The fundamental matrix of the Hilbert problem in transport theory

A method of solving the homogeneous Hilbert problem encountered in the two-media case of the one-dimensional transport equation with degenerate scattering kernels is developed. The existence of the corresponding fundamental matrix is proved when the fundamental matrices for both the full-range probl...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Transport theory and statistical physics Jg. 3; H. 4; S. 175 - 197
1. Verfasser: Rajamäki, Markku
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Taylor & Francis Group 01.01.1973
ISSN:0041-1450, 1532-2424
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Abstract A method of solving the homogeneous Hilbert problem encountered in the two-media case of the one-dimensional transport equation with degenerate scattering kernels is developed. The existence of the corresponding fundamental matrix is proved when the fundamental matrices for both the full-range problems exist and the elements of these matrices and of their inverses belong to a class L p , p < 2, and the zeroes of the dispersion functions are of integer order. Matrices related to the fundamental matrices satisfy a Fredholm equation whose free term involves unknown polynomial elements. Means of constructing the fundamental matrix from certain solutions of this equation are investigated. Sufficient supplementary conditions are presented in the form of an algebraic equation. Since the kernel is simple in form and its norm approaches zero with the cross sections of both media approaching each other, the Fredholm equation is very suitable for computation.
AbstractList A method of solving the homogeneous Hilbert problem encountered in the two-media case of the one-dimensional transport equation with degenerate scattering kernels is developed. The existence of the corresponding fundamental matrix is proved when the fundamental matrices for both the full-range problems exist and the elements of these matrices and of their inverses belong to a class L p , p < 2, and the zeroes of the dispersion functions are of integer order. Matrices related to the fundamental matrices satisfy a Fredholm equation whose free term involves unknown polynomial elements. Means of constructing the fundamental matrix from certain solutions of this equation are investigated. Sufficient supplementary conditions are presented in the form of an algebraic equation. Since the kernel is simple in form and its norm approaches zero with the cross sections of both media approaching each other, the Fredholm equation is very suitable for computation.
Author Rajamäki, Markku
Author_xml – sequence: 1
  givenname: Markku
  surname: Rajamäki
  fullname: Rajamäki, Markku
  organization: Department of Technical Physics Helsinki , University of Technology
BookMark eNp1kFFLwzAUhYNMsJv-AN_yB6r3pmmTgi9jqBMGvsznkqUJVtqkJBHdvzdjvolPF84534VzlmThvDOE3CLcIUi4B-CIvBYVSMZFWzUXpMC6YiXjjC9IcfLLHIArsozxAwAlcl6Q9f7dUPvpejUZl9RIJ5XC8E29pSk722E8mJDoHPxhNBMdHE1BuTj7LOaAD8drcmnVGM3N712Rt6fH_WZb7l6fXzbrXamZEKm0DZO1QQlVbRgy0Kq1ve5ZK7mBRhuDrWZcWw1gWSNyid72UtdCaEB2wGpF8PxXBx9jMLabwzCpcOwQutMG3Z8NMvNwZgZnfZjUlw9j3yV1HH2wuYceYlf9j_8A28djLQ
Cites_doi 10.13182/NSE71-A19085
10.1016/B978-0-08-010067-8.50007-4
10.13182/NSE68-A18013
10.13182/NSE71-A21262
10.1086/151525
10.1063/1.1666105
ContentType Journal Article
Copyright Copyright Taylor & Francis Group, LLC 1973
Copyright_xml – notice: Copyright Taylor & Francis Group, LLC 1973
DBID AAYXX
CITATION
DOI 10.1080/00411457308247936
DatabaseName CrossRef
DatabaseTitle CrossRef
