A New Approach to Least Squares Support Vector Regression Using the Legendre Operational Matrix for Solving Fractional Fredholm Integro‐Differential Equations

In this study, a novel hybrid technique is introduced to address linear and nonlinear fractional Fredholm integro‐differential equations (FFIDEs) by utilizing least squares support vector regression (LS‐SVR) along with the Legendre operational matrix of fractional derivatives (OMFD). The unknown function is represented as a linear combination of shifted Legendre polynomials. Also, the Galerkin and collocation spectral methods are used. We then formulate an optimization problem and convert it into a set of algebraic equations. To address a linear FFIDE, we introduced a matrix representation of the dual problem and demonstrated the resulting linear systems. Also, a discussion on error estimates is included. Finally, to demonstrate the validity and applicability of the suggested methods, some test examples are presented.