mittag-leffler wavelet-based numerical method for fractional pantograph delay differential equations

This paper proposes a robust numerical framework for solving fractional pantograph delay differential equations. the approach leverages the riemann–liouville fractional integral operator, represented through mittag-leffler wavelet functions within a collocation-based scheme. to facilitate computation, an operational matrix is constructed, enabling the transformation of the fractional differential system into a system of algebraic equations. the proposed method's accuracy, stability, and convergence are rigorously validated through comprehensive numerical experiments.