Applications of Solvable Lie Algebras to a Class of Third Order Equations

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2022-01Department
MatemáticasSource
Mathematics 2022, 10(2), 254Abstract
A family of third-order partial differential equations (PDEs) is analyzed. This family
broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and
the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic
models for semiconductors considering quantum effects—for example, models for the transmission of
electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family.
For this family, all point symmetries have been derived. These symmetries are used to determine
group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry
group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation
(ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking
into account the Type-II hidden symmetries that appear in the reduced second-order ODEs.
Subjects
third-order partial differential equations; lie symmetries; solvable symmetry algebras; group invariant solutionsCollections
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