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Coercive estimates for the solutions of some singular differential equations and their applications
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
2013 (Engelska)Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

This Licentiate thesis deals with the study of existence and uniqueness together with coercive estimates for solutions of certain differential equations. The thesis consists of four papers (papers A, B, C and D) and an introduction, which put these papers into a more general frame and which also serves as an overview of this interesting field of mathematics. In the text below the functions r(x), q(x), m(x) etc. are functions on (-∞,+∞), which are different but well defined in each paper. In paper A we study the separation and approximation properties for the differential operator ly=-y″+r(x)y′+q(x)y in the Hilbert space L2 :=L2(R), R=(-∞,+∞), as well as the existence problem for a second order nonlinear differential equation in L2 . Paper B deals with the study of separation and approximation properties for the differential operator ly=-y″+r(x)y′+s(x)‾y′ in the Hilbert spaceL2:=L2(R), R=(-∞,+∞), (here ¯y is the complex conjugate of y). A coercive estimate for the solution of the second order differential equation ly =f is obtained and its applications to spectral problems for the corresponding differential operatorlis demonstrated. Some sufficient conditions for the existence of the solutions of a class of nonlinear second order differential equations on the real axis are obtained. In paper C we study questions of the existence and uniqueness of solutions of the third order differential equation (L+λE)y:=-m(x)(m(x)y′)″+[q(x)+ir(x)+λ]y=f(x), (0.1) and conditions, which provide the following estimate: ||m(x)(m(x)y′)″||pp+||(q(x)+ir(x)+λ)y||pp≤ c||f(x)||pp for a solution y of (0.1). Paper D is devoted to the study of the existence and uniqueness for the solutions of the following more general third order differential equation with unbounded coefficients: -μ1(x)(μ2(x)(μ1(x)y′)′)′+(q(x)+ir(x)+λ)y=f(x). Some new existence and uniqueness results are proved and some normestimates of the solutions are given.

Ort, förlag, år, upplaga, sidor
Luleå: Luleå tekniska universitet, 2013. , s. 105
Serie
Licentiate thesis / Luleå University of Technology, ISSN 1402-1757
Nationell ämneskategori
Matematisk analys
Forskningsämne
Matematik
Identifikatorer
URN: urn:nbn:se:ltu:diva-16970Lokalt ID: 0f123f49-ab51-4486-b8e3-53e81e7156c8ISBN: 978-91-7439-560-0 (tryckt)OAI: oai:DiVA.org:ltu-16970DiVA, id: diva2:989962
Anmärkning

Godkänd; 2013; 20130208 (rayakh); Tillkännagivande licentiatseminarium 2013-02-26 Nedanstående person kommer att hålla licentiatseminarium för avläggande av teknologie licentiatexamen. Namn: Raya Akhmetkaliyeva Ämne: Matematik/Mathematics Uppsats: Coercive Estimates for the Solutions of some Singular Differential Equations and their Applications Examinator: Professor Lars-Erik Persson, Institutionen för teknikvetenskap och matematik, Luleå tekniska universitet Diskutant: Associate Professor Gabriela Holubová, Department of Mathematics, University of West Bohemia, Czech Republic Tid: Onsdag den 20 mars 2013 kl 10.00 Plats: E246, Luleå tekniska universitet

Tillgänglig från: 2016-09-29 Skapad: 2016-09-29 Senast uppdaterad: 2018-04-19Bibliografiskt granskad

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