Advanced Numerical Methods for the Fisher-Kolmogorov Reaction-Diffusion Equation: A Quintic Hermite Splines Approach with Richardson Extrapolation

<ns3:p>This study introduces a robust numerical method for addressing the fourth-order Fisher-Kolmogorov reaction-diffusion equation, employing a combination of quintic Hermite splines and Richardson extrapolation. The proposed technique effectively transformed the original equation into a system of linear and nonlinear equations, achieving sixth-order convergence in space and fourth-order convergence in time. A thorough stability analysis confirmed the unconditional stability of the method, ensuring that the numerical solutions remained accurate throughout various perturbation scenarios. Extensive numerical experiments validated the performance of the method, revealing significant improvements in accuracy compared to existing approaches. The findings underscore the applicability of the method in mathematical biology and related fields, where the precise modelling of dynamic processes is crucial. Overall, this study contributes to the advancement of numerical techniques for complex differential equations, demonstrating enhanced reliability and precision in computational simulations.</ns3:p>