โ Scribed by G. Bluman
No coin nor oath required. For personal study only.
Group-theoretic methods based on local symmetries are useful to construct invariant solutions of PDEs and to linearize nonlinear PDEs by invertible mappings. Local symmetries include point symmetries, contact symmetries and, more generally, Lie-Biicklund symmetries. An obvious limitation in their utility for particular PDEs is the non-existence of local symmetries. A given system of PDEs with a conserved form can be embedded in a related auxiliary system of PDEs. A local symmetry of the auxiliary system can yield a nonlocal symmetry (potential symmetry) of the given system. The existence of potential symmetries leads to the construction of corresponding invariant solutions ss well as to the linearization of nonlinear PDBs by non-invertible mappings. Recent work considers the problem of finding all potential symmetries of given systems of PDEs. Examples include linear wave equations with variable wave speeds ss well ss nonlinear diffusion, reaction-diffusion, and gas dynamics equations.
๐ SIMILAR VOLUMES
We consider systems of two pure one-dimensional diffusion equations that have considerable interest in Soil Science and Mathematical Biology. We construct non-local symmetries for these systems. These are determined by expressing the equations in a partially and wholly conserved form, and then by pe
A construction of orthogonal wavelet bases in \(L_{2}\left(R^{d}\right)\) from a multiresolution analysis is given when the scaling function is skew-symmetric about an integer point. These wavelets are real-valued when the scaling function is real-valued. As an application, orthogonal wavelets gener