TY - GEN A1 - Anco, Stephen A1 - Ali, Sajid A1 - Wolf, Thomas T1 - Exact solutions of nonlinear partial differential equations by the method of group foliation reduction N2 - 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 is based on group foliation reduction and employs 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 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. T3 - ZIB-Report - 11-25 KW - semilinear heat equation, similarity reduction, exact solutions, group foliation, symmetry Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13171 ER - TY - GEN A1 - Anco, Stephen A1 - Ali, Sajid A1 - Wolf, Thomas T1 - Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions N2 - A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in $n>1$ dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity reduction of this heat equation gives a semilinear ODE that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors. T3 - ZIB-Report - 11-26 KW - semilinear heat equation, similarity reduction, exact solutions, group foliation, symmetry Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13213 ER -