Traveling-Wave Solutions of the Kolmogorov–Petrovskii–Piskunov Equation
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We consider quasi-stationary solutions of a problem without initial conditions for the Kolmogorov–Petrovskii–Piskunov (KPP) equation, which is a quasilinear parabolic one arising in the modeling of certain reaction–diffusion processes in the theory of combustion, mathematical biology, and other areas of natural sciences. A new efficiently numerically implementable analytical representation is constructed for self-similar plane traveling-wave solutions of the KPP equation with a special right-hand side. Sufficient conditions for an auxiliary function involved in this representation to be analytical for all values of its argument, including the endpoints, are obtained. Numerical results are obtained for model examples.
KeywordsKolmogorov–Petrovskii–Piskunov equation generalized Fisher equation Abel’s equation of the second kind Fuchs–Kowalewski–Painlevé test self-similar solutions traveling waves intermediate asymptotic regime
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