Numerical solution of a class of nonlinear singular tempered fractional integro-differential equations

In this paper, a new class of singular fractional integro-differential equations is defined. The Caputo tempered differentiation is considered to define these equations. By employing the fixed point theorem, the existence and uniqueness of a solution for such equations is proved. The orthonormal discrete Hahn polynomials are considered to generate a computational strategy for these tempered fractional equations. To enable this methodology, a matrix representation of the tempered integral operator is constructed for the polynomial basis. Subsequently, the fractional operators in the governing equation are discretized using the polynomial approximation framework. By leveraging the precomputed integral operator matrix and solving the resulting algebraic system, a numerical approximation to the solution is efficiently computed. The convergence of the introduced strategy is proved, theoretically. The precision of the scheme is checked by solving three examples.