Hyers-Ulam Stability Analysis of Nonlinear Volterra-Fredholm Integro-Differential Equation with Caputo Derivative

The main aim of this study is to examine the Hyers-Ulam stability of fractional derivatives in Volterra-Fredholm integro-differential equations using Caputo fractional derivatives. We explore the existence and uniqueness of solutions for the proposed integro-differential equation using Banach and Krasnoselskii's fixed-point techniques. Furthermore, we examine the Hyers-Ulam stability of the equation under the Caputo fractional derivative by deriving suitable sufficient conditions. We analyze the graphical behavior of the obtained results to demonstrate the efficiency of the analytical method, highlighting its ability to deliver accurate and precise approximate numerical solutions for fractional differential equations. Finally, numerical applications are presented to validate the stability of the proposed integro-differential equation.