Numerical Solution of Integro-diffrential Equation Using Legendre Collocation Method

Integro-differential equations (IDEs) seamlessly blend differential and integral operators, offering a versatile mathematical framework for modelling various physical, biological, and engineering systems. Analytical solutions to these equations are often unattainable due to their complexity, necessitating efficient numerical approaches. This study introduces the Legendre Collocation Method (LCM), leveraging shifted Legendre polynomials to approximate solutions for Volterra and Fredholm IDEs. By transforming the IDEs into a system of linear equations, the proposed method achieves high accuracy and computational efficiency. Numerical experiments on representative IDEs validate the approach, showing excellent agreement with exact solutions and outperforming existing methods in terms of absolute error. These results underscore the potential of the LCM in solving complex IDEs with broad applicability across disciplines.