| Summary: | In this article, effective techniques are described to solve numerically the Fredholm integro-differential equations of multi-fractional order that lie in (0,1] in the Caputo sense (FIFDEs). The approach uses finite difference approximation to Caputo derivative utilizing collocation points and is based on the quadrature rule, Trapezoidal, and Simpson process. 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 trapezoidal and Simpson rules, to create a finite difference formula for our fractional equation. Additionally, numerical examples are provided to demonstrate the validity and use of the approach as well as comparisons with earlier findings. The aforementioned procedure has been used to construct algorithms for treating FIFDEs. A MATLAB program is created to express these solutions. Furthermore, some numerical tests are provided to demonstrate the method's accuracy.
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