Indian Journal of Pure and Applied Mathematics

, Volume 43, Issue 6, pp 591–600

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

## Authors

• B. Mayil Vaganan
• Department of Applied Mathematics and StatisticsMadurai Kamaraj University
• M. Senthilkumaran
• Department of MathematicsThiagarajar College
• T. Shanmuga Priya
• Department of Applied Mathematics and StatisticsMadurai Kamaraj University
Article

DOI: 10.1007/s13226-012-0035-y

Mayil Vaganan, B., Senthilkumaran, M. & Shanmuga Priya, T. Indian J Pure Appl Math (2012) 43: 591. doi:10.1007/s13226-012-0035-y
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## Abstract

The direct method is applied to the two dimensional Burgers equation with a variable coefficient (ut + uuxuxx)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 + uuxuxx)x + t−3/2uyy to the parabolic equation $$U_r = U_{\rho \rho } + \left( {\tfrac{\rho } {2} - 1} \right)U_\rho$$.

### Key words

Generalized Cole-Hopf transformationstwo-dimensional Burgers equation with variable coefficientRiccati equation