On the numerical solution of direct and inverse problems for the heat equation in a semi-infinite region

We consider the initial boundary value problem for the heat equation in a region with infinite and finite boundaries (direct problem) and the related problem to reconstruct the finite boundary from Cauchy data on the infinite boundary (inverse problem). The numerical solution of the direct problem i...

Full description

Saved in:

Bibliographic Details
Published in:	Journal of computational and applied mathematics Vol. 108; no. 1; pp. 41 - 55
Main Author:	Chapko, Roman
Format:	Journal Article
Language:	English
Published:	Amsterdam Elsevier B.V 15.08.1999 Elsevier
Subjects:	Collocation method Exact sciences and technology Green's function Heat equation Initial boundary value problem Integral equation Inverse boundary problem Mathematical analysis Mathematics Newton method Numerical analysis Numerical analysis. Scientific computation Ordinary differential equations Partial differential equations Partial differential equations, initial value problems and time-dependant initial-boundary value problems Regularization Sciences and techniques of general use Semi-infinite region Trigonometric interpolation Trigonometric interpolation Initial boundary value problem Heat equation Semi-infinite region Inverse boundary problem Green's function Integral equation Collocation method Newton method Regularization Sobolev space Initial value problem Green function Uniqueness theorem Partial differential equation Convergence Inverse problem Direct problem Interpolation Boundary value problem
ISSN:	0377-0427, 1879-1778
Online Access:	Get full text
Tags:	Add Tag No Tags, Be the first to tag this record!

Abstract	We consider the initial boundary value problem for the heat equation in a region with infinite and finite boundaries (direct problem) and the related problem to reconstruct the finite boundary from Cauchy data on the infinite boundary (inverse problem). The numerical solution of the direct problem is realized by a boundary integral equation method. For an approximate solution of the inverse problem we use a regularized Newton method based on numerical approach for the direct problem. Numerical examples illustrating our results are presented.
AbstractList	We consider the initial boundary value problem for the heat equation in a region with infinite and finite boundaries (direct problem) and the related problem to reconstruct the finite boundary from Cauchy data on the infinite boundary (inverse problem). The numerical solution of the direct problem is realized by a boundary integral equation method. For an approximate solution of the inverse problem we use a regularized Newton method based on numerical approach for the direct problem. Numerical examples illustrating our results are presented.
Author	Chapko, Roman
Author_xml	– sequence: 1 givenname: Roman surname: Chapko fullname: Chapko, Roman email: chapko@franko.lviv.ua organization: Department of Applied Mathematics and Computer Science, Lviv University, 290602 Lviv, Ukraine
BackLink	http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=1907048$$DView record in Pascal Francis
BookMark	eNqFkE9LwzAYh4NMcE4_gpCDBz1Uk7ZrEjyIDP_BYAf1HJL0rYu0yUyygd_erhMFLzvlkOd54fcco5HzDhA6o-SKElpdv5CCsYyUObsQ4pIQIkRGDtCYciYyyhgfofEvcoSOY_zooUrQcoyWC4fTErBbdxCsUS2Ovl0n6x32Da5tAJOwcjW2bgMhAl4Fr1voIm58GMwlqIThc60GyTqscITOZtY11tkEOMB7_3OCDhvVRjj9eSfo7eH-dfaUzRePz7O7eWaKokpZpUrgvKi5YibnQmnD6lxToYUCQflUF9xArqum1JxMdT9I5CrPjQGgqtK6mKDz3d2Viv2cJihnbJSrYDsVviQVhJGS99h0h5ngYwzQ_BFEbqvKoarcJpNCyKGqJL13888zNg3TU1C23Wvf7mzoC2wsBBmNBWdgF1rW3u658A170JUQ
