SIMILARITY SOLUTIONS AND CONSERVATION LAWS FOR THE BEAM EQUATIONS: A COMPLETE STUDY

Authors

  • Amlan Kanti Halder PONDICHERRY UNIVERSITY
  • Andronikos Paliathanasis Instituto de Ciencias F´ısicas y Matem´aticas, Universidad Austral de Chile, Valdivia, Chile and Institute of Systems Science, Durban University of Technology Durban 4000, Republic of South Africa.
  • Peter Gavin Lawrence Leach School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa. Institute of Systems Science, Durban University of Technology Durban 4000, Republic of South Africa.

DOI:

https://doi.org/10.14311/AP.2020.60.0098

Keywords:

Symmetry analysis, singularity analysis, conservation laws, beam equation

Abstract

We study the similarity solutions and we determine the conservation laws of various forms of beam equations, such as Euler-Bernoulli, Rayleigh and Timoshenko-Prescott. The travelling-wave reduction leads to solvable fourth-order odes for all the forms. In addition, the reduction based on the scaling symmetry for the Euler-Bernoulli form leads to certain odes for which there exists zero symmetries. Therefore, we conduct the singularity analysis to ascertain the integrability. We study two reduced odes of second and third orders. The reduced second-order ode is a perturbed form of Painlevé-Ince equation, which is integrable and the third-order ode falls into the category of equations studied by Chazy, Bureau and Cosgrove. Moreover, we derived the symmetries and its corresponding reductions and conservation laws for the forced form of the abovementioned beam forms. The Lie Algebra is mentioned explicitly for all the cases.

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Published

2020-04-30

Issue

Vol. 60 No. 2 (2020)

Section

Articles

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How to Cite

Halder, A. K., Paliathanasis, A., & Leach, P. G. L. (2020). SIMILARITY SOLUTIONS AND CONSERVATION LAWS FOR THE BEAM EQUATIONS: A COMPLETE STUDY. Acta Polytechnica, 60(2), 98-110. https://doi.org/10.14311/AP.2020.60.0098