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Eigenvalues of the Laplacian on an elliptic domain
The importance of eigenvalue problems concerning the Laplacian is well documented in classical and modern literature. Finding the eigenvalues for various geometries of the domains has posed many challenges which include infinite systems of algebraic equations, asymptotic methods, integral equations...
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| Published in: | Computers & mathematics with applications (1987) 2008-03, Vol.55 (6), p.1129-1136 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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| container_end_page | 1136 |
| container_issue | 6 |
| container_start_page | 1129 |
| container_title | Computers & mathematics with applications (1987) |
| container_volume | 55 |
| creator | Wu, Yan Shivakumar, P.N. |
| description | The importance of eigenvalue problems concerning the Laplacian is well documented in classical and modern literature. Finding the eigenvalues for various geometries of the domains has posed many challenges which include infinite systems of algebraic equations, asymptotic methods, integral equations etc. In this paper, we present a comprehensive account of the general solutions to Helmholtz’s equations (defined on simply connected regions) using complex variable techniques. We consider boundaries of the form
z
z
̄
=
f
(
z
±
z
̄
)
or its inverse
z
±
z
̄
=
g
(
z
z
̄
)
. To illustrate the theory, we reduce the problem on elliptic domains to equivalent linear infinite algebraic systems, where the coefficients of the infinite matrix are known polynomials of the eigenvalues. We compute truncations of the infinite system for numerical values. These values are compared to approximate values and some inequalities available in literature. |
| doi_str_mv | 10.1016/j.camwa.2007.06.017 |
| format | article |
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z
z
̄
=
f
(
z
±
z
̄
)
or its inverse
z
±
z
̄
=
g
(
z
z
̄
)
. To illustrate the theory, we reduce the problem on elliptic domains to equivalent linear infinite algebraic systems, where the coefficients of the infinite matrix are known polynomials of the eigenvalues. We compute truncations of the infinite system for numerical values. These values are compared to approximate values and some inequalities available in literature.</description><identifier>ISSN: 0898-1221</identifier><identifier>EISSN: 1873-7668</identifier><identifier>DOI: 10.1016/j.camwa.2007.06.017</identifier><language>eng</language><publisher>Elsevier Ltd</publisher><subject>Algebra ; Approximation ; Asymptotic methods ; Boundaries ; Eigenvalues ; Helmholtz ; Infinite systems ; Inverse ; Laplacian ; Mathematical analysis ; Mathematical models ; Simply connected</subject><ispartof>Computers & mathematics with applications (1987), 2008-03, Vol.55 (6), p.1129-1136</ispartof><rights>2007 Elsevier Ltd</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c412t-93396de2781a21107e0866422cf600c233838cb12cd6897f7d9f2850157b27423</citedby><cites>FETCH-LOGICAL-c412t-93396de2781a21107e0866422cf600c233838cb12cd6897f7d9f2850157b27423</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,777,781,27905,27906</link.rule.ids></links><search><creatorcontrib>Wu, Yan</creatorcontrib><creatorcontrib>Shivakumar, P.N.</creatorcontrib><title>Eigenvalues of the Laplacian on an elliptic domain</title><title>Computers & mathematics with applications (1987)</title><description>The importance of eigenvalue problems concerning the Laplacian is well documented in classical and modern literature. Finding the eigenvalues for various geometries of the domains has posed many challenges which include infinite systems of algebraic equations, asymptotic methods, integral equations etc. In this paper, we present a comprehensive account of the general solutions to Helmholtz’s equations (defined on simply connected regions) using complex variable techniques. We consider boundaries of the form
z
z
̄
=
f
(
z
±
z
̄
)
or its inverse
z
±
z
̄
=
g
(
z
z
̄
)
