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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

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