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.

The monkeypox virus causes a respiratory illness called monkeypox, which belongs to the Poxviridae virus family and the Orthopoxvirus genus. Although initially endemic in Africa, it has recently become a global threat with cases worldwide. Using the Antangana–Baleanu fractional order approach, this study aims to propose a new monkeypox transmission model that represents the interaction between the infected human and rodent populations. An iterative method and the fixed-point theorem are used to prove the existence and uniqueness of the symmetry model’s system of solutions. It shows that the symmetry model has equilibrium points when there are epidemics and no diseases. As well as the local asymptotic stability of the disease-free equilibrium point, conditions for the endemic equilibrium point’s existence have also been demonstrated. For this purpose, the existence of optimal control is first ensured. The aim of the proposed optimal control problem is to minimize both the treatment and prevention costs, and the number of infected individuals. Optimal conditions are acquired Pontryagin’s maximum principle is used. Then, the stability of the symmetry model is discussed at monkeypox-free and endemic equilibrium points with treatment strategies to control the spread of the disease. Numerical simulations clearly show how necessary and successful the proposed combined control strategy is in preventing the disease from becoming epidemic.