Preliminary group classification and exact solutions of Smoluchowski equation with a source (Q6556714)

This paper is devoted to study the homogeneous Smoluchowski equation \N\[\NSf = \frac{\partial f(x,t)}{\partial t} - \frac{1}{2} x^{\gamma +1} \int \limits ^1_0 K(1-s,s) f(x(1-s),t) f(xs,t)ds \]\N\[+ f(x,t) \int \limits ^{\infty}_0 K(x,y) f(y,t) dy =0,\N\]\Nwith kernel \N\[\NK(x,y) =k_0, \; k_1(x +y), \;k_2 xy, \; k_3 \Bigl(\frac{1}{x}+\frac{1}{y}\Bigr), \; \frac{k_4}{xy}.\N\]\NHere the kinetic coefficients \(k_i\), \(i=0,1,2,3,4\) are positive constants.\N\NAt first, an algorithm to find a symmetry Lie group of the above equation is investigated. Then the corresponding Lie algebra is considered and the authors classify symmetry subalgebras. \N\NFurther, the nonhomogeneous equation\N\(\NSf = G(x,t,f)\N\)\Nis considered\Nand a preliminary symmetry group classification is given. \NFinally, the invariant solutions and exact solutions for a special choice of \(G\) are given.