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[{"key": "dc.contributor.author", "value": "Halinen, Hanna", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2014-06-29T17:45:26Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2014-06-29T17:45:26Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2014", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1438226", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/43861", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Fraktaaliderivaatta on derivaatta, jonka kertaluku on reaali- tai kompleksiluku. Fraktaaliderivaatta voidaan m\u00e4\u00e4ritell\u00e4 usealla eri tavalla, mutta mik\u00e4\u00e4n m\u00e4\u00e4ritelm\u00e4 ei ole selke\u00e4sti muita parempi. Koska fraktaaliderivaatan ominaisuudet riippuvat valitusta m\u00e4\u00e4ritelm\u00e4st\u00e4, ominaisuuksia ei voida suoraan yleist\u00e4\u00e4 kaikille fraktaaliderivaatoille. T\u00e4m\u00e4n tutkielman tarkoitus on antaa lukijalle perustiedot reaalilukukertaisista fraktaaliderivaatoista ja niiden m\u00e4\u00e4ritelm\u00e4sidonnaisista ominaisuuksista. \n\nTutkielmassa esitell\u00e4\u00e4n kolme yleisimmin viitattua m\u00e4\u00e4ritelm\u00e4\u00e4: Gr\u00fcnwald-Letnikov, Riemann-Liouville ja Caputo. Gr\u00fcnwald-Letnikovin m\u00e4\u00e4ritelm\u00e4 yleist\u00e4\u00e4 klassisen derivaatan m\u00e4\u00e4ritelm\u00e4n suoraan reaali- ja kompleksiluvuille, mink\u00e4 vuoksi se on helpoiten ymm\u00e4rrett\u00e4viss\u00e4 analyysin perustietojen pohjalta. Riemann-Liouvillen m\u00e4\u00e4ritelm\u00e4 yhdist\u00e4\u00e4 fraktaaliderivaatan ja fraktaali-integraalin k\u00e4sitteet. Caputon fraktaaliderivaatta on taas kehitetty sovellusten n\u00e4k\u00f6kulmasta. \n\nVaikka fraktaaliderivaatan ominaisuudet riippuvat valitusta m\u00e4\u00e4ritelm\u00e4st\u00e4, joitakin ominaisuuksia voidaan yleist\u00e4\u00e4. Ensiksi, fraktaaliderivaatta on yht\u00e4pit\u00e4v\u00e4 klassisen derivaatan kanssa, kun sen kertaluku on kokonaisluku. Toiseksi, fraktaalinen differintegraalioperaattori (engl. fractional differintegral operator) on lineaarinen. M\u00e4\u00e4ritelm\u00e4st\u00e4 riippuvia ominaisuuksia ovat esimerkiksi Riemann-Liouvillen fraktaali-integraalien vaihdannaisuus sek\u00e4 Riemann-Liouvillen / Caputon fraktaaliderivaatan additiivisuus klassisen derivaatan kanssa. Riemann-Liouvillen ja Caputon m\u00e4\u00e4ritelmien v\u00e4lill\u00e4 on kuitenkin se ero, ett\u00e4 additiivisuus p\u00e4tee toiselle p\u00e4invastaisessa j\u00e4rjestyksess\u00e4. Siten fraktaaliderivaatat eiv\u00e4t kommutoi. Riemann-Liouvillen differintegraalioperaattorin t\u00e4rke\u00e4 ominaisuus on my\u00f6s se, ett\u00e4 derivointioperaattori on samaa kertalukua olevan integrointioperaattorin vasen k\u00e4\u00e4nteisoperaatio. \n\nM\u00e4\u00e4ritett\u00e4ess\u00e4 funktioiden fraktaaliderivaatan lausekkeita Riemann-Liouvillen ja Gr\u00fcnwald-Letnikovin m\u00e4\u00e4ritelm\u00e4t antavat samat tulokset. Caputon m\u00e4\u00e4ritelm\u00e4 ei kuitenkaan ole yht\u00e4pit\u00e4v\u00e4 edellisten m\u00e4\u00e4ritelmien kanssa muulloin kuin erikoistapauksissa. Merkitt\u00e4vin ero n\u00e4iden m\u00e4\u00e4ritelmien v\u00e4lill\u00e4 on, ett\u00e4 vakion Gr\u00fcnwald-Letnikovin ja Riemann-Liouvillen fraktaaliderivaatat eiv\u00e4t ole nollia, kun taas Caputon fraktaaliderivaatta on nolla.\n\nFraktaaliderivaatan sovelluksia ovat erilaiset fraktaalidifferentiaaliyht\u00e4l\u00f6t. Fraktaalidifferentiaaliyht\u00e4l\u00f6 saadaan, kun klassisen differentiaaliyht\u00e4l\u00f6n derivaatta korvataan fraktaaliderivaatalla. Alkuarvoteht\u00e4viss\u00e4 Riemann-Liouvillen fraktaaliderivaatan k\u00e4ytt\u00f6 on kuitenkin ongelmallista, sill\u00e4 se tuottaa alkuehdoiksi reaalilukukertaisia derivaattoja, joille ei ole keksitty fysikaalista tulkintaa. Caputon fraktaaliderivaattaa k\u00e4ytett\u00e4ess\u00e4 vastaavaa ongelmaa ei ole. Fraktaalidifferentiaaliyht\u00e4l\u00f6t ovat nyky\u00e4\u00e4n t\u00e4rke\u00e4 tutkimuskohde muun muassa fysiikassa. T\u00e4ss\u00e4 tutkielmassa esitell\u00e4\u00e4n yksi esimerkki fysiikan sovelluksesta: fraktaaliv\u00e4r\u00e4htelij\u00e4n differentiaaliyht\u00e4l\u00f6. Numeerisin menetelmin on havaittu, ett\u00e4 fraktaaliv\u00e4r\u00e4htelij\u00e4ll\u00e4 on sis\u00e4inen vaimenemismekanismi, joten se ei voi muodostaa lainkaan eristetty\u00e4 systeemi\u00e4. Toistaiseksi on kuitenkin ep\u00e4selv\u00e4\u00e4, mist\u00e4 fraktaaliv\u00e4r\u00e4htelij\u00e4n sis\u00e4inen vaimenemismekanismi johtuu.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Hanna Halinen (hamakaha) on 2014-06-29 17:45:25.918234. 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 2014-06-29T17:45:26Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201406292177.pdf: 483439 bytes, checksum: b951391c001e73c2c14047b8a31c5714 (MD5)\nlicense.html: 4811 bytes, checksum: 5616cd42e09a451c8592a823808d0ad6 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2014-06-29T17:45:26Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201406292177.pdf: 483439 bytes, checksum: b951391c001e73c2c14047b8a31c5714 (MD5)\nlicense.html: 4811 bytes, checksum: 5616cd42e09a451c8592a823808d0ad6 (MD5)\n Previous issue date: 2014", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (52 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.subject.other", "value": "fraktaaliderivaatta", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "fraktaali-integraali", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "fraktaalidifferentiaaliyht\u00e4l\u00f6", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Johdatus fraktaaliderivaattoihin ja niiden sovelluksiin", "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-201406292177", "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": "2014-06-29T17:45:27Z", "language": null, "element": "date", "qualifier": "updated", "schema": "dc"}, {"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": "fraktaalit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "differentiaaliyht\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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