Legendre Collocation approach for Integro-Differential equations

Abstract

This study presents the application of the Legendre Collocation Method (LCM) for solving Integro-Differential Equations (IDEs), which model a range of scientific and engineering problems.IDEs, involving both differential and integral terms, often require numerical methods for their solutions due to the complexity of obtaining exact solutions. The proposed approach transforms IDEs into systems of linear algebraic equations using shifted Legendre polynomials. By collocating the resulting equations, approximate solutions are efficiently computed. The accuracy of the method is validated through several numerical examples, including Volterra and Fredholm types of IDEs, and the results are compared with known exact solutions. The effectiveness and robustness of LCM are demonstrated through high-order approximations. The theoretical uniqueness of the method is established using relevant theorems, including the Banach Contraction Principle. Overall, the LCM provides a reliable and efficient technique for solving a wide class of IDEs with high accuracy.