A new and unified semi-analytical method with a convergence acceleration parameter for linear and nonlinear fragmentation equations

In recent years, turbulent multiphase flows, including bubbles and droplet breakup, have been commonly modelled using linear fragmentation population balance equations (PBEs) to capture droplet distribution effectively. These models often rely on binary breakup kernels, where two particles form due to external forces, while particle–particle collisions are neglected to simplify the physics. The mathematical complexities of collisional breakage models stem from the intricate structures of the collisional and fragmentation kernels, along with nonlinear integrals, which limit their applicability in real-world scenarios. These complex kernel structures also constrain the availability of analytical solutions for simple collisional and fragmentation kernels. To address this, a unified approximation method with a convergence acceleration parameter is proposed for both linear and nonlinear fragmentation problems. By optimizing the acceleration parameter, the method extends the convergence zone of series solutions over longer time domains and resolves issues present in existing approaches. A detailed theoretical convergence analysis of the proposed method in Banach space is provided. Various analytically tractable and physically relevant kernels, with different initial conditions (ICs) such as Dirac delta, exponential, gamma and Gaussian distributions, are used to validate the accuracy of this approach against the homotopy perturbation method and BLUES function method.