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[{"key": "dc.contributor.advisor", "value": "V\u00e4h\u00e4kangas, Antti", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Puustinen, Ipa", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-06-24T11:58:12Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-06-24T11:58:12Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2019", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/64802", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on muodostaa ratkaisufunktiot Laplacen ja Poissonin yht\u00e4l\u00f6ille avaruudessa $\\mathbb{R}^n$ ja sen avoimen osajoukon tapauksessa. Laplacen yht\u00e4l\u00f6n avulla voidaan m\u00e4\u00e4ritell\u00e4 muun muassa harmoniset funktiot, joita etsitt\u00e4v\u00e4t ratkaisufunktiotkin tulevat olemaan. Siten tutkielman aluksi tutkitaan harmonisia funktioita ja osoitetaan niit\u00e4 koskevien lauseiden avulla, ett\u00e4 tiettyjen differentiaaliyht\u00e4l\u00f6iden ratkaisut ovat yksik\u00e4sitteisi\u00e4. T\u00e4m\u00e4n j\u00e4lkeen ryhdyt\u00e4\u00e4n tutkimaan Laplacen ja Poissonin yht\u00e4l\u00f6it\u00e4 ensiksi koko avaruudessa.\n\nAvoin osajoukko asettaa puolestaan uusia vaatimuksia ratkaisufunktioille, sill\u00e4 joukon reunalla niiden on toteutettava er\u00e4it\u00e4 ehtoja. T\u00e4t\u00e4 varten m\u00e4\u00e4ritell\u00e4\u00e4n ensiksi korjausfunktio tietyn differentiaaliyht\u00e4l\u00f6n ratkaisuna ja sit\u00e4 hy\u00f6dynt\u00e4en m\u00e4\u00e4ritell\u00e4\u00e4n Greenin funktio. Yleisen avoimen osajoukon tapauksessa varsinkin korjausfunktiolle on hyvinkin vaikea saada yksik\u00e4sitteist\u00e4 esitysmuotoa eik\u00e4 sen olemassaolokaan ole aina selv\u00e4\u00e4. T\u00e4st\u00e4 huolimatta esitet\u00e4\u00e4n Laplacen ja Poissonin yht\u00e4l\u00f6iden ratkaisukaavat avoimen osajoukon tapauksessa ja todistetaan ne. T\u00e4t\u00e4 varten joudutaan olettamaan korjausfunktio olemassaolevaksi, eli tutkittava joukko on oltava suotuisa.\n\nKorjausfunktio on kuitenkin hyvin keskeisess\u00e4 roolissa Greenin funktion muodostamisessa ja siten my\u00f6s ratkaisufunktioiden esitt\u00e4misess\u00e4. T\u00e4st\u00e4 syyst\u00e4 tutkielman lopussa k\u00e4sitell\u00e4\u00e4n kaksi joukkoa, joissa korjausfunktiolle l\u00f6ydet\u00e4\u00e4n konkreettinen esitys: yksikk\u00f6pallo ja puoliavaruus. Lopuksi esitet\u00e4\u00e4n viel\u00e4 ratkaisufunktiot n\u00e4iss\u00e4 joukoissa mahdollisimman konkreettisessa muodossa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2019-06-24T11:58:12Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-06-24T11:58:12Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "42", "language": "", "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.subject.other", "value": "Poisson", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Laplace", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Differentiaaliyht\u00e4l\u00f6t", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Harmonisuus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Green", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Poissonin yht\u00e4l\u00f6 ja Greenin funktio", "language": "", "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-201906243384", "language": "", "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": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "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": null, "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": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "yht\u00e4l\u00f6t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "funktiot", "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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