CODEN	JCAMDI
CitedBy_id	crossref_primary_10_1007_s10891_005_0050_5 crossref_primary_10_1016_S1365_1609_03_00013_3 crossref_primary_10_1016_S0377_0427_01_00595_7 crossref_primary_10_1016_j_amc_2003_10_035 crossref_primary_10_1007_s10958_012_0978_x crossref_primary_10_1007_s12190_019_01297_4 crossref_primary_10_1016_j_cam_2003_12_019
Cites_doi	10.1007/978-3-662-03537-5 10.1137/S0036139994277828 10.1007/978-3-642-61527-6 10.1007/BF01385870 10.1216/jiea/1181075987 10.1007/BF01210400 10.1090/qam/998101 10.1142/9789812798886_0010 10.1088/0266-5611/14/4/006 10.1007/978-3-642-97146-4 10.1007/BF01385694 10.1007/978-3-642-48860-3 10.1002/mma.1670160203 10.1088/0266-5611/6/6/003 10.1007/978-1-4612-2706-9
ContentType	Journal Article
Copyright	1999 Elsevier Science B.V. 1999 INIST-CNRS
Copyright_xml	– notice: 1999 Elsevier Science B.V. – notice: 1999 INIST-CNRS
DBID	6I. AAFTH AAYXX CITATION IQODW
DOI	10.1016/S0377-0427(99)00099-0
DatabaseName	ScienceDirect Open Access Titles Elsevier:ScienceDirect:Open Access CrossRef Pascal-Francis
DatabaseTitle	CrossRef
DatabaseTitleList
DeliveryMethod	fulltext_linktorsrc
Discipline	Mathematics
EISSN	1879-1778
EndPage	55
ExternalDocumentID	1907048 10_1016_S0377_0427_99_00099_0 S0377042799000990
GroupedDBID	--K --M -~X .~1 0R~ 1B1 1RT 1~. 1~5 29K 4.4 457 4G. 5GY 5VS 6I. 7-5 71M 8P~ 9JN AABNK AACTN AAEDT AAEDW AAFTH AAFWJ AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AAXUO ABAOU ABEFU ABFNM ABJNI ABMAC ABTAH ABVKL ABXDB ABYKQ ACAZW ACDAQ ACGFS ACRLP ADBBV ADEZE ADMUD AEBSH AEKER AENEX AEXQZ AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AIEXJ AIGVJ AIKHN AITUG AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ ARUGR ASPBG AVWKF AXJTR AZFZN BKOJK BLXMC CS3 D-I DU5 EBS EFJIC EFLBG EJD EO8 EO9 EP2 EP3 F5P FDB FEDTE FGOYB FIRID FNPLU FYGXN G-2 G-Q G8K GBLVA HVGLF HZ~ IHE IXB J1W KOM LG9 M26 M41 MHUIS MO0 N9A NCXOZ NHB O-L O9- OAUVE OK1 OZT P-8 P-9 P2P PC. Q38 R2- RIG RNS ROL RPZ SDF SDG SDP SES SEW SPC SPCBC SSW SSZ T5K TN5 UPT WUQ XPP YQT ZMT ZY4 ~02 ~G- 9DU AATTM AAXKI AAYWO AAYXX ABDPE ABWVN ACLOT ACRPL ACVFH ADCNI ADNMO ADVLN AEIPS AEUPX AFJKZ AFPUW AGQPQ AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP CITATION EFKBS ~HD AFXIZ AGCQF AGRNS BNPGV IQODW SSH
ID	FETCH-LOGICAL-c336t-6a4e883d8a7c289abc7d2b19b9ae9185b38ce2b6f4b805b42792a22ccee1a6bb3
ISICitedReferencesCount	8
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000081945900004&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN	0377-0427
IngestDate	Mon Jul 21 09:11:18 EDT 2025 Sat Nov 29 06:06:48 EST 2025 Tue Nov 18 21:08:29 EST 2025 Fri Feb 23 02:33:25 EST 2024
IsDoiOpenAccess	true
IsOpenAccess	true
IsPeerReviewed	true
IsScholarly	true
Issue	1
Keywords	Trigonometric interpolation Initial boundary value problem Heat equation Semi-infinite region Inverse boundary problem Green's function Integral equation Collocation method Newton method Regularization Sobolev space Initial value problem Green function Uniqueness theorem Partial differential equation Convergence Inverse problem Direct problem Interpolation Boundary value problem
Language	English
License	http://www.elsevier.com/open-access/userlicense/1.0 https://www.elsevier.com/tdm/userlicense/1.0 https://www.elsevier.com/open-access/userlicense/1.0 CC BY 4.0
