A new approach to least squares support vector regression using the Legendre operational matrix for solving fractional Fredholm integro-differential equations

Abstract

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.