This paper explores the investigation of a Volterra-Fredholm integro-differential equation that incorporates Caputo fractional derivatives and adheres to specific order conditions. The study rigorously establishes both the existence and uniqueness of analytical solutions by applying the Banach principle. Additionally, it presents a unique outcome regarding the existence of at least one solution, supported by exacting conditions derived from the Krasnoselskii fixed point theorem. Furthermore, the paper encompasses neutral Volterra-Fredholm integro-differential equations, thus extending the applicability of the findings. Additionally, the paper explores the concept of Ulam stability for the obtained solutions, providing valuable insights into their long-term behavior. To emphasis the practical significance and reliability of the results, an illustrative example is included, effectively demonstrating the applicability of the theoretical discoveries.

This paper delves into the investigation of a Volterra-Fredholm integro-differential equation enhanced with Caputo fractional derivatives subject to specific order conditions. The study rigorously establishes the existence of solutions through the application of the Schauder fixed-point theorem. Furthermore, it encompasses neutral Volterra-Fredholm integro-differential equations, thereby extending the applicability of the findings. In addition, the paper explores the concept of controllability for the obtained solutions, offering valuable insights into how these solutions behave over extended time periods.

This paper reveals the solutions to several families of fractional integro-differential equations through the application of a simple fractional calculus method. This approach results in various interesting consequences and also extends the classical Frobenius method. The provided approach is primarily based on established theorems concerning particular solutions of fractional integro-differential equations using the Laplace transform and the extension coefficients of binomial series. Additionally, an illustrative example of such fractional integro-differential equations is presented.

This study approaches some families of fractional integro-differential equations (FIDEs) using a simple fractional calculus method, which leads to several appealing consequences, including the classical Frobenius method, which is generalized. The method presented here is based mostly on certain general theorems on particular solutions of FIDEs using the Aboodh transform and binomial series extension coefficients. We additionally demonstrate techniques to solve FIDEs.

This investigation focuses on an impulsive Volterra-Fredholm integro-differential equation enriched with fractional Caputo derivatives and subject to specific order conditions. The study establishes both the existence and uniqueness of analytical solutions using the Banach principle. Moreover, it reveals a distinctive outcome regarding the existence of at least one solution, supported by conditions derived from the Krasnoselskii fixed point theorem. Additionally, the paper extends its examination to impulsive neutral Volterra-Fredholm integro-differential equations, providing insights into their long-term behavior through Ulam stability. The inclusion of an illustrative example emphasizes the practical significance and reliability of the results.