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New results on group classification of nonlinear diffusionconvection equations
, 2003
"... Using a new method and transformations of conditional equivalence, we carry out group classification in a class of variable coefficient (1 + 1)dimensional nonlinear diffusionconvection equations of the general form f(x)ut = (D(u)ux)x + K(u)ux. We obtain new interesting cases of such equations with ..."
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Cited by 22 (6 self)
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Using a new method and transformations of conditional equivalence, we carry out group classification in a class of variable coefficient (1 + 1)dimensional nonlinear diffusionconvection equations of the general form f(x)ut = (D(u)ux)x + K(u)ux. We obtain new interesting cases of such equations with localized density f, having large invariance algebra. Examples of Lie ansätze and exact solutions of these equations are constructed.
A discussion on the different notions of symmetry of differential equations
 in Proceedinds of Fifth International Conference ”Symmetry in Nonlinear Mathematical Physics” (June 2329, 2003, Kyiv), Editors A.G. Nikitin, V.M. Boyko, R.O. Popovych and I.A. Yehorchenko, Proceedings of Institute of Mathematics, Kyiv
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Linear determining equations for differential constraints Sb
 Math
, 1998
"... A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical determining equations used in the search for admissible Lie oper ..."
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Cited by 5 (1 self)
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A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical determining equations used in the search for admissible Lie operators. As applications of this approach nonlinear heat equations and GibbonsTsarev’s equation are discussed. We introduce the notion of an invariant solution under an involutive distribution and give sufficient conditions for existence of such a solution.
ThirdOrder Conditional LieBäcklund Symmetries of Nonlinear ReactionDiffusion Equations
"... The thirdorder conditional LieBäcklund symmetries of nonlinear reactiondiffusion equations are constructed due to the method of linear determining equations. As a consequence, the exact solutions of the resulting equations are derived due to the compatibility of the governing equations and the a ..."
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The thirdorder conditional LieBäcklund symmetries of nonlinear reactiondiffusion equations are constructed due to the method of linear determining equations. As a consequence, the exact solutions of the resulting equations are derived due to the compatibility of the governing equations and the admitted differential constraints, which are resting on the characteristic of the admitted conditional LieBäcklund symmetries to be zero.
MAIK “Nauka
"... Exact solutions of nonlinear heat and masstransfer equations have always played an important role in the formation of a correct understanding of qualitative features of various processes in chemical engineering, thermophysics, and power engineering. Exact solutions of nonlinear equations vividly ..."
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Exact solutions of nonlinear heat and masstransfer equations have always played an important role in the formation of a correct understanding of qualitative features of various processes in chemical engineering, thermophysics, and power engineering. Exact solutions of nonlinear equations vividly demonstrate and allow one to understand the mechanism of such complex nonlinear effects as the spatial localization of heattransfer processes, the multiplicity or absence of steady states under certain conditions, the existence of blowup modes, the presence or absence of periodic modes, etc. Simple solutions are widely used to illustrate the theoretical material, and some applications in lecture courses in universities and technical institutes (on heatand masstransfer theory, chemical engineering, hydro