A numerical method based on the shifted Jacobi polynomials for a class of tempered fractional quadratic integro-differential equations

This paper introduces a new class of tempered fractional quadratic integro-differential equations using the Caputo fractional derivative. The existence and uniqueness of solutions to these equations are analyzed. A numerical method based on the shifted Jacobi polynomials is developed to solve these equations. To execute the proposed method, two operational matrices corresponding to the ordinary and Riemann–Liouville tempered fractional integrals of these polynomials are extracted. In the developed method, the tempered fractional derivative term is initially represented as a linear combination of the aforementioned polynomials with some unknown coefficients. Then, by applying the Riemann–Liouville tempered fractional integral to the expressed polynomials and utilizing their fractional integral operational matrix, an approximation of the unknown solution is defined based on these polynomials and the introduced coefficients. Subsequently, by substituting these approximations into the problem under consideration, and applying the operational matrix of ordinary integral to the shifted Jacobi polynomials, along with utilizing their orthogonality, an approximate solution to the original problem is obtained by solving a nonlinear system of algebraic equations. The convergence of the proposed method is analyzed theoretically and demonstrated through numerical examples. Furthermore, the stability of the solutions is analyzed.