On the existence and the approximation of solutions of linear and nonlinear fractional order delay-integro-differential equations with vanishing variable delays

In this study, we investigate the application of the meshless collocation technique using radial basis functions (RBF) to approximate a class of linear and nonlinear fractional order delay-integro-differential equations with vanishing variable delays. Such equations, formulated within the framework of fractional calculus, are particularly useful for modeling processes with memory and hereditary characteristics, providing a more accurate representation of many physical and engineering phenomena than classical integer-order models. First, we prove the existence and uniqueness of solutions of the studied equation for both linear and nonlinear cases. Then, we develop an RBF interpolation combined with the Gauss-Legendre quadrature formula, enabling the proposed approach to solve the equations without relying on background approximation cells. The developed method is computationally efficient, stable, and requires low memory. Furthermore, the error analysis of this scheme is discussed. Several computational tests are presented to demonstrate the reliability and precision of the proposed technique for solving the considered equations. The obtained results are compared with analytical solutions, the moving least squares method, and other existing approaches to confirm the effectiveness and applicability of the proposed scheme.