SIGMA 7 (2011), 066, 11 pages arXiv:1105.5303
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”
Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction
Stephen C. Anco a, Sajid Ali b and Thomas Wolf a
a) Department of Mathematics, Brock University, St. Catharines, ON L2S 3A1 Canada
b) School of Electrical Engineering and Computer Sciences, National University of Sciences and Technology,
H-12 Campus, Islamabad 44000, Pakistan
Received March 05, 2011, in final form July 03, 2011; Published online July 12, 2011;
Typos in the solutions are corrected August 02, 2013
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.
semilinear heat equation; similarity reduction; exact solutions; group foliation; symmetry.
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