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postgraduate thesis: Exact meromorphic solutions of complex algebraic differential equations

TitleExact meromorphic solutions of complex algebraic differential equations
Authors
Advisors
Advisor(s):Ng, TW
Issue Date2012
PublisherThe University of Hong Kong (Pokfulam, Hong Kong)
Citation
Wong, K. [黃國堅]. (2012). Exact meromorphic solutions of complex algebraic differential equations. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4833021
AbstractFor any given complex algebraic ordinary differential equation (ODE), one major task of both pure and applied mathematicians is to find explicit meromorphic solutions due to their extensive applications in science. In 2010, Conte and Ng in [12] proposed a new technique for solving complex algebraic ODEs. The method consists of an idea due to Eremenko in [20] and the subequation method of Conte and Musette, which was first proposed in [9]. Eremenko’s idea is to make use of the Nevanlinna theory to analyze the value distribution and growth rate of the solutions, from which one would be able to show that in some cases, all the meromorphic solutions of the studied differential equation are in a class of functions called “class W”, which consists of elliptic functions and their degenerates. The establishment of solutions is then achieved by the subequation method. The main idea is to build subequations which have solutions that also satisfy the original differential equation, hoping that the subequations will be easier to solve. As in [12], the technique has been proven to be very successful in obtaining explicit particular meromorphic solutions as well as giving complete classification of meromorphic solutions. In this thesis, the necessary theoretical background, including the Nevanlinna theory and the subequation method, will be developed. The technique will then be applied to obtain all meromorphic stationary wave solutions of the real cubic Swift-Hohenberg equation (RCSH). This last part is joint work with Conte and Ng and will appear in Studies in Applied Mathematics [13]. RCSH is important in several studies in physics and engineering problems. For instance, RCSH is used as modeling equation for Rayleigh- B?nard convection in hydrodynamics [43] as well as in pattern formation [16]. Among the explicit stationary wave solutions obtained by the technique used in this thesis, one of them appears to be new and could be written down as a rational function composite with Weierstrass elliptic function.
DegreeMaster of Philosophy
SubjectDifferential-algebraic equations.
Dept/ProgramMathematics

 

DC FieldValueLanguage
dc.contributor.advisorNg, TW-
dc.contributor.authorWong, Kwok-kin.-
dc.contributor.author黃國堅.-
dc.date.issued2012-
dc.identifier.citationWong, K. [黃國堅]. (2012). Exact meromorphic solutions of complex algebraic differential equations. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4833021-
dc.description.abstractFor any given complex algebraic ordinary differential equation (ODE), one major task of both pure and applied mathematicians is to find explicit meromorphic solutions due to their extensive applications in science. In 2010, Conte and Ng in [12] proposed a new technique for solving complex algebraic ODEs. The method consists of an idea due to Eremenko in [20] and the subequation method of Conte and Musette, which was first proposed in [9]. Eremenko’s idea is to make use of the Nevanlinna theory to analyze the value distribution and growth rate of the solutions, from which one would be able to show that in some cases, all the meromorphic solutions of the studied differential equation are in a class of functions called “class W”, which consists of elliptic functions and their degenerates. The establishment of solutions is then achieved by the subequation method. The main idea is to build subequations which have solutions that also satisfy the original differential equation, hoping that the subequations will be easier to solve. As in [12], the technique has been proven to be very successful in obtaining explicit particular meromorphic solutions as well as giving complete classification of meromorphic solutions. In this thesis, the necessary theoretical background, including the Nevanlinna theory and the subequation method, will be developed. The technique will then be applied to obtain all meromorphic stationary wave solutions of the real cubic Swift-Hohenberg equation (RCSH). This last part is joint work with Conte and Ng and will appear in Studies in Applied Mathematics [13]. RCSH is important in several studies in physics and engineering problems. For instance, RCSH is used as modeling equation for Rayleigh- B?nard convection in hydrodynamics [43] as well as in pattern formation [16]. Among the explicit stationary wave solutions obtained by the technique used in this thesis, one of them appears to be new and could be written down as a rational function composite with Weierstrass elliptic function.-
dc.languageeng-
dc.publisherThe University of Hong Kong (Pokfulam, Hong Kong)-
dc.relation.ispartofHKU Theses Online (HKUTO)-
dc.rightsThe author retains all proprietary rights, (such as patent rights) and the right to use in future works.-
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.source.urihttp://hub.hku.hk/bib/B48330218-
dc.subject.lcshDifferential-algebraic equations.-
dc.titleExact meromorphic solutions of complex algebraic differential equations-
dc.typePG_Thesis-
dc.identifier.hkulb4833021-
dc.description.thesisnameMaster of Philosophy-
dc.description.thesislevelMaster-
dc.description.thesisdisciplineMathematics-
dc.description.naturepublished_or_final_version-
dc.identifier.doi10.5353/th_b4833021-
dc.date.hkucongregation2012-

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