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

B. Mayil Vaganan, M. Senthilkumaran and T. Shanmuga Priya
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B. Mayil Vaganan: Madurai Kamaraj University
M. Senthilkumaran: Thiagarajar College
T. Shanmuga Priya: Madurai Kamaraj University

Indian Journal of Pure and Applied Mathematics, 2012, vol. 43, issue 6, 591-600

Abstract: 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: Generalized Cole-Hopf transformations; two-dimensional Burgers equation with variable coefficient; Riccati equation (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/s13226-012-0035-y

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