‎This paper obtains the exact solutions of the wave equation as a second-order partial differential equation (PDE)‎. ‎We are going to calculate polynomial and non-polynomial exact solutions by using Lie point symmetry‎. ‎We demonstrate the generation of such polynomial through the medium of the group theoretical properties of the equation‎. ‎A generalized procedure for polynomial solution is pr...

We consider a left-invariant dynamics on an arbitrary Lie group. We show that it is possible, when the output map is compatible with right multiplications, to build non-linear observers such that the error equation is autonomous. The theory is illustrated by an inertial navigation example. Index Terms Lie group, symmetries, invariance, nonlinear asymptotic observers, inertial navigation.