A Midpoint Quadratic Approach for Solving Numerically Multi-Order Fractional Integro-Differential Equation

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.