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

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 semiconductor...

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Bibliographic Details
Published in:Mathematics (Basel) 2022-01, Vol.10 (2), p.254
Main Authors: Bruzón, María S., de la Rosa, Rafael, Gandarias, María L., Tracinà, Rita
Format: Article
Language:English
Subjects:
Algebra
Burgers equation
Exact solutions
Food science
Generators
group invariant solutions
Group theory
Lie groups
lie symmetries
Macroscopic models
Mathematical models
Methods
Nonlinear differential equations
Partial differential equations
solvable symmetry algebras
Subgroups
Symmetry
third-order partial differential equations
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math10020254