A Numerical Solution of Mixed-Order Fredholm Integro-Differential Equations by Using Trapezoidal, Two-Block, and Modified Two-Block Methods
DOI:
https://doi.org/10.29304/jqcsm.2026.18.12418Keywords:
Classical and fractional order integro-differential equations, Fredholm type, Caputo fractional derivatives, Trapezoidal method, Two-point block method, Modify two-point block method, Finite difference approximation and mixed boundary conditions.Abstract
The purpose of this article is to use the trapezoidal rule, two-point block schemes, and a modified two-point block approach to numerically solve classical and fractional-order (Mixed-orders in the Caputo sense) Fredholm integro-differential equations with variable coefficients under limit conditions (IFDEs-CF). Three new algorithms are proposed to find approximate solutions to these equations, firstly transforming them into systems of linear algebraic equations using the trapezoidal method aided with finite difference approximation, then applying two-block techniques with the predictor value obtained, and finally modifying the results by modification technique to obtain correcting values for these equations at each fixed point. This approach is computationally attractive, and illustrative examples with usage explanations are provided. Additionally, we provide specific examples to showcase the accuracy of the method, and we employ the least-squares error methodology to minimize error terms within the given domain. Finally, the most common application suggested for the numerical approaches is developed in the Python program.
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