A globally convergent Carleman-Picard method for an inverse initial-value problem for a nonlinear diffusive coagulation-fragmentation equation coagulation-fragmentation equation

We study an inverse initial-density problem for a nonlinear diffusive coagulation--fragmentation equation with known coagulation and fragmentation kernels. The objective is to recover the unknown initial particle-size distribution on a finite interval from time-dependent boundary observations of the solution and its size derivative. To solve this inverse problem, we develop a globally convergent numerical method based on a Legendre--exponential time reduction and a Carleman--Picard iteration. The time reduction transforms the original problem into a nonlinear coupled system for the spatial mode coefficients, while the Carleman weight and the corresponding Carleman estimate guaranty the global convergence of the Picard iteration without requiring a good initial guess. We prove the convergence of the proposed method and obtain a complete reconstruction procedure for the initial density. Numerical experiments with noisy boundary data demonstrate that the method yields accurate and stable reconstructions for several representative test profiles.