Asymptotic solutions to the Smoluchowski's coagulation equation with singular gamma distributions as initial size spectra

Smoluchowski's coagulation equation is studied for the kernel K(u,v)=E(uαvβ+uβvα)K(u,v)=E(uαvβ+uβvα) with real, non-negative α, β and E, using gamma distributions with a singularity at zero volume as initial size spectra. As the distribution parameter of the gamma distribution, p  , approaches its lower limit (p→0p→0) the distribution becomes ∼pvp−1∼pvp−1 for small v  . Asymptotic solutions to the coagulation equation are derived for the two cases p→0p→0 and v→0v→0. The constant kernel (α=β=0α=β=0) is shown to be unique among the studied kernels in the sense that the p→0p→0 asymptote and the v→0v→0 asymptote differ.

Graphical abstractAsymptotic solutions to the Smoluchowski's coagulation equation using singular gamma distributions (i.e., the distribution parameter 0