Numerical solutions of integro-differential equations via the finite difference method

This study investigates numerical methods for solving Fredholm and Volterra integro-differential equations. A finite difference method on a uniform grid is first presented for second-order linear Fredholm integro-differential equations. The construction of the method is based on the use of integral identities and basis functions, along with the application of quadrature formulas with remainder terms for the approximation of integral components. Additionally, a factorization technique is utilized in the development of the numerical algorithm. Error analysis and convergence properties of the proposed method are examined, and the results are supported by numerical examples.Next, the proposed method is extended to handle nonlinear Volterra integro-differential equations. An efficient linearization technique is introduced for solving these equations. The effectiveness of the method is demonstrated through several theoretical and numerical examples, showing consistency with the analytical findings.