Numerical Solution of Fractional Order Integro Differential with Dirichlet Boundary Conditions Using Shifted Legendre Collocation Method

The paper develops and implements a numerical method for solving fractional order Fredholm Volterra integro differential equations with Dirichlet boundary conditions using the shifted Legendre collocation method. The proposed method is formulated by first obtaining the integral form of the given model equation, followed by applying the collocation technique to generate a system of nonlinear equations. These nonlinear equations are then solved using Newton-Raphson’s iterative method. The accuracy and efficiency of the developed method are analyzed, demonstrating that the obtained solutions are continuous and exhibit convergence. The uniqueness of the solution is established, further validating the reliability of the approach. To assess the effectiveness of the method, several numerical examples are presented, comparing the obtained results with existing techniques. The numerical experiments confirm that the proposed approach yields highly accurate solutions while maintaining computational efficiency. This study shows the applicability of the shifted Legendre collocation method in solving complex integro-differential equations.