{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T18:24:38Z","timestamp":1777487078316,"version":"3.51.4"},"reference-count":9,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,4,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>In this article, we formalized in Mizar [4], [1] simple partial differential equations. In the first section, we formalized partial differentiability and partial derivative. The next section contains the method of separation of variables for one-dimensional wave equation. In the last section, we formalized the superposition principle.We referred to [6], [3], [5] and [9] in this formalization.<\/jats:p>","DOI":"10.2478\/forma-2019-0003","type":"journal-article","created":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T05:33:51Z","timestamp":1558071231000},"page":"25-34","source":"Crossref","is-referenced-by-count":3,"title":["A Simple Example for Linear Partial Differential Equations and Its Solution Using the Method of Separation of Variables"],"prefix":"10.2478","volume":"27","author":[{"given":"Sora","family":"Otsuki","sequence":"first","affiliation":[{"name":"Shinshu University , Nagano , Japan"}]},{"given":"Pauline N.","family":"Kawamoto","sequence":"additional","affiliation":[{"name":"Shinshu University , Nagano , Japan"}]},{"given":"Hiroshi","family":"Yamazaki","sequence":"additional","affiliation":[{"name":"Shinshu University , Nagano , Japan"}]}],"member":"374","published-online":{"date-parts":[[2019,5,16]]},"reference":[{"key":"2026042801430100108_j_forma-2019-0003_ref_001_w2aab3b7b2b1b6b1ab1ab1Aa","doi-asserted-by":"crossref","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9\u201332, 2018. doi:10.1007\/s10817-017-9440-6.10.1007\/s10817-017-9440-6604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026042801430100108_j_forma-2019-0003_ref_002_w2aab3b7b2b1b6b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"[2] Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Higher-order partial differentiation. Formalized Mathematics, 20(2):113\u2013124, 2012. doi:10.2478\/v10037-012-0015-z.10.2478\/v10037-012-0015-z","DOI":"10.2478\/v10037-012-0015-z"},{"key":"2026042801430100108_j_forma-2019-0003_ref_003_w2aab3b7b2b1b6b1ab1ab3Aa","unstructured":"[3] John Fritz. Nonlinear Wave Equations, Formulation of Singularities. American Mathematical Society, 1990. ISBN 978-0-8218-7001-3."},{"key":"2026042801430100108_j_forma-2019-0003_ref_004_w2aab3b7b2b1b6b1ab1ab4Aa","doi-asserted-by":"crossref","unstructured":"[4] Adam Grabowski, Artur Korni\u0142owicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191\u2013198, 2015. doi:10.1007\/s10817-015-9345-1.10.1007\/s10817-015-9345-1","DOI":"10.1007\/s10817-015-9345-1"},{"key":"2026042801430100108_j_forma-2019-0003_ref_005_w2aab3b7b2b1b6b1ab1ab5Aa","unstructured":"[5] Mitsuhiro Nakao. Bibun-sekibun-gaku (Japanese). Kindai-kagaku-sha, pages 52\u201353, 1992."},{"key":"2026042801430100108_j_forma-2019-0003_ref_006_w2aab3b7b2b1b6b1ab1ab6Aa","unstructured":"[6] Ian Naismith Sneddon. Elements of Partial Differential Equations. Tokyo McGraw-Hill Kogakusha, pages 209\u2013273, 1957."},{"key":"2026042801430100108_j_forma-2019-0003_ref_007_w2aab3b7b2b1b6b1ab1ab7Aa","unstructured":"[7] Hiroshi Yamazaki, Yoshinori Fujisawa, and Yatsuka Nakamura. On replace function and swap function for finite sequences. Formalized Mathematics, 9(3):471\u2013474, 2001."},{"key":"2026042801430100108_j_forma-2019-0003_ref_008_w2aab3b7b2b1b6b1ab1ab8Aa","unstructured":"[8] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255\u2013263, 1998."},{"key":"2026042801430100108_j_forma-2019-0003_ref_009_w2aab3b7b2b1b6b1ab1ab9Aa","unstructured":"[9] Kentaro Yano. Kaiseki-gaku-gairon (Japanese). Shokabo Co., Ltd., 1982."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/27\/1\/article-p25.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2019-0003","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T14:35:49Z","timestamp":1777386949000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2019-0003"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,1]]},"references-count":9,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,5,16]]},"published-print":{"date-parts":[[2019,4,1]]}},"alternative-id":["10.2478\/forma-2019-0003"],"URL":"https:\/\/doi.org\/10.2478\/forma-2019-0003","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2019,4,1]]}}}