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Abstract
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The extended Fisher–Kolmogorov (EFK) equation plays a vital role in describing nonlinear reaction–diffusion processes, phase transitions, biological pattern formation, and population dynamics, yet its higher-order nonlinear nature makes it analytically intractable in most cases. Traditional numerical methods often face challenges such as reduced accuracy, stability issues, and high computational cost, particularly when capturing steep gradients, wavefronts, or long-time dynamics inherent in the EFK equation. This highlights the necessity for more efficient and accurate numerical schemes capable of handling its complexity. The improved quintic B-spline collocation method offers a powerful solution, as quintic B-splines provide high-order smoothness, local flexibility, and excellent approximation properties, while the collocation framework ensures computational efficiency. The importance of this research lies in establishing a robust, accurate, and stable numerical approach for solving the EFK equation, thereby contributing significantly to both theoretical studies and practical applications in physics, biology, and engineering where such models are central.
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