Symmetries and invariances of differential equations with applications to fluid dynamics

Frysalis, Christos; Φρυσάλης Χρήστος

dc.contributor.author	Frysalis, Christos	en
dc.contributor.author	Φρυσάλης Χρήστος	el
dc.date.accessioned	2025-09-15T11:28:26Z
dc.date.available	2025-09-15T11:28:26Z
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/62451
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.30147
dc.description	Εθνικό Μετσόβιο Πολυτεχνείο--Μεταπτυχιακή Εργασία. Διεπιστημονικό-Διατμηματικό Πρόγραμμα Μεταπτυχιακών Σπουδών (Δ.Π.Μ.Σ.) “Εφαρμοσμένες Μαθηματικές Επιστήμες”	el
dc.rights	Default License
dc.subject	Lie group analysis	en
dc.subject	Differential equations	en
dc.subject	Symmetries	en
dc.subject	Group-invariant solutions	en
dc.subject	Conservation laws	en
dc.title	Symmetries and invariances of differential equations with applications to fluid dynamics	en
heal.type	masterThesis
heal.classification	Differential Equations	en
heal.language	en
heal.access	free
heal.recordProvider	ntua	el
heal.publicationDate	2024-11-05
heal.abstract	In this thesis, we explore the role of symmetries in the analysis of ordinary and partial differential equations. Specifically, we describe the systematic derivation of symmetry groups for the equations under consideration, and demonstrate their use in deriving explicit solutions and constructing conservation laws. These methods are applied to a range of specific, primarily nonlinear, problems. Beginning with the theory of Lie groups of transformations and infinitesimal generators, we develop key concepts that are essential for several applications. Through infinitesimal methods, we describe a systematic approach for deriving symmetry groups for systems of differential equations by employing a necessary and sufficient condition known as the determining equation for symmetries. Any known symmetry group of a differential equation or a system of differential equations can be used to obtain explicit solutions which remain invariant under the action of the group. We demonstrate how a reduction of the number of independent variables can be achieved, thus facilitating the search for exact solutions to partial differential equations. We then focus on the problem of constructing conservation laws. An alternative formulation of Noether’s theorem is presented, known as Boyer’s formulation. We also present the multipliers method for constructing conservation laws and prove that for systems that can be written in Cauchy-Kovalevskaya form, all conservation laws (up to equivalence) arise from a set of multipliers, with a one-to-one correspondence between equivalent classes of conservation laws and sets of multipliers. Throughout the thesis, several applications are explored. For instance, we derive the full set of symmetries for Euler’s equations in three-dimensional space for an inviscid, incompressible fluid. We also obtain explicit solutions and an infinite hierarchy of generalized symmetries for the KdV equation. Finally, we analyze a Boussinesq equation, deriving its symmetries, invariant solutions, and conservation laws.	en
heal.advisorName	Athanassoulis, Gerassimos	en
heal.committeeMemberName	Athanassoulis, Gerassimos	en
heal.committeeMemberName	Charalampopoulos, Antonios	en
heal.committeeMemberName	Fellouris, Anargyros	en
heal.academicPublisher	Εθνικό Μετσόβιο Πολυτεχνείο. Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών	el
heal.academicPublisherID	ntua
heal.numberOfPages	162 σ.	el
heal.fullTextAvailability	false