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Lie symmetries of finite-difference equations. (English) Zbl 0859.39008
For the 2-dimensional Helmholtz equation, the 2- and 3-dimensional heat equation, and the 3- and 4-dimensional wave equation, respectively, discretizations on uniform lattices are considered, which keep the Lie algebras of the differential equations invariant as symmetry algebras. Hence, the usual techniques can be used to find solutions of the corresponding difference equations with definite transformation properties and, in particular, to find discrete versions of hypergeometric and Bessel functions as well as Hermite, Laguerre and Gegenbauer polynomials.
Reviewer: L.Berg (Rostock)
MSC:
39A12Discrete version of topics in analysis
39A10Additive difference equations
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
35K05Heat equation
35L05Wave equation (hyperbolic PDE)
33C45Orthogonal polynomials and functions of hypergeometric type