Optimizing pantograph fractional differential equations: A Haar wavelet operational matrix method

In this study, we developed an operational matrix method of integration using Haar wavelets to solve both linear and nonlinear pantograph fractional differential equations by taking Atangana's beta derivative. The proposed method utilizes an operational matrix and truncated Haar wavelet series to transform linear and nonlinear problems into objective functions. These objective functions are then minimized using differential evolution optimization. Illustrative examples demonstrate convergence and validation of the proposed method. The proposed method yields exceptional numerical outcomes in terms of competitiveness and accuracy when compared to exact solutions. In addition, performance metrics, including the mean absolute deviation, root mean squared error, P2norm, P∞ norm, Theil's inequality coefficient, variance account for, coefficient of determination, and Nash-Sutcliffe efficiency, were calculated for a varying number of collocation points. The results indicate that the Haar wavelet operational matrix method is a straightforward and efficient method for solving pantograph fractional differential equations.