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(ualphavbeta+ubetavalpha) with real, non-negative alpha, beta 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-->0) the distribution becomes approximately pv(p-1) for small v. Asymptotic solutions to the coagulation equation are derived for the two cases p-->0 and v-->0. The constant kernel (alpha=beta=0) is shown to be unique among the studied kernels in the sense that the p-->0 asymptote and the v-->0 asymptote differ.