Non-linear partial differential equations are crucial in modeling natural processes due to their substantial relevance in real-world applications. This study presents a novel approach for determining the numerical solution of the non-linear fourth-order extended Fisher–Kolmogorov equation (EFKE). This article introduces a novel hybrid method utilizing a quartic non-polynomial spline (QNPS) and finite difference method to approximate the solution of EFKE. Specifically, the von-Neumann method is used to conclude the unconditional stability analysis of the current approach, and the truncation error is examined to investigate the order of convergence of the suggested method. Numerical examples demonstrate the practical utility of the suggested approach. The error norms L_2 and L_8, central processing unit (CPU) time and the order of convergence for each example are presented in tabular form to verify the accuracy of the proposed method and are compared with those reported in recent literature.
A hybrid numerical method combining the quartic QNPS technique with the finite difference method is developed to obtain approximate solutions of the EFKE. The unconditional stability of the proposed scheme is established using the von Neumann stability analysis, while the truncation error analysis is carried out to determine the order of convergence. The effectiveness and accuracy of the method are further demonstrated through numerical examples.
Numerical examples demonstrate the practical utility of the suggested approach. The error norms , CPU time and order of convergence for each example are reported in tabular form to verify the accuracy of the proposed method and compared with others in recent literature.
This study presents a new hybrid QNPS–finite difference scheme for solving the EFKE. The proposed method offers an accurate, stable, and computationally efficient approach, with unconditional stability and proven convergence. Comparative numerical results demonstrate its improved performance over existing methods, highlighting its potential for solving higher-order nonlinear differential equations.
