@article{AncoAliWolf2011, author = {Anco, Stephen and Ali, Sajid and Wolf, Thomas}, title = {Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction}, journal = {SIGMA 7}, doi = {10.3842/SIGMA.2011.066}, year = {2011}, abstract = {A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetries for the reaction-diffusion equation. With this group-foliation reduction method, solutions of the reaction-diffusion equation are obtained in an explicit form, including group-invariant similarity solutions and travelling-wave solutions, as well as dynamically interesting solutions that are not invariant under any of the point symmetries admitted by this equation.}, language = {en} } @article{AncoAliWolf2011, author = {Anco, Stephen and Ali, Sajid and Wolf, Thomas}, title = {Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions}, volume = {379}, journal = {Journal of Mathematical Analysis and Applications}, number = {2}, doi = {10.1016/j.jmaa.2011.01.073}, pages = {748 -- 763}, year = {2011}, language = {en} }