Kraštinių uždavinių su įvairių tipų nelokaliosiomis sąlygomis tyrimas ir skaitinė analizė ; Investigation and numerical analysis of boundary problems with various types nonlocal conditions

Jolanta Bak

Description

Sigita PE ČIULYT Ė INVESTIGATION AND NUMERICAL ANALYSIS OF BOUNDARY PROBLEMS WITH VARIOUS TYPES NONLOCAL CONDITIONS Summary of Doctoral Dissertation Physical Sciences, Mathematics (01P) 1339 Vilnius 2006 VILNIUS GEDIMINAS TECHNICAL UNIVERSITY INSTITUTE OF MATHEMATICS AND INFORMATICS Sigita PE ČIULYT Ė INVESTIGATION AND NUMERICAL ANALYSIS OF BOUNDARY PROBLEMS WITH VARIOUS TYPES NONLOCAL CONDITIONS Summary of Doctoral Dissertation Physical Sciences, Mathematics (01P) Vilnius 2006 Doctoral dissertation was prepared at Vytautas Magnus University in 2002–2006 The dissertation is defended as an external work Scientific Advisor Assoc Prof Dr Art ūras ŠTIKONAS (Institute of Mathematics and Informatics, Physical Sciences, Mathematics – 01P) The dissertation is being defended at the Council of Scientific Field of Mathematics at Vilnius Gediminas Technical University: Chairman Prof Dr Habil Raimondas ČIEGIS (Vilnius Gediminas Technical University, Physical Sciences, Mathematics – 01P) Members: Prof Dr Habil Juozas AUGUTIS (Vytautas Magnus University, Technological Sciences, Energetics and Thermal Engineering – 06T) Prof Dr Habil Feliksas IVANAUSKAS (Vilnius University, Physical Sciences, Mathematics – 01P) Assoc Prof Dr Me čislavas MEIL ŪNAS (Vilnius Gediminas Technical University, Physical Sciences, Mathematics – 01P) Prof Dr Habil Mifodijus

Sujets

• Mathematics

Informations

• Publié par: vilnius_gediminas_technical_university
• Publié le: 01 janvier 2007
• Nombre de lectures: 27

Extrait

 

 

               Sigita PEČIULYTĖ           INVESTIGATION AND NUMERICAL ANALYSIS OF BOUNDARY PROBLEMS WITH VARIOUS TYPES NONLOCAL CONDITIONS         Summary of Doctoral Dissertation Physical Sciences, Mathematics (01P)          

Vilnius 2006  

1339

 

 

 

 

VILNIUS GEDIMINAS TECHNICAL UNIVERSITY INSTITUTE OF MATHEMATICS AND INFORMATICS                   Sigita PEČIULYTĖ           INVESTIGATION AND NUMERICAL ANALYSIS OF BOUNDARY PROBLEMS WITH VARIOUS TYPES NONLOCAL CONDITIONS           Summary of Doctoral Dissertation Physical Sciences, Mathematics (01P)          

Vilnius 2006

 

Doctoral dissertation was prepared at Vytautas Magnus University in 20022006 The dissertation is defended as an external work   Scientific Advisor Assoc Prof Dr Artūras TIKONAS of Mathematics and (Institute Informatics, Physical Sciences, Mathematics  01P)   The dissertation is being defended at the Council of Scientific Field of Mathematics at Vilnius Gediminas Technical University: Chairman Prof Dr Habil RaimondasČIEGIS(Vilnius Gediminas Technical University, Physical Sciences, Mathematics  01P)   Members: Prof Dr Habil Juozas AUGUTIS (Vytautas Magnus University, Technological Sciences, Energetics and Thermal Engineering  06T)   Prof Dr Habil Feliksas IVANAUSKAS (Vilnius University, Physical Sciences, Mathematics  01P) Assoc Prof Dr Mečislavas MEILŪNAS Gediminas Technical (Vilnius University, Physical Sciences, Mathematics  01P) Prof Dr Habil Mifodijus SAPAGOVAS(Institute of Mathematics and Informatics, Physical Sciences, Mathematics  01P) Opponents: Assoc Prof Dr Vytautas KLEIZA University of Technology, (Kaunas Physical Sciences, Mathematics  01P) Assoc Prof Dr Aleksandras KRYLOVAS(Vilnius Gediminas Technical University, Physical Sciences, Mathematics  01P)   The dissertation will be defended at the public meeting of the Council of Scientific Field of Mathematics in the Senate Hall of Vilnius Gediminas Technical University at 2 p. m. on 23 January 2007. Address: Saulėtekio al. 11, LT-10223 Vilnius, Lithuania Tel.: [phone]; fax [phone]; e-mail [e-mail] The summary of the doctoral dissertation was   distributed on 22 December 2006. A copy of the doctoral dissertation is available for review at the Library of Vilnius Gediminas Technical University (Saulėtekio al. 14, Vilnius, Lithuania) and the Library of the Institute of Mathematics and Informatics (Akademijos 4, Vilnius, Lithuania).   © Sigita Pečiulytė, 2006

