arXiv 2004.004952020-04-01math-phmath.MP

梁方程的相似解和守恒律：完整的研究

Similarity solutions and conservation laws for the Beam Equations: a complete study

Amlan K Halder, Andronikos Paliathanasis, PGL Leach

备注：: 14 pages and accepted for publication by Acta Polytechnica

摘要：

我们研究了梁方程的相似解，并确定了各种形式的梁方程的守恒律，如Euler-Bernoulli，Rayleigh和Eulshenko-Prescott。行波还原导致可解的四阶odes的所有形式。此外，基于Euler-Bernoulli形式的标度对称性的约化导致存在零对称性的某些节点。因此，我们进行奇异性分析，以确定可积性。我们研究两个减少颂歌的第二和第三阶。约化的二阶微分方程是Painlevé-Ince方程的扰动形式，它是可积的，三阶微分方程属于Chazy，Bureau和Cosgrove研究的方程范畴.此外，我们还导出了上述梁形式的强迫形式的对称性及其相应的约化和守恒律。李代数是明确提到的所有情况。

英文摘要：

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 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 above mentioned beam forms. The Lie Algebra is mentioned explicitly for all the cases.

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摘要：

英文摘要：

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