An Efficient Symmetric Operational Matrix Method for Solving Tempered Fractional Differential Equations with Respect to Another Function

In this paper, we introduce a novel extension of the symmetry operational matrix method specifically designed to tackle tempered fractional differential equations (FDE) that incorporate an additional function. Our approach leverages the framework of shifted Legendre polynomials (SLP), which are well-suited for this context. While the operational matrix method has been widely recognized for its efficacy in addressing a range of problems within fractional calculus, its application to tempered fractional differential equations remains relatively uncharted territory. To bridge this gap, we begin by deriving the analytical expression for the tempered fractional derivative (TFD) of the term τ^p. This crucial step paves the way for the formulation of a new operational matrix that captures the behavior of fractional derivatives in conjunction with another function. We use a method that combines a limited number of terms from the shifted Legendre polynomial basis. This allows us to accurately solve tempered fractional differential equations that include an additional function. We show that our approach works well through several numerical examples, demonstrating how effective and accurate our results are in tackling these complex equations.