Existence, Uniqueness, and Stability Results for Tempered Fractional Integro-Differential Equations via Fixed Point Techniques

In this chapter, we consider the most important and problems used frequently in random walks and tempered fractional equation that describes the particle plume shape. This work investigates the basic results of the tempered fractional integro-differential equation (TF-IDE) via the concept of Caputo fractional derivative. The existence and uniqueness of a solution for BVP-FDEs are discussed by utilizing the Banach fixed point theorem and Schaefer’s fixed point theorem. Then, we provide the Ulam–Hyers stability results under the Lipschitz condition. These theoretical results are significant before solving the problems and help in finding the solutions to the problems.