Symmetries and integrability of a fourth-order Euler–Bernoulli beam equation

Abstract

The complete symmetry group classification of the fourth-order Euler–Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solutions via the Noether integrals. The Lie reduction gives a second-order ordinary differential equation which has nonlocal symmetry.

Symmetry analysis of a class of autonomous even-order ordinary differential equations

“…On the other hand, the same theorem generalizes the results of [Govinder and Leach (2007), Bokhari et al(2010), Fatima et al (2013)] for semilinear ODEs admitting the sl(2, R) symmetry Lie algebra. Therefore, this paper not only extends the results of [Svirshchevskii (1993), Bozhkov(2006)] to the equation (1), but also generalizes the results of [Bokhari et al(2010), Fatima et al (2013), Freire et al (2013)] regarding fourth-order equation to an arbitrary even-order semilinear ODEs.…”

“…an equation first obtained in [Bokhari et al(2010)] and later discussed in [Fatima et al (2013)]. We observe that from (65) it is possible to obtain a three-parameter family of solutions of the considered equation, which does not imply that it is an easy task.…”

“…As it was previously pointed out, it is interesting to observe that all Lie symmetries of the considered equation are also Noether symmetries. This fact was observed in the literature for certain equations, see [Bozhkov(2005), Bozhkov and Mitidieri(2007)]. However, for equations of the type (1), this fact was known for the case n = 1 and, more recently, for the case n = 2.…”

“…We, however, will not proceed with further information on these transformations in this paper once we are interested in understanding the onedimensional version of the results obtained in [Svirshchevskii (1993), Bozhkov(2006)]. Moreover, it is well known that for semilinear equations such as (1), in order to carry out a complete group classification, up to equivalence transformations, one must take the functions f as considered in the Theorem 1, see [Bokhari et al(2010), Bozhkov(2006), Bozhkov and Freire (2008a), Bozhkov and Freire (2010), Freire et al (2013), Svirshchevskii (1993)]. They, in fact, arise from a compatibility condition that will be deduced on section 4.…”

“…However, there are some examples in the literature where both groups coincide. In [Bozhkov(2005)] this fact was firstly discussed. Later, in [Bozhkov and Mitidieri(2007)], this point was retaken and many examples were analysed.…”