Convergence analysis for the fractional decomposition method applied to class of nonlinear fractional Fredholm integro-differential equation

For scientists conducting research, fractional integral differential equation analysis is crucial. Therefore, in this study, we investigate analysis utilizing a novel method called the fractional decomposition method, which is applicable to fractional nonlinear fractional Fredholm integro-differential equations. Then, we apply the approach to five test problems for a general fractional derivative beta involving fractional Fredholm integro-differential equations. To the best of our knowledge, we are the first to ever do so because of the very complicated calculations involved when dealing with the general case beta. For fractional Fredholm integral-differential equations, we provide both exact and approximate solutions. Throughout this work, the fractional Caputo derivative is discussed. This technique leads us to say that the method is precise, accurate, and efficient, according to the theoretical analysis.