Numerical treatment of the spherically symmetric solutions of a generalized Fisher–Kolmogorov–Petrovsky–Piscounov equation

摘要

In the present work, the connection of the generalized Fisher–KPP equation to physical and biological fields is noted. Radially symmetric solutions to the generalized Fisher–KPP equation are considered, and analytical results for the positivity and asymptotic stability of solutions to the corresponding time-independent elliptic differential equation are quoted. An energy analysis of the generalized theory is carried out with further physical applications in mind, and a numerical method that consistently approximates the energy of the system and its rate of change is presented. The method is thoroughly tested against analytical and numerical results on the classical Fisher–KPP equation, the Heaviside equation, and the generalized Fisher–KPP equation with logistic nonlinearity and Heaviside initial profile, obtaining as a result that our method is highly stable and accurate, even in the presence of discontinuities. As an application, we establish numerically that, under the presence of suitable initial conditions, there exists a threshold for the relaxation time with the property that solutions to the problems considered are nonnegative if and only if the relaxation time is below a critical value. An analytical prediction is provided for the Heaviside equation, against which we verify the validity of our computational code, and numerical approximations are provided for several generalized Fisher–KPP problems.