Variational lambda-symmetries and exact solutions to Euler-Lagrange equations lacking standard symmetries
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SourceMath Meth Appl Sci. 2022;1–13
Variational lambda-symmetries are used to find exact solutions to second- and fourth-order Euler-Lagrange equations associated to variational problems for which standard procedures fail. A one-parameter family of exact solutions in terms of Bessel functions is obtained for a first-order variational problem whose Euler-Lagrange equation does not admit Lie symmetries. A family of second- order equations, involving arbitrary functions and parameters, is first written in variational form. The variational lambda-symmetry method successes in finding one-parameter families of exact solutions, despite the lack of Lie point and variational symmetries. A three-parameter family of exact solutions for a fourth-order equation with absence of Lie point symmetries is also deduced.
SubjectsEuler-Lagrange equation; variational lambda-symmetry; variational problem; variational symmetries
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