梁方程的相似解和守恒定律:一个完整的研究

Amlan K. Halder, A. Paliathanasis, P. Leach
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引用次数: 2

摘要

我们研究了相似解,并确定了各种形式的梁方程,如Euler-Bernoulli, Rayleigh和Timoshenko-Prescott的守恒定律。行波约简导致所有形式的可解四阶谱。此外,基于欧拉-伯努利形式的尺度对称性的约简导致某些零对称性存在的ode。因此,我们通过奇点分析来确定其可积性。我们研究了二阶和三阶的约简阶。约简二阶密码是Painleve-Ince方程的摄动形式,它是可积的,三阶密码属于Chazy、Bureau和Cosgrove研究的方程范畴。此外,我们还推导了上述梁形式的对称及其相应的约简和守恒律。李代数在所有情况下都被明确地提到。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
SIMILARITY SOLUTIONS AND CONSERVATION LAWS FOR THE BEAM EQUATIONS: A COMPLETE STUDY
We study the similarity solutions and we determine the conservation laws of the various forms of beam equation, 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 order second and third. The reduced second-order ode is a perturbed form of Painleve-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 above mentioned beam forms. The Lie Algebra is mentioned explicitly for all the cases.
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