authors' approximate method. It is interesting that the new alternative approach (RG technique) is more useful in practice than the approximate variational approach, in spite of being less accurate for high values of the delay time.