Analytical study of the time-fractional Smoluchowski coagulation equation in light of different integrodifferential operators

This article presents the derivation of the fractional Smoluchowski coagulation equation via the variational principles technique. We use the variational iteration method to solve the Caputo-type fractional coagulation equation. Furthermore, we analyze the time-fractional coagulation equation using the homotopy perturbation transform approach, considering three different fractional operators: Caputo, Caputo-Fabrizio, and Atangana-Baleanu. Our findings demonstrate that the solutions for the total number of particles during coagulation align well with existing literature, particularly in the short time limit. Additionally, we examine the impact of the time-fractional order on the dynamics of particle coagulation for each fractional operator.