Numerical Solutions for Linear Integro-Differential Equations Using Shifted Legendre Basis Functions

This research explore the use of Shifted Legendre Basis functions for the numerical solution of a specific class of integro-differential equations. These equations are known for their analytical complexity,making it challenging to derive exact solutions. To address this, we employ an approximate methodusing Legendre polynomials as basis functions, which provides an efficient approach to finding solutions for these complex problems. The proposed method is computationally efficient, requiringminimal computational resources and storage. The results obtained demonstrate strong agreementwith existing solutions found in the literature, validating the accuracy and effectiveness of the approach. This study highlights the potential of Shifted Legendre Basis functions in solving challenging integro-differential equations, offering a reliable alternative to more computationally intensivemethods.