. To illustrate the theory, we reduce the problem on elliptic domains to equivalent linear infinite algebraic systems, where the coefficients of the infinite matrix are known polynomials of the eigenvalues. We compute truncations of the infinite system for numerical values. These values are compared to approximate values and some inequalities available in literature.</description><subject>Algebra</subject><subject>Approximation</subject><subject>Asymptotic methods</subject><subject>Boundaries</subject><subject>Eigenvalues</subject><subject>Helmholtz</subject><subject>Infinite systems</subject><subject>Inverse</subject><subject>Laplacian</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Simply connected</subject><issn>0898-1221</issn><issn>1873-7668</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2008</creationdate><recordtype>article</recordtype><recordid>eNp9kLtOAzEQRS0EEiHwBTRbIZpdZsYbPwoKFIWHFIkGasvxesHRPsJ6E8Tf4xDqNDPNuaM7h7FrhAIBxd26cLb9tgUByAJEAShP2ASV5LkUQp2yCSitciTCc3YR4xoASk4wYbQIH77b2WbrY9bX2fjps6XdNNYF22V9l6XpmyZsxuCyqm9t6C7ZWW2b6K_-95S9Py7e5s_58vXpZf6wzF2JNOaacy0qT1KhJUSQHpQQJZGrBYAjzhVXboXkKqG0rGWla1IzwJlckSyJT9nN4e5m6L9SvdG0IbpUxna-30bDiSvSWiTw9iiIoAi1mEmZUH5A3dDHOPjabIbQ2uEnQWav0qzNn0qzV2lAmKQype4PKZ_e3QU_mOiC75yvwuDdaKo-HM3_AuOHel0</recordid><startdate>20080301</startdate><enddate>20080301</enddate><creator>Wu, Yan</creator><creator>Shivakumar, P.N.</creator><general>Elsevier Ltd</general><scope>6I.</scope><scope>AAFTH</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20080301</creationdate><title>Eigenvalues of the Laplacian on an elliptic domain</title><author>Wu, Yan ; Shivakumar, P.N.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c412t-93396de2781a21107e0866422cf600c233838cb12cd6897f7d9f2850157b27423</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2008</creationdate><topic>Algebra</topic><topic>Approximation</topic><topic>Asymptotic methods</topic><topic>Boundaries</topic><topic>Eigenvalues</topic><topic>Helmholtz</topic><topic>Infinite systems</topic><topic>Inverse</topic><topic>Laplacian</topic><topic>Mathematical analysis</topic><topic>Mathematical models</topic><topic>Simply connected</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Wu, Yan</creatorcontrib><creatorcontrib>Shivakumar, P.N.</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Computers & mathematics with applications (1987)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Wu, Yan</au><au>Shivakumar, P.N.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Eigenvalues of the Laplacian on an elliptic domain</atitle><jtitle>Computers & mathematics with applications (1987)</jtitle><date>2008-03-01</date><risdate>2008</risdate><volume>55</volume><issue>6</issue><spage>1129</spage><epage>1136</epage><pages>1129-1136</pages><issn>0898-1221</issn><eissn>1873-7668</eissn><abstract>The importance of eigenvalue problems concerning the Laplacian is well documented in classical and modern literature. Finding the eigenvalues for various geometries of the domains has posed many challenges which include infinite systems of algebraic equations, asymptotic methods, integral equations etc. In this paper, we present a comprehensive account of the general solutions to Helmholtz’s equations (defined on simply connected regions) using complex variable techniques. We consider boundaries of the form
z
z
̄
=
f
(
z
±
z
̄
)
or its inverse
z
±
z
̄
=
g
(
z
z
̄
)
. To illustrate the theory, we reduce the problem on elliptic domains to equivalent linear infinite algebraic systems, where the coefficients of the infinite matrix are known polynomials of the eigenvalues. We compute truncations of the infinite system for numerical values. These values are compared to approximate values and some inequalities available in literature.</abstract><pub>Elsevier Ltd</pub><doi>10.1016/j.camwa.2007.06.017</doi><tpages>8</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0898-1221 |
| ispartof | Computers & mathematics with applications (1987), 2008-03, Vol.55 (6), p.1129-1136 |
| issn | 0898-1221 1873-7668 |
| language | eng |
| recordid | cdi_proquest_miscellaneous_32382996 |
| source | ScienceDirect Journals |
| subjects | Algebra Approximation Asymptotic methods Boundaries Eigenvalues Helmholtz Infinite systems Inverse Laplacian Mathematical analysis Mathematical models Simply connected |
| title | Eigenvalues of the Laplacian on an elliptic domain |
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