Innovative analytical and semi-analytical approaches to the nonlinear Fisher–KPP model: Accurate computational solutions and applications

This study investigates the nonlinear Fisher–Kolmogorov–Petrovsky–Piskunov (KPP) model, which is widely applied to phenomena such as population dynamics, combustion processes, wave propagation in excitable media, and other systems characterized by reaction–diffusion behavior. The primary objective is to examine the intricate interplay between diffusion and nonlinear growth, crucial for understanding processes where spatial spreading is coupled with local growth or decay, such as species invasion, disease transmission, and chemical reactions. The problem addressed is the challenge of obtaining accurate, analytical solutions to the nonlinear Fisher–KPP equation, which often resists conventional solution techniques due to its complex, coupled dynamics. This study employs two advanced analytical methods — the modified Khater (MKhat) and unified (UF) techniques — to derive new, exact solutions to the Fisher–KPP model. These solutions provide deeper insights into the underlying dynamics of the reaction–diffusion systems, showcasing the balance between diffusion-driven spreading and nonlinear growth effects. Numerical validation of the obtained solutions is carried out using He’s variational iteration (HVI) scheme, ensuring the reliability and accuracy of the analytical results. This combination of exact and numerical solutions strengthens the study’s findings and offers robust tools for future research. The expected results include a set of novel, exact analytical solutions that contribute significantly to the understanding of the diffusion and nonlinear growth interplay in reaction–diffusion systems. These solutions have practical applications in forecasting and modeling processes such as population growth, epidemic spread, combustion dynamics, and chemical reaction rates. The study’s innovative use of MKhat and UF methods introduces new approaches for solving nonlinear differential equations, expanding the toolkit available to researchers in applied mathematics and related fields. In conclusion, this research provides a comprehensive exploration of the nonlinear Fisher–KPP model, offering both theoretical advancements and practical applications. The findings have wide-ranging interdisciplinary implications, benefiting fields such as biology, physics, chemistry, and engineering, where reaction–diffusion systems play a central role.