Fractional-order clique functions to solve left-sided Bessel fractional integro-differential equations

In this study, we consider a new class of nonlinear integro-differential equations with the Bessel fractional integral-derivative. For solving the considered equations, fractional-order clique functions (FCFs), and some of their properties are introduced. First, we approximate the unknown function and its derivatives/integrals in terms of the FCFs. Then, we substitute these approximations and their derivatives/integrals into the considered equation. The left-sided Bessel fractional derivative/integral (LSBFD/I) of the unknown function is approximated using the properties of the FCFs and LSBFD/I. By collocating the resulting residual function at the well-known shifted Legendre points, we derive a system of nonlinear algebraic equations. In addition, convergence analysis of the proposed approach is discussed. Finally, the presented strategy is applied to some numerical experiments to verify its applicability and accuracy.