DatabaseTitleList
DeliveryMethod fulltext_linktorsrc
Discipline Physics
EISSN 1532-2424
EndPage 197
ExternalDocumentID 10_1080_00411457308247936
8247936
GroupedDBID 07S
123
1TA
4.4
5VS
AATVF
ABZIJ
ACGEJ
ACKDS
ADCVX
ADXPE
AENEX
AFFNX
AFGMD
AKOOK
ALMA_UNASSIGNED_HOLDINGS
AQTUD
ASQZU
BVUPT
C09
C5L
CS3
DU5
EBS
ECKKI
EJD
EXXQB
F5P
H13
JHRKR
MVM
NA5
NUSFT
NZ-
P2P
TAZ
TCJPB
TDBHL
TFL
TFMCV
TFT
TFW
UC4
UU9
V3L
VOH
WH7
.7F
.QJ
0BK
29Q
AAGDL
AALDU
AAMIU
AANMX
AAPUL
AAQRR
AAYXX
ABCES
ABFIM
ABGQB
ABHAV
ABLIJ
ABPAQ
ABPEM
ABXUL
ABXYU
ACLZH
ACTIO
ADGTB
ADPDO
AEISY
AFDUV
AFKVX
AFRVT
AGDLA
AGMYJ
AHDZW
AHWVO
AIJEM
AIPZZ
AIYEW
AJWEG
AKBVH
ALQZU
AQRUH
AVBZW
BLEHA
CAG
CCCUG
CE4
CITATION
COF
DGEBU
DGEYW
DKSSO
DWNMW
EBKLY
E~A
E~B
HF~
H~P
IPNFZ
J.P
KYCEM
M4Z
QSQFL
RIG
RNANH
ROSJB
RTWRZ
S-T
SNACF
TASJS
TBQAZ
TEX
TTHFI
TUROJ
TWF
UT5
UU3
ZGOLN
~S~
ID FETCH-LOGICAL-c277t-f6285e18035e2120ca9fdcd2984e06cee19c24cfc00f267247dfd8c577c012b13
IEDL.DBID TFW
ISICitedReferencesCount 0
ISICitedReferencesURI http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=10_1080_00411457308247936&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN 0041-1450
IngestDate Sat Nov 29 06:44:20 EST 2025
Mon Oct 20 23:46:25 EDT 2025
IsPeerReviewed false
IsScholarly false
Issue 4
Language English
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c277t-f6285e18035e2120ca9fdcd2984e06cee19c24cfc00f267247dfd8c577c012b13
PageCount 23
ParticipantIDs crossref_primary_10_1080_00411457308247936
informaworld_taylorfrancis_310_1080_00411457308247936
PublicationCentury 1900
PublicationDate 1/1/1973
1973-01-00
PublicationDateYYYYMMDD 1973-01-01
PublicationDate_xml – month: 01
  year: 1973
  text: 1/1/1973
  day: 01
PublicationDecade 1970
PublicationTitle Transport theory and statistical physics
PublicationYear 1973
Publisher Taylor & Francis Group
Publisher_xml – name: Taylor & Francis Group
References Pahor S. (CIT0006) 1968; 31
Muskhelishvili M. I. (CIT0010) 1953
Vekua N. P. (CIT0005) 1967
Burniston E. E. (CIT0011) 1971; 45
Gakhov F. D. (CIT0012) 1966
Rajamäki M. (CIT0014) 1973
Case K. M. (CIT0009) 1967
Shultis J. K. (CIT0007) 1968
Mandzavidze G. F. (CIT0004) 1958; 123
Silvennonien P. (CIT0001) 1971
Rajamaki M. (CIT0003) 1972
CIT0013
Jauho P. (CIT0002) 1971; 43
CIT0008
References_xml – volume-title: Singular Integral Equations
  year: 1953
  ident: CIT0010
– year: 1968
  ident: CIT0007
– volume: 45
  start-page: 331
  year: 1971
  ident: CIT0011
  publication-title: Nucl. Sci. Eng.
  doi: 10.13182/NSE71-A19085
– volume-title: Linear Transport Theory
  year: 1967
  ident: CIT0009
– volume-title: Systems of Singular Integral Equations
  year: 1967
  ident: CIT0005
– volume-title: Boundary Value Problems
  year: 1966
  ident: CIT0012
  doi: 10.1016/B978-0-08-010067-8.50007-4
– volume: 31
  start-page: 110
  year: 1968
  ident: CIT0006
  publication-title: Nucl. Sci. Eng.
  doi: 10.13182/NSE68-A18013
– start-page: 93
  year: 1973
  ident: CIT0014
  publication-title: Acta Polytech. Scand.
– volume: 43
  start-page: 145
  year: 1971
  ident: CIT0002
  publication-title: Nucl. Sci. Eng.
  doi: 10.13182/NSE71-A21262
– ident: CIT0008
  doi: 10.1086/151525
– ident: CIT0013
  doi: 10.1063/1.1666105
– start-page: 91
  year: 1972
  ident: CIT0003
  publication-title: Acta Polytechnic Scand.
– year: 1971
  ident: CIT0001
– volume: 123
  start-page: 791
  year: 1958
  ident: CIT0004
  publication-title: Dokl, Akad. Nauk.
SSID ssj0018144
Score 1.165014
Snippet A method of solving the homogeneous Hilbert problem encountered in the two-media case of the one-dimensional transport equation with degenerate scattering...
