Numerical-analytical method for nonlinear equations of Kolmogorov–Petrovskii–Piskunov type

This paper deals with an efficient solution of basic initial-boundary value problems for one-dimensional nonlinear parabolic equations describing reaction–diffusion processes. These equations include the Kolmogorov–Petrovskii–Piskunov and Burgers equations. A numerical-analytical method based on an implicit discretization of the differential operator in combination with the explicit Adams–Bashforth extrapolation for the nonlinear term of the equation is proposed for the problems under study. A new efficient algorithm relying on analytical representations using the fundamental system of solutions in explicit form is developed for solving the arising sequence of linear problems. The efficiency of the developed method and its advantages over some traditional algorithms are demonstrated on several complicated model examples.