A Midpoint Quadratic Approach for Solving Numerically Multi-Order Fractional Integro-Differential Equation - Zahir, Dashne Chapuk; Ahmed, Shazad Shawki; Mohammedfaeq, Shabaz Jalil (Majallat Jāmiʻat Bābil, 2025)
Background: This article presents a method for finding numerical solutions to Fredholm integro-differential equations (FIFDEs) with multi-fractional orders of one or less, using a useful algorithm. Materials and Methods:  A finite difference approximation to Caputo's derivative using collocation points is used to build the midpoint method for the quadrature rule, which forms the basis of the approach. Results: Our method simplifies the evaluation of treatments by transforming the FIFDEs into algebraic equations with operational matrices. After calculating the Caputo derivative at a specific point using the finite difference method, we use the quadrature method, which includes the midpoint rule, to create a finite difference formula for our fractional equation. Conclusions: Additionally, numerical examples are provided to demonstrate the validity and use of the approach as well as comparisons with earlier findings. The results are expressed using a program created in MATLAB.