A Bernstein Operational Matrix Approach for Solving Certain Nonlinear Volterra-Fredholm Integro-Differential Equations Involved by Caputo Fractional Derivative

Background: In this article, we present a numerical method that applies the Bernstein polynomial approach in matrix form to solve non-linear integro-fractional differential equations of the Volterra-Fredholm type (NIFDEs of V-F) that involve the Caputo fractional derivative with mixed conditions. Materials and Methods: The primary objective of the method is to convert the nonlinear functional equation and its associated conditions into matrix relations using normal and fractional Bernstein polynomials. using a strategy like Newton's method and obtaining the Bernstein coefficients. Results: This approach is appealing for computation, and examples and explanations of its use are given. Illustrative examples are used to show the method's correctness and efficiency, and the methodology is validated by comparing the approximate results with exact or reference solutions. Conclusion: A table compares both exact and approximate solutions. They also use the least-squares error technique to decrease error terms in the specified domain. As a result, the majority of general codes are written in Python.