Editorial Recent Advances in Symmetry Groups and Conservation...
3
Editorial Recent Advances in Symmetry Groups and Conservation Laws for Partial Differential Equations and Applications Maria Gandarias, 1 Mariano Torrisi, 2 Maria Bruzón, 1 Rita Tracinà, 2 and Chaudry Masood Khalique 3 1 Departamento de Matematicas, Universidad de C´ adiz, Puerto Real, 11510 C´ adiz, Spain 2 Dipartimento di Matematica e Informatica, Universit` a di Catania, 95125 Catania, Italy 3 Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical Modelling, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa Correspondence should be addressed to Maria Gandarias; [email protected] Received 25 August 2014; Accepted 25 August 2014; Published 22 December 2014 Copyright © 2014 Maria Gandarias et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Differential equations govern many natural phenomena and play an important role in the progress of engineering and technology. Essentially a lot of the fundamental equations are nonlinear and in general such nonlinear equations are oſten very difficult to solve explicitly. Symmetry group techniques provide methods to obtain solutions of these equations. ese methods have several applications in the studies of partial differential equation. ey are also useful in the search for conservation laws which arises in many fields of the applied sciences. Recent studies have shown that infinitely many nonlocal symmetries of various integrable models are related to their Lax pairs. Moreover symmetry method is one of the most powerful tools that give new integrable models from known ones. Integrable models have played an important role in applied sciences and are one of the central topics in soliton theory. In order to know if a system is integrable, it is very important to study Lax pairs of the system. A symmetry can be considered as an equivalence trans- formation which leaves invariant not only the differential structure of equation but also the form of the arbitrary elements. is fact made Ovsiannikov search for equivalence transformations in a systematic way by using an algorithm based on the extension of the Lie infinitesimal criterion. When an equation contains an arbitrary function, it reflects the individual characteristic of the phenomena belonging to a large class. In this sense, the knowledge of equivalence transformations can provide us with certain relations between the solutions of different phenomena of the same class. Nowadays several branches of the theoretical and applied sciences such as mathematics, physics, biology, economy, and finance rely on processes which are usually modeled by nonlinear differential equations. Oſten it is difficult to obtain reductions and exact solutions for these models. Our aim is to highlight applications of symmetry methods to nonlinear models in physics, engineering, and the applied science as well as to show recent theoretical developments in symmetry groups and geometric methods. e authors of this special issue had been invited to submit original research articles as well as review articles in the following topics: advanced researches and theoretical analyses in group transformations and differential equations; Noether symmetries, applications, and conservation laws; numerical algorithms concerning the symmetry groups for partial differential equations; new and direct methods to obtain exact explicit solutions for differential equations; applications: novel applications in sciences, including engi- neering, physics, biology, and finance; and reviews: lucid surveys and review articles dealing with modern and classical topics. However, we received 20 papers in these research fields. Aſter a rigorous reviewing process, twelve articles were finally Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 457145, 2 pages http://dx.doi.org/10.1155/2014/457145
Transcript of Editorial Recent Advances in Symmetry Groups and Conservation...
EditorialRecent Advances in Symmetry Groups and Conservation Lawsfor Partial Differential Equations and Applications
Maria Gandarias,1 Mariano Torrisi,2 Maria Bruzón,1
Rita Tracinà,2 and Chaudry Masood Khalique3
1Departamento de Matematicas, Universidad de Cadiz, Puerto Real, 11510 Cadiz, Spain2Dipartimento di Matematica e Informatica, Universita di Catania, 95125 Catania, Italy3Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical Modelling,North-West University, Mafikeng Campus, Mmabatho 2735, South Africa
Correspondence should be addressed to Maria Gandarias; [email protected]
Received 25 August 2014; Accepted 25 August 2014; Published 22 December 2014
Copyright © 2014 Maria Gandarias et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
Differential equations govern many natural phenomena andplay an important role in the progress of engineering andtechnology. Essentially a lot of the fundamental equations arenonlinear and in general such nonlinear equations are oftenvery difficult to solve explicitly. Symmetry group techniquesprovidemethods to obtain solutions of these equations.Thesemethods have several applications in the studies of partialdifferential equation. They are also useful in the search forconservation laws which arises in many fields of the appliedsciences. Recent studies have shown that infinitely manynonlocal symmetries of various integrable models are relatedto their Lax pairs. Moreover symmetry method is one of themost powerful tools that give new integrable models fromknown ones. Integrablemodels have played an important rolein applied sciences and are one of the central topics in solitontheory. In order to know if a system is integrable, it is veryimportant to study Lax pairs of the system.
A symmetry can be considered as an equivalence trans-formation which leaves invariant not only the differentialstructure of equation but also the form of the arbitraryelements.This fact made Ovsiannikov search for equivalencetransformations in a systematic way by using an algorithmbased on the extension of the Lie infinitesimal criterion.
When an equation contains an arbitrary function, itreflects the individual characteristic of the phenomenabelonging to a large class. In this sense, the knowledge
of equivalence transformations can provide us with certainrelations between the solutions of different phenomena of thesame class.
Nowadays several branches of the theoretical and appliedsciences such as mathematics, physics, biology, economy,and finance rely on processes which are usually modeled bynonlinear differential equations. Often it is difficult to obtainreductions and exact solutions for these models. Our aim isto highlight applications of symmetry methods to nonlinearmodels in physics, engineering, and the applied science aswell as to show recent theoretical developments in symmetrygroups and geometric methods.
The authors of this special issue had been invited tosubmit original research articles as well as review articlesin the following topics: advanced researches and theoreticalanalyses in group transformations and differential equations;Noether symmetries, applications, and conservation laws;numerical algorithms concerning the symmetry groups forpartial differential equations; new and direct methods toobtain exact explicit solutions for differential equations;applications: novel applications in sciences, including engi-neering, physics, biology, and finance; and reviews: lucidsurveys and review articles dealing withmodern and classicaltopics.
However, we received 20 papers in these research fields.After a rigorous reviewing process, twelve articles were finally
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 457145, 2 pageshttp://dx.doi.org/10.1155/2014/457145
2 Abstract and Applied Analysis
accepted for publication.These articles contain somenew andinnovative techniques and ideas that may stimulate furtherresearches in several branches of theory and applications ofthe transformation groups.
Acknowledgments
We would like to thank all the authors who sent their worksand all the referees for the time spent in reviewing thepapers. Their contributions and their efforts have been veryimportant for the publication of this special issue.
Maria GandariasMariano TorrisiMaria BruzonRita Tracina
Chaudry Masood Khalique
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Stochastic AnalysisInternational Journal of
