Haar wavelet method tuned with hippopotamus optimization: Modeling pantograph differential equations

Numerous physical models require dynamic analysis to study the processes involved, and wavelet-based methods are important for this purpose. This study introduces a wavelet-based metaheuristic optimization method that involves the Atangana-Baleanu-Caputo fractional operator. The proposed method assimilates the Haar wavelet and the stochastic computing hippopotamus optimization algorithm (HOA), stimulated by the behavior of the hippopotamus. The Haar wavelet, which is recognized for its piecewise constant nature, is used to develop the AB integral operational matrix to solve a special class of delay differential equations, that is, the pantograph differential equation. These differential equations are transformed into an algebraic system, and then articulated into an optimization function, which is minimized by the HOA solver. To validate the reliability and accuracy of the proposed method, the numerical solutions are benchmarked against the exact solutions, N-homotopy perturbation solutions, and other methods published in the literature. The methodology is validated and assessed for precision using the absolute error and several other statistical performance indicators. Furthermore, the use of modern metaheuristic optimization techniques enhanced the accuracy of the computed solutions and enabled the exploration of other complex fractional-order dynamical processes.