Analytical and numerical techniques for initial‐boundary value problems of Kolmogorov–Petrovsky–Piskunov equation

Abstract

In this work, we are interested in finding exact and numerical solutions of well‐known Kolmogorov–Petrovsky–Piskunov (KPP) equation. Here, we introduce modified extended tanh method for finding the exact solutions of KPP equation and an average linear finite difference scheme for numerical investigations. It has been observed that the numerical scheme so employed is stable and accurate enough to produce stable results.

A comparative investigation of a time‐dependent mesh method and physics‐informed neural networks to analyze the generalized Kolmogorov–Petrovsky–Piskunov equation

“…The Dirichlet BCs are u(0, t) = u(1, t) = 0 and the initial condition is u(y, 0) = sin(2𝜋y). 6,21 Computed solutions illustrated in Figure 1 using MMPDE6:…”

“…5 Ben et al established exact and numerical estimations of the KPP-Fisher equations using the tanh technique to find analytical approximations and an averaged finite difference method to get approximate solutions. 6 The Fitzhugh-Nagumo equation's wave approximations were produced by Yokus et al using the auto-Backlund transformation method and forward difference schemes; in addition, the method's stability was attained using a Fourier von-Neumann investigation. 7 Browne et al looked into how the Fitzhugh-Nagumo (FN) equation was transformed into its generalized form and how numerical approximations obtained utilizing the method of lines were used.…”

“…Bhrawy et al introduced a Jacobi–Guass–Lobatto collection technique to obtain the numerical solution of the Fitzhugh–Nagumo equation in a broader form using Jacobi–Guass–Lobatto collocation points for space derivatives 5 . Ben et al established exact and numerical estimations of the KPP‐Fisher equations using the $$$ \mathit{\tanh} $$$ technique to find analytical approximations and an averaged finite difference method to get approximate solutions 6 . The Fitzhugh–Nagumo equation's wave approximations were produced by Yokus et al using the auto‐Backlund transformation method and forward difference schemes; in addition, the method's stability was attained using a Fourier von‐Neumann investigation 7 .…”

Study of Fuzzy Fractional Third-Order Dispersive KdV Equation in a Plasma under Atangana-Baleanu Derivative

“…Anyway, most nonlinear phenomena that can exist in plasma physics, nonlinear optics, quantum physics, etc. can only be investigated using useful techniques to solve their evolution equations [43][44][45][46][47][48][49]. In 1895, Korteweg and de Vries proposed a KdV model to design Russells soliton phenomenon, such as small and huge water waves.…”

“…Solitons are steady waves of solitary; this implies that these lone waves are particles. The mathematical model for exploring dispersive wave phenomena in several research areas is the KdV equations, such as quantum mechanics, fluid dynamics, optics, and plasma physics [44,45]. Fifth-order KdV/Kawahara form equations utilized to analyze different nonlinear phenomena in particle physics and in plasma physics [38][39][40].…”