 

 

 

  VILNIAUS GEDIMINO TECHNIKOS UNIVERSITETAS MATEMATIKOS IR INFORMATIKOS INSTITUTAS                   Sigita PEČIULYTĖ         KRATINIŲUDAVINIŲSUĮVAIRIŲTIPŲ   NELOKALIOSIOMIS SĄLYGOMIS TYRIMAS IR SKAITINĖANALIZĖ             Daktaro disertacijos santrauka Fiziniai mokslai, matematika (01P)          

Vilnius 2006 

 

Disertacija   rengta   20022006   metais   Vytauto Didiojo universitete. Disertacija ginama eksternu.   Mokslinis konsultantas doc. dr. Artūras TIKONAS (Matematikos ir informatikos institutas, fiziniai mokslai, matematika  01P).   Disertacija ginama Vilniaus Gedimino technikos universiteto Matematikos mokslo krypties taryboje: Pirmininkas prof. habil. dr. RaimondasČIEGIS (Vilniaus Gedimino technikos universitetas, fiziniai mokslai, matematika  01P). Nariai: prof. habil. dr. Juozas AUGUTIS Didiojo universitetas, (Vytauto technologijos mokslai, energetika ir termoininerija  06T), prof. habil. dr. Feliksas IVANAUSKAS(Vilniaus universitetas, fiziniai mokslai, matematika  01P), doc. dr. Mečislovas MEILŪNAS (Vilniaus Gedimino technikos universitetas, fiziniai mokslai, matematika  01P), prof. habil. dr. Mifodijus SAPAGOVAS (Matematikos ir informatikos institutas, fiziniai mokslai, matematika  01P). Oponentai: doc. dr. Vytautas KLEIZA (Kauno technologijos universitetas, fiziniai mokslai, matematika  01P), doc. dr. Aleksandras KRYLOVAS(Vilniaus Gedimino technikos universitetas, fiziniai mokslai, matematika  01P).   Disertacija bus ginama vieame Matematikos mokslo krypties tarybos posėdyje 2007 m. sausio 23 d. 14 val. Vilniaus Gedimino technikos universiteto senato posėdiųsalėje. Adresas: Saulėtekio al. 11, LT-10223 Vilnius-40, Lietuva. Tel.: [phone]; faksas [phone]; el. patas [e-mail]   Disertacijos santrauka isiuntinėta 2006 m. gruodio 22 d. Disertaciją galima periūrėti Vilniaus Gedimino technikos universiteto bibliotekoje (Saulėtekio al. 14, Vilnius, Lithuania) ir Matematikos ir informatikos instituto bibliotekoje (Akademijos 4, Vilnius, Lithuania). VGTU leidyklos Technika 1339 mokslo literatūros knyga. © Sigita Pečiulytė, 2006

 

 

GENERAL DESCRIPTION  Topicality of the problem. problems with nonlocal conditions Boundary are an area of the fast developing differential equations theory. Nonlocal conditions come up when the value of a function or derivative on the boundary is connected with the values inside the domain. Such problems occur when we cannot measure values directly on the domain boundary of the investigated problem. Theoretical investigation of problems with various type of nonlocal boundary conditions is actual problem and recently much attention has been paid to them in scientific literature. These problems are investigated in the papers of both foreign and Lithuanian scientists. Problems with nonlocal conditions of Samarskii-Bitsadze type are investigated widely in scientific literature. Generalizations on these problems are presented in many papers (authors: A. Samarskii, A. Bitsadze, D. Gordeziani, V. Ilyin, M. Sapagovas, R.Čiegis, G. Infante, R. Ma). Problems with nonlocal boundary conditions of integral type for parabolic problems have been investigated by N. Borovykh, A. Bouziani, G. Fairweather, N. Ionkin, L. Kamynin, B. Liu, V. Makarov, S. Sangwon, M. Sapagovas, Y. Yin, N. Yurchuk, for elliptic problems A. Gushin, A. Scubachevski, Y. Wang and for hyperbolic problems S. Beilin, A. Bouziani, D. Gordeziani. Investigation of the spectrum of differential equations with nonlocal conditions is quite a new area related with problems of this type. Eigenvalue problems for differential operators with nonlocal conditions are considerably less investigated than classical boundary conditions cases. In chapters one and two of this dissertation we present the analysis of the spectrum of Sturm- Liouville problem with one nonlocal boundary condition of integral or Samarskii-Bitsadze type. We investigate how the spectrum of this problem depends on the parameters of some nonlocal boundary condition. In chapter three of the dissertation eigenfunctions of the Sturm-Liouville problem, which is a separate case of the problem analysed by G. Infante, with nonlocal two- point and integral boundary conditions are investigated. Differential and discrete (finite difference) problems are analysed in R.Čiegis, A. tikonien O. tikonas,ė O. Subo andč How the existence papers. of the solution of these problems depends on nonlocal conditions parameters is the main focus of their attention. The researchers found the domains of existence of only one nonnegative solution, they got necessary and sufficient conditions of solution correctness of finite difference scheme in maximum norm for stationary problem, and in the case of parabolic problem sufficient conditions of correctness were received. In chapter four of the dissertation we