LinkModel	OpenURL
MergedId	FETCHMERGED-LOGICAL-c336t-6a4e883d8a7c289abc7d2b19b9ae9185b38ce2b6f4b805b42792a22ccee1a6bb3
OpenAccessLink	https://dx.doi.org/10.1016/S0377-0427(99)00099-0
PageCount	15
ParticipantIDs	pascalfrancis_primary_1907048 crossref_primary_10_1016_S0377_0427_99_00099_0 crossref_citationtrail_10_1016_S0377_0427_99_00099_0 elsevier_sciencedirect_doi_10_1016_S0377_0427_99_00099_0
PublicationCentury	1900
PublicationDate	1999-08-15
PublicationDateYYYYMMDD	1999-08-15
PublicationDate_xml	– month: 08 year: 1999 text: 1999-08-15 day: 15
PublicationDecade	1990
PublicationPlace	Amsterdam
PublicationPlace_xml	– name: Amsterdam
PublicationTitle	Journal of computational and applied mathematics
PublicationYear	1999
Publisher	Elsevier B.V Elsevier
Publisher_xml	– name: Elsevier B.V – name: Elsevier
References	Hsiao, Saranen (BIB15) 1993; 16 A.J. Nowak, BEM approach to inverse thermal problems, in: D.B. Ingham, L.C. Wrobel (Eds.), Boundary Integral Formulations for Inverse Analysis, Computational Mechanics Publications, Southampton, 1997, pp. 259–298. Lubich, Schneider (BIB20) 1992; 63 Bryan, Caudill (BIB5) 1996; 56 N.N. Lebedev, Special Functions and Their Applications, Prentice-Hall, Englewood Cliffs, 1965. C.A. Brebbia, J.C.F. Telles, L.C. Wrobel, Boundary Element Techniques, Springer, New York, 1984. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd Edition, Springer, Berlin, 1998. R. Kress, Linear Integral Equations, Springer, Berlin, 1989. R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 1, Physical Origins and Classical Methods, Springer, Berlin, 1990. Costabel (BIB10) 1990; 13 Kress, Sloan (BIB17) 1993; 66 A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964. Banks, Kojima, Winfree (BIB2) 1990; 6 R.B. Guenther, J.W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Prentice-Hall, Englewood Cliffs, 1988. Chapko, Kress, Yoon (BIB8) 1998; 14 R. Chapko, R. Kress, On a quadrature method for a logarithmic integral equation of the first kind, in: R.P. Agarwal (Ed.), Contributions in Numerical Mathematics, World Scientific, Singapore, 1993, pp. 127–140. Banks, Kojima (BIB1) 1989; 47 Z.S. Berezhanskaya, I.V. Lyudkevich, Numerical solution of a spatial boundary value problem for the heat equation by integral equations, Vychisl. Prikl. Mat. 41 (1980) 3–7 (in Russian). V. Galazyuk, R. Chapko, The Chebyshev-Laguerre transformation and integral equations methods for initial boundary value problems for the telegraph equation, Dokl. Akad. Nauk UkrSSR 8 (1990) 11–14 (in Russian). Chapko, Kress (BIB7) 1997; 9 F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, Berlin, 1993. O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS Publications, Providence, 1968. 10.1016/S0377-0427(99)00099-0_BIB6 Hsiao (10.1016/S0377-0427(99)00099-0_BIB15) 1993; 16 10.1016/S0377-0427(99)00099-0_BIB9 Chapko (10.1016/S0377-0427(99)00099-0_BIB8) 1998; 14 Costabel (10.1016/S0377-0427(99)00099-0_BIB10) 1990; 13 10.1016/S0377-0427(99)00099-0_BIB3 10.1016/S0377-0427(99)00099-0_BIB4 Kress (10.1016/S0377-0427(99)00099-0_BIB17) 1993; 66 Lubich (10.1016/S0377-0427(99)00099-0_BIB20) 1992; 63 Chapko (10.1016/S0377-0427(99)00099-0_BIB7) 1997; 9 10.1016/S0377-0427(99)00099-0_BIB19 10.1016/S0377-0427(99)00099-0_BIB18 10.1016/S0377-0427(99)00099-0_BIB16 Banks (10.1016/S0377-0427(99)00099-0_BIB1) 1989; 47 10.1016/S0377-0427(99)00099-0_BIB14 Banks (10.1016/S0377-0427(99)00099-0_BIB2) 1990; 6 10.1016/S0377-0427(99)00099-0_BIB13 10.1016/S0377-0427(99)00099-0_BIB12 10.1016/S0377-0427(99)00099-0_BIB11 10.1016/S0377-0427(99)00099-0_BIB22 Bryan (10.1016/S0377-0427(99)00099-0_BIB5) 1996; 56 10.1016/S0377-0427(99)00099-0_BIB21
