Haar wavelet Arctic Puffin optimization method (HWAPOM): Application to logistic models with fractal-fractional Caputo-Fabrizio operator

This study introduces a novel hybrid numerical methodology for approximating differential equations involving the fractal-fractional Caputo-Fabrizio (FFCF) operator, which is an essential tool for modelling complex dynamical systems involving memory effects. The proposed method integrates the Haar wavelet with the Arctic Puffin optimization (APO) algorithm, a meta-heuristic optimization inspired by the foraging behavior of Arctic Puffins. The Haar wavelet, well-known for its compact support and piecewise constant characteristics, is based on the Haar basis functions used to construct an operational matrix for the FFCF operator. These matrices transform the differential equations into a system of algebraic equations involving unknown coefficients, and then optimize them using the APO algorithm, ensuring efficient and accurate solutions. Two nonlinear quadratic and cubic logistic models were examined to demonstrate the effectiveness of this method. The accuracy of the designed method was validated by comparing its results with those obtained using the modified Homotopy Perturbation method (MHPM). Error metrics, such as mean absolute error, maximum absolute error, and the experimental convergence rate, are calculated at various collocation points and presented in a tabular format. The findings revealed the method's high accuracy, rapid convergence, and computational efficiency. Overall, the proposed method offers a powerful tool for solving complex differential equations, as evidenced by its strong agreement with MHPM results. The study results were further reinforced through statistical performance metrics and their visual representations, confirming the reliability of the method, low computational cost, and its potential for broad application in numerical computations.