 

5

generalize the results when in boundary problem are various boundary conditions published in these papers. The main aims and tasks of the work are    1.   To investigate real eigenvalues of the Sturm-Liouville problem with one classical boundary condition and other integral or two-point nonlocal boundary condition, to analyse the dependence of spectrums of these problems on nonlocal boundary conditions parameters. 2.   To find eigenfunctions positiveness intervals of the Sturm-Liouville problem with one classical boundary condition and other integral or two-point nonlocal boundary condition. 3.   To investigate the stability and convergence in energy norm of finite difference schemes that approximate stationary problem with various nonlocal boundary conditions. 4.   To apply Greens function which allow to express the values of the solution inside the domain with values on the boundary for parabolic problem with one classical and other nonlocal boundary condition. Scientific novelty. In the dissertation the dependence of the spectrum of Sturm-Liouville problem with various nonlocal integral and two-point boundary conditions on nonlocal boundary conditions parameters is investigated. Meanwhile only separate cases of spectrums of stationary problems have been investigated in scientific literature till now. The methodology for investigation of spectrums of Sturm-Liouville problems with nonlocal conditions is developed in the dissertation. Some special cases for this problem were found when two real negative eigenvalues exist, or when two eigenvalues become complex and then become real again. This work presents the investigation of eigenfunctions of Sturm-Liouville problem with various nonlocal integral and two-point boundary conditions widely. Eigenfunctions positiveness intervals subject to nonlocal conditions parameters are found. The results presented in the dissertation correct the described results presented in the scientific literature till now and allow to get the necessary and sufficient conditions for existence of these intervals. In scientific literature parabolic and stationary problems with nonlocal conditions have been investigated quite widely. This dissertation summarizes results of stationary problem when in boundary problem are various boundary conditions presented by R.Čiegis, A. tikonas, O. tikonienėand O. Suboč. In this work we investigated the problems with second or third type boundary conditions. Thus we can not apply the maximum principle directly and we get estimates in energy norm. In the dissertation the solution method of a parabolic problem with nonlocal conditions when nonlocal condition is reduced to second order Voltera

 

6

integral equation is proposed. Then instead of nonlocal condition we get the values on the domain boundary points and we can solve classical problem. Significance of the results. of qualitative analysis of Sturm- Results Liouville problem with various nonlocal conditions spectrum might be applied solving multidimensional and parabolic problems, or investigating the stability of those methods, or in iterative methods. We can use the obtained results on positive eigenfunctions existence intervals as a test for separate cases of a Hammerstein integral equation or other problems. Necessary and sufficient existence conditions for finite difference schemes for stationary problem with various nonlocal boundary conditions and stability estimates allow to use these finite difference schemes for solving such problems.can be applied investigating multidimensional problems or approximating these problems by a finite difference method for increasing the order of convergence. Problem with nonlocal boundary condition is reduced to integral equation or the solution of the problem with nonlocal boundary conditions as a linear combination of fundamental solutions is searched. For these problems we can use theoretical results and methods (finite difference, finite elements, splines an so on) for classical problems. The scope of the scientific work.The doctoral dissertation consists of the introduction, five chapters, general results and conclusions and the list of references. The total scope of the doctoral dissertation is 125 pages and 44 figures. The language of the doctoral dissertation is Lithuanian.