References_xml	– reference: R.B. Guenther, J.W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Prentice-Hall, Englewood Cliffs, 1988. – reference: V. Galazyuk, R. Chapko, The Chebyshev-Laguerre transformation and integral equations methods for initial boundary value problems for the telegraph equation, Dokl. Akad. Nauk UkrSSR 8 (1990) 11–14 (in Russian). – reference: N.N. Lebedev, Special Functions and Their Applications, Prentice-Hall, Englewood Cliffs, 1965. – reference: R. Chapko, R. Kress, On a quadrature method for a logarithmic integral equation of the first kind, in: R.P. Agarwal (Ed.), Contributions in Numerical Mathematics, World Scientific, Singapore, 1993, pp. 127–140. – volume: 6 start-page: 897 year: 1990 end-page: 921 ident: BIB2 article-title: Boundary estimation problems arising in thermal tomography publication-title: Inverse Problems – volume: 63 start-page: 455 year: 1992 end-page: 481 ident: BIB20 article-title: Time discretizations of parabolic boundary integral equations publication-title: Numer. Math. – volume: 16 start-page: 87 year: 1993 end-page: 114 ident: BIB15 article-title: Boundary integral solution of the two-dimensional heat equation publication-title: Math. Meth. Appl. Sci. – reference: O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS Publications, Providence, 1968. – volume: 56 start-page: 715 year: 1996 end-page: 735 ident: BIB5 article-title: An inverse problem in thermal imaging publication-title: SIAM J. Appl. Math. – reference: F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, Berlin, 1993. – reference: C.A. Brebbia, J.C.F. Telles, L.C. Wrobel, Boundary Element Techniques, Springer, New York, 1984. – reference: D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd Edition, Springer, Berlin, 1998. – volume: 13 start-page: 498 year: 1990 end-page: 552 ident: BIB10 article-title: Boundary integral operators for the heat equation publication-title: Integral Equations Operator Theory – reference: R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 1, Physical Origins and Classical Methods, Springer, Berlin, 1990. – reference: Z.S. Berezhanskaya, I.V. Lyudkevich, Numerical solution of a spatial boundary value problem for the heat equation by integral equations, Vychisl. Prikl. Mat. 41 (1980) 3–7 (in Russian). – reference: R. Kress, Linear Integral Equations, Springer, Berlin, 1989. – volume: 14 start-page: 853 year: 1998 end-page: 867 ident: BIB8 article-title: On the numerical solution of an inverse boundary value problem for the heat equation publication-title: Inverse Problems – volume: 66 start-page: 199 year: 1993 end-page: 214 ident: BIB17 article-title: On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation publication-title: Numer. Math. – volume: 9 start-page: 47 year: 1997 end-page: 69 ident: BIB7 article-title: Rothe's method for heat equation and boundary integral equations publication-title: J. Integral Equations Appl. – reference: A.J. Nowak, BEM approach to inverse thermal problems, in: D.B. Ingham, L.C. Wrobel (Eds.), Boundary Integral Formulations for Inverse