Symmetries of partial differential equations - Applications to initial and boundary value problems

Μπεληγιάννης, Γιώργος; Beligiannis, Georgios

dc.contributor.author	Μπεληγιάννης, Γιώργος	el
dc.contributor.author	Beligiannis, Georgios	en
dc.date.accessioned	2025-10-30T08:44:17Z
dc.date.available	2025-10-30T08:44:17Z
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/62800
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.30496
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	Partial Differential Equations	en
dc.subject	Symmetries	en
dc.subject	Lie Groups	en
dc.subject	Lie Algebra	en
dc.title	Symmetries of partial differential equations - Applications to initial and boundary value problems	en
heal.type	masterThesis
heal.classification	Μαθηματικά	el
heal.classification	Mathematics	en
heal.language	en
heal.access	free
heal.recordProvider	ntua	el
heal.publicationDate	2025-02-28
heal.abstract	Symmetry analysis of differential equations (DEs) is originated from the Norwegian mathematician Sophus Lie, in the 1870s. Lie research work on continuous group analysis and their role in discovering symmetries of DEs provides an indispensable framework for the systematic and efficient study of DEs, both ordinary and partial. By revealing the admitted symmetries of these equations, the Lie-group approach provides exact solutions in many cases, and also a deep understanding of the structural properties underlying these equations, whence a variety of ad hoc methods of integration for DEs fail to do so. The first part of this thesis examines the fundamental theory of Lie groups and Lie algebras, and their significance in the study of DEs. We highlight the importance of the First Fundamental Theorem of Lie, which allows us to reduce the problem of identifying symmetries, by transistioning to the corresponding tangent vector space, the Lie algebra. Building on this foundation, and treating a DE as a geometric surface, we introduce a systematic approach for determining symmetry groups through an algorithmic procedure based on the determining equations. In this part of our thesis, we also utilize all this theoretical framework in order to reduce the order of an ODE, admitting a solvable Lie algebra, and finally obtain the general solution. We explain how to derive first integrals by utilizing the corresponding symmetries. These first integrals lead us (under appropriate conditions) in the consecutive reduction of order, and thus in the derivation of solutions of ODEs, usually unsolvable through the classical integrating methods. We also formulate a classification of second-order ODEs according to the admitted symmetries that span two-dimensional Lie algebras. On the second part of the thesis, we turn our attention in PDEs. We first provide a presentation about a class of solutions, characterized by their invariance under the action of a corresponding symmetry group, also known as invariant solutions. These solutions are derived by solving a reduced system of equations with fewer independent variables than the original PDEs. In many cases of nonlinear PDEs, where we cannot obtain an exact solution with classical integration methods, we can still derive invariant solutions. To illustrate the methods discussed, we focus on the Heat equation as a primary example, followed by Burgers’ equation and a system of gas-dynamic equations. Initial and boundary conditions are also imposed to the Heat equation, to demonstrate how to solve initial and boundary value problems (IBVPs) of PDEs, invariant under the action of a Lie group. Building on this introductory example, we apply the developed methods to a physically significant IVP, governed by the Fokker-Planck equation with an odd forcing function, which reflects systems with restoring forces symmetric about the origin, and is initially undetermined. By leveraging the symmetry conditions provided in the first part of this thesis, alongside the properties of odd functions, we can construct the general form of this function, so that we can derive non-trivial symmetries for the IVP. Then, by utilizing these symmetries, we simplify the problem in order to obtain an invariant solution.	en
heal.advisorName	Charalampopoulos, Antonios	en
heal.advisorName	Athanassoulis, Gerasimos	en
heal.committeeMemberName	Athanassoulis, Gerasimos	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	176 σ.	el
heal.fullTextAvailability	false