An Invariance and Closed Form Analysis of the Nonlinear Biharmonic Beam Equation.

In this paper, we study the one-parameter Lie groups of point transformations that leave invariant the biharmonic partial differential equation (PDE) uxxxx + 2uxxyy + uyyyy = f(u). To this end, we construct the Lie and Noether symmetry generators and present reductions of biharmonic PDE. When f is arbitrary function of u, we obtain the solution of biharmonic equation in terms of Green function. The equation is further analysed when f is exponential function and for general power law. Furthermore, we use Noether's theorem and the 'multiplier approach' to construct conservation laws of the PDE. [ABSTRACT FROM AUTHOR]