Analysis, Computational Mechanics Publications, Southampton, 1997, pp. 259–298. – volume: 47 start-page: 273 year: 1989 end-page: 293 ident: BIB1 article-title: Boundary shape identification problems in two-dimensional domains related to thermal testing of materials publication-title: Quart. J. Appl. Math. – reference: A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964. – ident: 10.1016/S0377-0427(99)00099-0_BIB9 doi: 10.1007/978-3-662-03537-5 – volume: 56 start-page: 715 year: 1996 ident: 10.1016/S0377-0427(99)00099-0_BIB5 article-title: An inverse problem in thermal imaging publication-title: SIAM J. Appl. Math. doi: 10.1137/S0036139994277828 – ident: 10.1016/S0377-0427(99)00099-0_BIB21 – ident: 10.1016/S0377-0427(99)00099-0_BIB11 doi: 10.1007/978-3-642-61527-6 – volume: 63 start-page: 455 year: 1992 ident: 10.1016/S0377-0427(99)00099-0_BIB20 article-title: Time discretizations of parabolic boundary integral equations publication-title: Numer. Math. doi: 10.1007/BF01385870 – ident: 10.1016/S0377-0427(99)00099-0_BIB18 – volume: 9 start-page: 47 year: 1997 ident: 10.1016/S0377-0427(99)00099-0_BIB7 article-title: Rothe's method for heat equation and boundary integral equations publication-title: J. Integral Equations Appl. doi: 10.1216/jiea/1181075987 – volume: 13 start-page: 498 year: 1990 ident: 10.1016/S0377-0427(99)00099-0_BIB10 article-title: Boundary integral operators for the heat equation publication-title: Integral Equations Operator Theory doi: 10.1007/BF01210400 – volume: 47 start-page: 273 year: 1989 ident: 10.1016/S0377-0427(99)00099-0_BIB1 article-title: Boundary shape identification problems in two-dimensional domains related to thermal testing of materials publication-title: Quart. J. Appl. Math. doi: 10.1090/qam/998101 – ident: 10.1016/S0377-0427(99)00099-0_BIB6 doi: 10.1142/9789812798886_0010 – ident: 10.1016/S0377-0427(99)00099-0_BIB3 – volume: 14 start-page: 853 year: 1998 ident: 10.1016/S0377-0427(99)00099-0_BIB8 article-title: On the numerical solution of an inverse boundary value problem for the heat equation publication-title: Inverse Problems doi: 10.1088/0266-5611/14/4/006 – ident: 10.1016/S0377-0427(99)00099-0_BIB16 doi: 10.1007/978-3-642-97146-4 – volume: 66 start-page: 199 year: 1993 ident: 10.1016/S0377-0427(99)00099-0_BIB17 article-title: On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation publication-title: Numer. Math. doi: 10.1007/BF01385694 – ident: 10.1016/S0377-0427(99)00099-0_BIB19 – ident: 10.1016/S0377-0427(99)00099-0_BIB12 – ident: 10.1016/S0377-0427(99)00099-0_BIB13 – ident: 10.1016/S0377-0427(99)00099-0_BIB14 – ident: 10.1016/S0377-0427(99)00099-0_BIB4 doi: 10.1007/978-3-642-48860-3 – volume: 16 start-page: 87 year: 1993 ident: 10.1016/S0377-0427(99)00099-0_BIB15 article-title: Boundary integral solution of the two-dimensional heat equation publication-title: Math. Meth. Appl. Sci. doi: 10.1002/mma.1670160203 – volume: 6 start-page: 897 year: 1990 ident: 10.1016/S0377-0427(99)00099-0_BIB2 article-title: Boundary estimation problems arising in thermal tomography publication-title: Inverse Problems doi: 10.1088/0266-5611/6/6/003 – ident: 10.1016/S0377-0427(99)00099-0_BIB22 doi: 10.1007/978-1-4612-2706-9
SSID	ssj0006914
Score	1.6099039
Snippet	We consider the initial boundary value problem for the heat equation in a region with infinite and finite boundaries (direct problem) and the related problem...
