Touchard polynomials for solving multi-high order fractional differential equations with variable coefficients

This paper introduces an approach for approximating solutions to multi-high-order fractional-differential equations by employing the Caputo fractional derivative, along with initial conditions. The technique is based on standard collocation points and Touchard polynomials. The linear equation and its initial conditions can be transformed into matrix relations by the new method, making it easier to solve a linear algebraic equation with generalized Touchard coefficients as unknowns. Also, emphasizing computational efficiency, the method is illustrated with examples.