A Class of Fractional Order Nonlinear Volterra–Fredholm Integro-Differential IVPs and BVPs: Qualitative Analysis and Numerical Investigation

In this work, we have examined a class of fractionally ordered Volterra–Fredholm integro-differential equations (VFIDE), where the fractional derivative has been interpreted in the Riemann–Liouville sense. We have focused separate attention on the boundary value problems (BVPs) and initial value problems (IVPs). Each of these problems has been reduced to a fractional order Volterra–Fredholm integral equation of the second kind using Leibnitz’s formula to simplify the analysis. We have derived sufficient conditions for the existence and uniqueness of the solutions to the IVPs and BVPs. We have used Banach’s fixed point and Schaefer’s fixed point theorems to prove the existence and uniqueness results for the considered problems. An operator-based approach using Laplace discrete modified Adomian decomposition methods based on Bernstein polynomials has been considered to approximate their solutions. The convergence and error analysis of the proposed method has also been investigated. We have compared the method with the Homotopy perturbation method for the considered problems. Some numerical examples have been provided to validate the theoretical results.