On nonlinear diffusion equations with x-dependent convection and absorption |
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Institution: | 1. Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada;2. Department of Fluid Dynamics, Technical University of Darmstadt, Darmstadt, Germany;1. Instituto de Matemática, Estatística e Computação Científica, IMECC Universidade Estadual de Campinas - UNICAMP, Rua Sérgio Buarque de Holanda 651 13083-859, Campinas, SP, Brasil;2. Centro de Matemática, Computação e Cognição, CMCC Universidade Federal do ABC - UFABC, Av. dos Estados 5001, Bairro Bangú 09210-580, Santo André, SP, Brasil |
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Abstract: | For the 1+1-dimensional nonlinear diffusion equations with x-dependent convection and source terms ut=(D(u)ux)x+Q(x,u)ux+P(x,u), we obtain conditions under which the equations admit the second-order generalized conditional symmetries η(x,u)=uxx+H(u)ux2+G(x,u)ux+F(x,u) and the first-order sign-invariants J(x,u)=ut?A(u)ux2?B(x,u)ux?C(x,u) on the solutions u(x,t). Several different generalized conditional symmetries and first-order sign-invariants for equations in which the diffusion term offers different possibilities (power-law, exponential, Mullin, Fujita) are presented. Exact solutions to the resulting equations corresponding to the generalized conditional symmetries and the first-order sign-invariants are constructed. |
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