Explicit and approximate solutions for the fragmentation equation in the presence of source and efflux terms: A coupled meshfree approach and its convergence analysis

In this study, a novel coupled approach is presented for extracting analytical solutions of a fragmentation equation with source and efflux terms, specifically focusing on long time dynamics. This method holds significant potential in understanding complex phenomena such as polymer degradation (depolymerization), droplet break-up, grinding, and rock crushing in extended time frames. Obtaining analytical solutions for number density functions associated with complex structured Austin kernels remains a highly challenging and open problem. The series solutions are constructed using a semi-analytical Laplace decomposition method, followed by Padé approximation to extend the validity of the time domain. The detailed convergence is performed in order to enhance the understanding of the new approach using the Banach contraction principle. The new approach shows promising potential in finding series and even closed form solutions for the fragmentation equation with high precision, corresponding to analytically tractable and physically relevant kernels with various selection functions, starting from exponential and gamma initial distributions. In order to show the applicability of this approach, various numerical examples are considered and results are compared with the exact solutions or other numerical approximations (for kernels whose closed form solutions are not available). Remarkably, this method achieves the high accuracy by utilizing only a few series terms of the truncated form, and can be made uniform for all time if the limiting equilibrium distribution is known. Moreover, in the majority of cases, closed-form solutions for the number density functions associated with binary and multiple breakage kernels with different source and efflux terms are derived for the first time.