SourceID	pascalfrancis crossref elsevier
SourceType	Index Database Enrichment Source Publisher
StartPage	41
SubjectTerms	Collocation method Exact sciences and technology Green's function Heat equation Initial boundary value problem Integral equation Inverse boundary problem Mathematical analysis Mathematics Newton method Numerical analysis Numerical analysis. Scientific computation Ordinary differential equations Partial differential equations Partial differential equations, initial value problems and time-dependant initial-boundary value problems Regularization Sciences and techniques of general use Semi-infinite region Trigonometric interpolation
Title	On the numerical solution of direct and inverse problems for the heat equation in a semi-infinite region
URI	https://dx.doi.org/10.1016/S0377-0427(99)00099-0
Volume	108
WOSCitedRecordID	wos000081945900004&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: PRVESC databaseName: ScienceDirect Freedom Collection - Elsevier customDbUrl: eissn: 1879-1778 dateEnd: 20180131 omitProxy: false ssIdentifier: ssj0006914 issn: 0377-0427 databaseCode: AIEXJ dateStart: 19950220 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1baxQxFA7r1geLiFesWsmDBWWJnXuSxyIVFW0FK-zbkGQzuLg7u-1uSx_88Z7MSTKWRaqCL8PukMuS79ucbzLnQsiLSV4mlTScGTCerKgKy4RpUqaALZmwShZNh_RHfnQkxmP5eTD4EWJhLma8bcXlpVz-V6jhHoDtQmf_Au44KNyAzwA6XAF2uP4R8MfouNie47uY2ShM53QhWjCfcsl5ZLg4qa6kzCo6HLrteWRPMQe4Ow9Ro5WdTxn8uKlTqCNXzMGjuSlrTVcmIhwxdqlgvdCdxwyxUce76IfvC_TwnnueTnxMnnRnqhiEiediG7ExGI_FOXOVPNDS4PYquGQpx6I9cf9NxAbRcDfFlFjeLmM2340dHw8fvsTZ9twT5F4mO-3Lkt7MRefDrq1rKrFRcoNsZbyUYki2Dt4fjj9ES15JzA0fxu4jwPb7CV9K-cpP9jttc3upVoB3g6VSftEvJ3fJHY8QPUDC3CMD294n2596TB6Qb8ctha80UocG6tBFQ5E6FBClnjo0UIcCdbqejjo0UAeaUUWvUIcidR6Sr28PT968Y74SBzN5Xq1ZpQorRD4RihtYX6UNn2Q6ldqldgfFp3NhbKarptAiKbVb20xlmQEFlqpK6_wRGbaL1j4mtFAWnpkbVSZZU5QgGFVTGSM02I3UTpJqhxRhDWvj09S7aimzuvdHzF1-epiklrLulr5Odsjr2G2JeVqu6yACQLUXm7iONTDruq67VwDtJ5RgQAvx5N-HfkpuhT9Ykj4jw_XZud0lN83Fero6e-75-RMbQKxt
linkProvider	Elsevier
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=On+the+numerical+solution+of+direct+and+inverse+problems+for+the+heat+equation+in+a+semi-infinite+region&rft.jtitle=Journal+of+computational+and+applied+mathematics&rft.au=Chapko%2C+Roman&rft.date=1999-08-15&rft.pub=Elsevier+B.V&rft.issn=0377-0427&rft.eissn=1879-1778&rft.volume=108&rft.issue=1&rft.spage=41&rft.epage=55&rft_id=info:doi/10.1016%2FS0377-0427%2899%2900099-0&rft.externalDocID=S0377042799000990
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0377-0427&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0377-0427&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0377-0427&client=summon