A generalized Cole-Hopf transformation for a two-dimensional burgers equation with a variable coefficient |
| |
Authors: | B Mayil Vaganan M Senthilkumaran T Shanmuga Priya |
| |
Institution: | 1. Department of Applied Mathematics and Statistics, Madurai Kamaraj University, Madurai, 625 021, India 2. Department of Mathematics, Thiagarajar College, Madurai, 625 009, India
|
| |
Abstract: | The direct method is applied to the two dimensional Burgers equation with a variable coefficient (u t + uu x ? u xx ) x + s(t)u yy = 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 (u t + uu x ? u xx ) x + t ?3/2 u yy to the parabolic equation $U_r = U_{\rho \rho } + \left( {\tfrac{\rho } {2} - 1} \right)U_\rho$ . |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|