Malaysian Journal of Mathematical Sciences, June 2023, Vol. 17, No. 2
An Invariance and Closed Form Analysis of the Nonlinear Biharmonic Beam Equation
Masood, Y., Kara, A. H., and Zaman, F. D.
Corresponding Email: Abdul.Kara@wits.ac.za
Received date: 22 November 2022
Accepted date: 6 April 2023
Abstract:
In this paper, we study the one-parameter Lie groups of point transformations that leave invariant the biharmonic partial differential equation (PDE) $u_{xxxx}+2u_{xxyy}+u_{yyyy}=f(u)$. 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.
Keywords: biharmonic equation; transformation groups; Lie symmetries; conservation laws
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