Numerical similarity solutions for a class of Kolmogorov-type equations

We consider a class of generalized nonlinear variable coefficient Kolmogorov-type equations. The equations in question are nonlinear time-dependent partial differential equations in two spatial dimensions. With the aid of Lie symmetries, suitable transformations are derived, the successive application of which in the end reduce the original (2+1) partial differential equation into nonlinear ordinary differential equations. Some of the latter are solved numerically, using the finite-difference method and a parametric study is conducted. Equivalence transformations are used to connect two different types of generalized Kolmogorov equations. Then, the computed numerical solution of one, is mapped via the equivalence transformation to the solution of the other. It is found that the solutions in question are of travelling-wave form, for all the cases examined.