CONTENT  In the introduction the topicality of the problem is defined, the goals of the research are formulated, the scientific novelty of the dissertation and the significance of the results are presented. Some definitions, notations and concepts are presented as well.   Chapter 1. Sturm-Liouville problem for stationary differential operator with one nonlocal boundary condition of integral type      In the first chapter of the dissertation we investigate the Sturm-Liouville problem with one classical condition on the left boundary (in pointt=0 ) −u ′ =u,t∈(0,1), (1)   u(0)=0,  (2) and an other nonlocal condition of integral type:

 

7

u(1)= γ ∫ 0 ξ u(t)t,  (3  1) (Case 1 ) ∫ 1 t t (Case 2)  (3 2 ) u(1)γ ξ u( ), with parametersγ∈C:=C∪ ∞ and∈[0,1] . Two cases of nonlocal boundary conditions are investigated. In the general case eigenvalues∈C    and eigenfunctionsu(t) are complex functions. However much attention is paid to investigation of spectrum of this problem when∈,γR. We investigate how the spectrum depends on the boundary condition in the case parameters and . In the classical case when=0 or=0 in problem (1)(3 1 ), and=0 or= problem (1)(2), (31 in 2 ), then eigenvalues and eigenfunctions do not depend on the parameter : λ k =(πk) 2 u k (t)=sin(πkt)k∈N.  (4) , , Lemma 1.The eigenvalue=0 exists if, and only ifγ = 2 2  in Case 1, γ = 2 in Case 2. 1− 2     In the general case, for≠ eigenfunctions are0 ,u=csin(qt) and eigenvalues =λq 2 , whereq∈C q \ {0} ,C q :={q∈C|Req>0 or Req=0 , Imq>0 , orq=0} . We call eigenvalues that do not depend on the parameterconstant eigenvalues. Letξr= nm ∈Q (0. For,1) we suppose thatm andn   (n>m>0) are we suppose positive coprime integer numbers. If 0m0 , n= if1 and=1 we supposem=1 ,n Let us denote subset1 . N m :={n∈N|n=km,k∈N} of integer positive numbers, N e ={k∈N 2 |k≤n}∪{0} are even numbers andN o :={n∈N\N 2 |k≤n} are odd numbers. Lemma 2.Constant eigenvalues exist only for rational numbersξ = nm ∈[0,1], and those eigenvalues are given by:λ k =(nπk) 2 ,k∈N, mN e  and λ k =(2nπk) 2 ,k∈N , mN o  in Case 1; λ k =(nπk) 2 ,k∈N , n−m∈N e  and λ k =(2nπk) 2 ,k∈N, n−m∈N o in Case 2.     

 

8

Real eigenvalues case for the problem with integral nonlocal boundary condition      Lemma 3.For> 0 one negative eigenvalue exists, and for 0 there are not negative eigenvalues (γ 0 = ξ 2 2 ,∈ξ(0,1](Case 1),γ 1 0 − 2 ξ 2 ξ∈,[0,1)(Case 2)).    Let us define 2 ≈0.639421 , 3 ≈0.393743 , 2 ≈ −2.4072 , 3 ≈3.95836 . Lemma 4.If 2 ≤ ≤ 3 , then all eigenvalues of the problem(1)–(3 1 )are real for all∈(0,1), and limitary cases are realizable when= 2 and= 3 . If 2 < ≤2eigenvalues are positive and simple for allthen all ∈(0,1).    Lemma 5.All eigenvalues of the problem(1)–(2),(3 2 ) are real.with real      Chapter 2. Sturm-Liouville problem for stationary differential operator with one two-point nonlocal boundary condition      In this chapter we investigate the Sturm-Liouville problem with one classical boundary condition on the left side of interval (0,1) −u ′ =u,t∈(0,1),  (5) u(0) 0,  (6) and another nonlocal two-point boundary condition of Samarskii-Bitsadze type   u(1)=u( ),  (7  1) (Case 1 )   u(1)=u ′ ( ),  (7  (Case 2) 2 )   u(1)u( ). 3) (Case  (7 3 ) = When=0 in the problem (5)(7), we get a problem with classical boundary conditions. Then eigenvalues and eigenfunctions are equal: λ k = π 2 (k− 2 1 ) 2 ,u k (t)=sin(π(k−) 2 1 t),k∈N,  (8 1,2 ) λ k (πk) 2 ,u k (t)=sin(πkt),k∈N.  (8 3 ) Lemma 6.The eigenvalue0exists if and only if:γ 1 in Case 1;1   = in Cases 2, 3.     Lemma 7.Constant eigenvalues do not exist for irrational , while for rationalξ =r= m n ∈[0,1]they exist in the following cases:

 

9