On fractional linear multi-step methods for fractional order multi-delay nonlinear pantograph equation

This paper presents the development of a series of fractional multi-step linear nite di erence methods (FLMMs) designed to address fractional multi-delay pantograph di erential equations of order 0 1. These p-FLMMs are constructed using fractional backward di erentiation formulas of rst and second orders, thereby facilitating the numerical solution of fractional di erential equations. Notably, we employ accurate approximations for the delayed components of the equation, guaranteeing the retention of stability and convergence characteristics in the proposed p-FLMMs. To substantiate our theoretical ndings, we o er numerical examples that corroborate the e cacy and reliability of our approach..