SourceID crossref
informaworld
SourceType Index Database
Publisher
StartPage 175
Title The fundamental matrix of the Hilbert problem in transport theory
URI https://www.tandfonline.com/doi/abs/10.1080/00411457308247936
Volume 3
WOSCitedRecordID wos10_1080_00411457308247936&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
journalDatabaseRights – providerCode: PRVAWR
  databaseName: Taylor & Francis Journals Complete
  customDbUrl:
  eissn: 1532-2424
  dateEnd: 20130919
  omitProxy: false
  ssIdentifier: ssj0018144
  issn: 0041-1450
  databaseCode: TFW
  dateStart: 19710101
  isFulltext: true
  titleUrlDefault: https://www.tandfonline.com
  providerName: Taylor & Francis
link http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LS8NAEB6kKHjxLdYXe_AkBLJJ9pFjEUMPUjxU7C0k-4AcTKWJov_e2U1aWtGL3rNhM5nk-2Z25huAm1QhqhQYqXLko26EWRxIbcNAsQLZB2cYj_mpJQ9iMpGzWfrY1-Y0fVmli6FtJxTh_9Xu4y7KZlkR5_R4kMUz4ZRWXGLICW4jq3f-Pc2eV2cIkiadBnNCA7x8dab50x02UGlDs3QNbbL9f-7zAPZ6mklGnV8cwpapj2DHl3uq5hhG6B3EuiaQTtufvDil_g8ytwQJIRlXTvmqJf20GVLVpF2KoBPf-vh5Ak_Z_fRuHPTDFAIVCdEG1vVKGirDmBmEq1AVqdVKR6lMTMgRKim-tkRZFYY24gI3rK2WigmhEMNKGp_CoJ7X5gwIPgvGsFazArEtUomMkQOWnOskMkaUfAi3S2Pmr51mRk5XUqTfbDIEtm7uvPWJCttNFcnjX9ed_3HdBexS55E-i3IJg3bxZq5gW723VbO49h70Bc6mwNQ
linkProvider Taylor & Francis
linkToHtml http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LS8NAEB60KnrxLdbnHjwJgWySfeRYxFKxFg8VewvpPqAHU2mj6L93dpOWVvSi92zYTCb5vpmd-QbgKlWIKjlGqhz5qBthFgdS2zBQLEf2wRnGY35qSVf0enIwSB_rhNu0Lqt0MbSthCL8v9p93C4ZPSuJc4I8SOOZcFIrLjPEV2GNIc66kr5--3l-iiBpUqkwJzTA6-enmj_dYgmXllRLF_CmvfPfne7Cds00SatyjT1YMcU-bPiKTzU9gBY6CLGuD6SS9ycvTqz_g4wtQU5IOiMnflWSeuAMGRWknOmgE9_9-HkIT-3b_k0nqOcpBCoSogysa5c0VIYxM4hYocpTq5WOUpmYkCNaUnxzibIqDG3EBW5YWy0VE0IhjA1pfASNYlyYYyD4LBjGWs1yhLdIJTJGGjjkXCeRMWLIm3A9s2b2WslmZHSuRvrNJk1gi_bOSp-rsNVgkSz-dd3JH9ddwman_9DNune9-1PYos5BfVLlDBrl5M2cw7p6L0fTyYV3py8O38T1
linkToPdf http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LS8NAEB60PvDiW6zPPXgSAtkku5scixoqltJDxd5Cug_owbS0UfTfO7tJSyt60Xs2bCaTfN_MznwDcJNIRJUcI1WOfNSOMAu9WBnfkyxH9sEZxmNuaklHdLvxYJD06tqcWV1WaWNoUwlFuH-1_bgnyswr4qweD7J4JqzSik0M8XXYcMJY6M799GVxiBDTqBJhjqiH1y8ONX-6xQosrYiWLsFNuvfPje7Dbs0zSatyjANY08UhbLl6Tzk7gha6BzG2C6QS9yevVqr_g4wNQUZI2iMrfVWSetwMGRWknKugE9f7-HkMz-lD_67t1dMUPBkIUXrGNktqGvsh04hXvswTo6QKkjjSPkespPjeImmk75uAC9ywMiqWTAiJIDak4Qk0inGhT4Hgs2AQaxTLEdwCGcUhksAh5yoKtBZD3oTbuTGzSSWakdGFFuk3mzSBLZs7K12mwlRjRbLw13Vnf1x3Ddu9-zTrPHafzmGHWu90GZULaJTTN30Jm_K9HM2mV86ZvgDGNsOZ
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+fundamental+matrix+of+the+Hilbert+problem+in+transport+theory&rft.jtitle=Transport+theory+and+statistical+physics&rft.au=Rajam%C3%A4ki%2C+Markku&rft.date=1973-01-01&rft.issn=0041-1450&rft.eissn=1532-2424&rft.volume=3&rft.issue=4&rft.spage=175&rft.epage=197&rft_id=info:doi/10.1080%2F00411457308247936&rft.externalDBID=n%2Fa&rft.externalDocID=10_1080_00411457308247936
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0041-1450&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0041-1450&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0041-1450&client=summon