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[{"key": "dc.contributor.advisor", "value": "Parviainen, Mikko", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Pakarinen, Tiina", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2015-05-10T06:48:03Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2015-05-10T06:48:03Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2015", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1473435", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/45828", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on tutustua kahteen t\u00e4rke\u00e4\u00e4n osittaisdifferentiaaliyht\u00e4l\u00f6\u00f6n; Laplacen yht\u00e4l\u00f6\u00f6n ja l\u00e4mp\u00f6yht\u00e4l\u00f6\u00f6n.\nN\u00e4it\u00e4 molempia hy\u00f6dynnet\u00e4\u00e4n fysiikan lis\u00e4ksi useiden muidenkin tieteenalojen sovelluksissa. \n\nL\u00e4mp\u00f6yht\u00e4l\u00f6, joka tunnetaan my\u00f6s diffuusioyht\u00e4l\u00f6n\u00e4, kuvaa jonkin suureen esimerkiksi l\u00e4mm\u00f6n johtumista aineissa ajan kuluessa. Pitk\u00e4n ajan kuluttua tilanne tasapainottuu, jolloin l\u00e4mm\u00f6n m\u00e4\u00e4r\u00e4 pysyy ajan suhteen vakiona tarkasteltavan alueen joka pisteess\u00e4. T\u00e4llaista t\u00e4ysin stabiloitunutta tilannetta voidaan mallintaa Laplacen yht\u00e4l\u00f6ll\u00e4. Laplacen yht\u00e4l\u00f6n toteuttavia funktioita kutsutaan harmonisiksi funktioiksi. Tutkielmassa johdetaan perusratkaisu ja keskiarvoperiaate molemmille osittaisdifferentiaaliyht\u00e4l\u00f6ille, sek\u00e4 tutustutaan yht\u00e4l\u00f6iden fysikaalisiin tulkintoihin. \n\nTutkielman motivaationa on oppia diskretisoimaan yksiulotteinen osittaisdifferentiaaliyht\u00e4l\u00f6 sek\u00e4 rakentamaan sen perusteella numeerinen ratkaisija l\u00e4mp\u00f6yht\u00e4l\u00f6lle Matlab-ohjelmaa hy\u00f6dynt\u00e4m\u00e4ll\u00e4. Diskretointien ja keskiarvoperiaatteiden v\u00e4lill\u00e4 havaitaan yhteys.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Tiina Pakarinen (tihepaka) on 2015-05-10 06:48:02.726953. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija@noreply.fi) on 2015-05-10T06:48:03Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201505101782.pdf: 1388359 bytes, checksum: 5f98d2c7bf0358bdd0551fa142c1ffa0 (MD5)\nlicense.html: 4822 bytes, checksum: 64cbf23d51a5c52ab8d9b3cd38cddd83 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2015-05-10T06:48:03Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201505101782.pdf: 1388359 bytes, checksum: 5f98d2c7bf0358bdd0551fa142c1ffa0 (MD5)\nlicense.html: 4822 bytes, checksum: 64cbf23d51a5c52ab8d9b3cd38cddd83 (MD5)\n Previous issue date: 2015", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (36 sivua)", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "L\u00e4mp\u00f6yht\u00e4l\u00f6n ja Laplacen yht\u00e4l\u00f6n ratkaisun ominaisuuksia", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201505101782", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2015-05-10T06:48:04Z", "language": null, "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": null, "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": "fi", "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "osittaisdifferentiaaliyht\u00e4l\u00f6t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "yht\u00e4l\u00f6t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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