A generalized Cole-Hopf transformation for a two-dimensional burgers equation with a variable coefficient

Abstract

The direct method is applied to the two dimensional Burgers equation with a variable coefficient (ut + uux − uxx)x + s(t)uyy = 0 is transformed into the Riccati equation \(H' - \tfrac{1} {2}H^2 + \left( {\tfrac{\rho } {2} - 1} \right)H = 0\) via the ansatz \(u\left( {x,y,t} \right) = \tfrac{1} {{\sqrt t }}H(\rho ) + \tfrac{y} {{2\sqrt t }}\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y\), provided that s(t) = t−3/2. Further, a generalized Cole-Hopf transformations \(u\left( {x,y,t} \right) = \tfrac{y} {{2\sqrt t }} - \tfrac{2} {{\sqrt t }}\tfrac{{U_\rho (\rho ,r)}} {{U(\rho ,r)}}\), \(\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y\), r(t) = log t is derived to linearize (ut + uux − uxx)x + t−3/2uyy to the parabolic equation \(U_r = U_{\rho \rho } + \left( {\tfrac{\rho } {2} - 1} \right)U_\rho\).