Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms

ROMAN CHERNIHA; MYKOLA SEROV

doi:10.1017/S0956792598003556

Abstract

New Lie and conditional symmetries of nonlinear diffusion equations with convection terms are constructed. Examples of new ansätze and exact solutions for some nonlinear equations (in particular, for the Murray and the Newell–Whitehead equations) are presented.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Cherniha, Roman and King, John R 2000. Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I. Journal of Physics A: Mathematical and General, Vol. 33, Issue. 2, p. 267.

Pucci, Edvige and Saccomandi, Giuseppe 2000. Evolution equations, invariant surface conditions and functional separation of variables. Physica D: Nonlinear Phenomena, Vol. 139, Issue. 1-2, p. 28.

Cherniha, Roman 2000. Lie symmetries of nonlinear two-dimensional reaction-diffusion systems. Reports on Mathematical Physics, Vol. 46, Issue. 1-2, p. 63.

Cherniha, Roman and Dutka, Vasyl' 2001. Exact and numerical solutions of the generalized fisher equation. Reports on Mathematical Physics, Vol. 47, Issue. 3, p. 393.

Cherniha, Roman M. 2001. Nonlinear Galilei-Invariant PDEs with Infinite-Dimensional Lie Symmetry. Journal of Mathematical Analysis and Applications, Vol. 253, Issue. 1, p. 126.

Estévez, P.G. Qu, Changzheng and Zhang, Shunli 2002. Separation of variables of a generalized porous medium equation with nonlinear source. Journal of Mathematical Analysis and Applications, Vol. 275, Issue. 1, p. 44.

Cherniha, Roman and King, John R 2003. Lie symmetries of nonlinear multidimensional reaction diffusion systems: II. Journal of Physics A: Mathematical and General, Vol. 36, Issue. 2, p. 405.

Cherniha, Roman and Serov, Mykola 2003. Nonlinear systems of the Burgers-type equations: Lie and Q-conditional symmetries, Ansätze and solutions. Journal of Mathematical Analysis and Applications, Vol. 282, Issue. 1, p. 305.

Zhang, Shun-Li Lou, S Y and Qu, Chang-Zheng 2003. New variable separation approach: application to nonlinear diffusion equations. Journal of Physics A: Mathematical and General, Vol. 36, Issue. 49, p. 12223.

Popovych, Roman O and Ivanova, Nataliya M 2004. New results on group classification of nonlinear diffusion–convection equations. Journal of Physics A: Mathematical and General, Vol. 37, Issue. 30, p. 7547.

Qu, Changzheng and Estévez, P.G. 2004. On nonlinear diffusion equations with x-dependent convection and absorption. Nonlinear Analysis: Theory, Methods & Applications, Vol. 57, Issue. 4, p. 549.

Cherniha, Roman and Henkel, Malte 2004. On non-linear partial differential equations with an infinite-dimensional conditional symmetry. Journal of Mathematical Analysis and Applications, Vol. 298, Issue. 2, p. 487.

Qu, Changzheng Ji, Lina and Dou, Jihong 2005. Exact solutions and generalized conditional symmetries to (n+1)-dimensional nonlinear diffusion equations with source term. Physics Letters A, Vol. 343, Issue. 1-3, p. 139.

Cherniha, Roman and King, John R. 2005. Non-linear reaction–diffusion systems with variable diffusivities: Lie symmetries, ansätze and exact solutions. Journal of Mathematical Analysis and Applications, Vol. 308, Issue. 1, p. 11.

Cherniha, Roman 2007. New Q-conditional symmetries and exact solutions of some reaction–diffusion–convection equations arising in mathematical biology. Journal of Mathematical Analysis and Applications, Vol. 326, Issue. 2, p. 783.

Plyukhin, O. H. 2007. Conditional symmetries and exact solutions of one reaction-diffusion-convection equation. Nonlinear Oscillations, Vol. 10, Issue. 3, p. 381.

Cherniha, Roman and Pliukhin, Oleksii 2007. New conditional symmetries and exact solutions of nonlinear reaction–diffusion–convection equations. Journal of Physics A: Mathematical and Theoretical, Vol. 40, Issue. 33, p. 10049.

Qu, Chang-zheng and Zhu, Chun-rong 2007. Invariant sets and solutions to the generalized thin film equation. Science in China Series A: Mathematics, Vol. 50, Issue. 6, p. 875.

Cherniha, Roman and Pliukhin, Oleksii 2007. Symmetries and solutions of reaction‐diffusion‐convection equations with power diffusivities. PAMM, Vol. 7, Issue. 1, p. 2040065.

Zhdanov, R and Lahno, V 2007. Group classification of the general second-order evolution equation: semi-simple invariance groups. Journal of Physics A: Mathematical and Theoretical, Vol. 40, Issue. 19, p. 5083.

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Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms

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This article has been cited by the following publications. This list is generated based on data provided by Crossref.

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Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms

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