A boundary value problem for algebraically accelerated frequency of mutations governed by Fisher–Kolmogorov–Petrovskii–Piskunov equation

A new generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation [Formula: see text] is introduced and subjected to Lie’s Classical Method to establish the existence of the traveling waves. Indeed, exact expressions for both the solution [Formula: see text], with [Formula: see text] and the wave speed [Formula: see text] are subsequently derived by the regular perturbation method and are shown to contain both the reports of Ablowitz and Zeppetella [Explicit solutions of fisher’s equation for a special wave speed, Bulletin of Mathematical Biology 41(6) (1979) 835–840] when [Formula: see text] and Kaliappan [An exact solution for the traveling waves of [Formula: see text], Physica D: Nonlinear Phenomena 11(3) (1984) 368–374] when [Formula: see text], respectively. Remarkably, the generalized Huxley equation is a special case of our model equation. The form of the traveling waves for the Fisher–Kolmogorov–Petrovskii–Piskunov equations is identified with the regular perturbation method and the Binomial series. Our study is in agreement with the conclusion of Fisher [The wave of advance of advantageous genes, Annals of Eugenics 7(4) (1937) 355–369] that the increase of frequency of mutant genes results in the diffusion of the mutant genes into the long range.