A numerical method based on the piecewise Chebyshev cardinal functions for a class of third-kind nonlinear fractional integro-differential equations

This paper introduces a new class of third-kind nonlinear fractional integro-differential equations, defined using the Caputo derivative. A numerical method based on the piecewise Chebyshev cardinal functions is proposed to solve these equations. In the developed approach, a substitution is applied to initially transform these integro-differential equations into an equivalent class of third-kind nonlinear fractional integral equations. Next, by approximating the solution of the problem as a linear combination of the given basis functions and applying both the ordinary and fractional integral operational matrices of these functions, the problem solving process is transformed into solving an algebraic system of equations to determine the coefficients of the linear combination. The theoretical convergence of the proposed method is thoroughly examined. The validity of the scheme is numerically evaluated through the solution of several numerical examples.