NUMERICAL-SOLUTION OF THE SMOLUCHOWSKI KINETIC-EQUATION AND ASYMPTOTICS OF THE DISTRIBUTION FUNCTION

We obtain the numerical solution of the Smoluchowski kinetic equation for model kernels U(M(1), M(2)) proportional to (M(1) + M(2))(lambda) and U(M(1), M(2)) proportional to (M(1) M(2))(lambda/2), 0 < lambda less than or equal to 2. We show that the behaviour of the solution for the kernel U proportional to (M(1) + M(2))(lambda) at 0 < lambda less than or equal to 1 and U proportional to (M(1) M(2))(lambda/2) at 0 < lambda less than or equal to 2 becomes self-similar after some time. The shape of the scaling function is analysed; in particular, a simple approximate expression for it at U proportional to (M(1) + M(2))(lambda) is found. An interesting result is obtained for U proportional to (M(1) M(2))(lambda/2), 0 < lambda < 1: the asymptotic behaviour of the scaling function proved to be non-power. We develop the procedure for determining t(cr), the critical indices and the exponent of the power-law asymptotics of the Smoluchowski equation solution. The concrete values of these quantities for the model kernels are obtained. The stability conditions of the algorithm for numerical solving are analysed. The possibilities afforded by numerical solution for investigation of the Smoluchowski equation are discussed.