{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T02:08:12Z","timestamp":1771466892025,"version":"3.50.1"},"reference-count":18,"publisher":"Wiley","license":[{"start":{"date-parts":[[2016,5,16]],"date-time":"2016-05-16T00:00:00Z","timestamp":1463356800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Numerical Analysis"],"published-print":{"date-parts":[[2016,5,16]]},"abstract":"<jats:p>The purpose of this paper is to develop a high-order compact finite difference method for solving one-dimensional (1D) heat conduction equation with Dirichlet and Neumann boundary conditions, respectively. A parameter is used for the direct implementation of Dirichlet and Neumann boundary conditions. The introduced parameter adjusts the position of the neighboring nodes very next to the boundary. In the case of Dirichlet boundary condition, we developed eighth-order compact finite difference method for the entire domain and fourth-order accurate proposal is presented for the Neumann boundary conditions. In the case of Dirichlet boundary conditions, the introduced parameter behaves like a free parameter and could take any value from its defined domain but for the Neumann boundary condition we obtained a particular value of the parameter. In both proposed compact finite difference methods, the order of accuracy is the same for all nodes. The time discretization is performed by using Crank-Nicholson finite difference method. The unconditional convergence of the proposed methods is presented. Finally, a set of 1D heat conduction equations is solved to show the validity and accuracy of our proposed methods.<\/jats:p>","DOI":"10.1155\/2016\/8376061","type":"journal-article","created":{"date-parts":[[2016,5,16]],"date-time":"2016-05-16T21:01:40Z","timestamp":1463432500000},"page":"1-12","source":"Crossref","is-referenced-by-count":9,"title":["Eighth-Order Compact Finite Difference Scheme for 1D Heat Conduction Equation"],"prefix":"10.1155","volume":"2016","author":[{"given":"Asma","family":"Yosaf","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Engineering and Technology, Lahore 54840, Pakistan"}]},{"given":"Shafiq Ur","family":"Rehman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Engineering and Technology, Lahore 54840, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2634-5130","authenticated-orcid":true,"given":"Fayyaz","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienza e Alta Tecnologia, Universita dell\u2019Insubria, Via Valleggio 11, 22100 Como, Italy"},{"name":"Departament de F\u00edsica i Enginyeria Nuclear, Universitat Polit\u00e8cnica de Catalunya, Comte d\u2019Urgell 187, 08036 Barcelona, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2944-0352","authenticated-orcid":true,"given":"Malik Zaka","family":"Ullah","sequence":"additional","affiliation":[{"name":"Departament de F\u00edsica i Enginyeria Nuclear, Universitat Polit\u00e8cnica de Catalunya, Comte d\u2019Urgell 187, 08036 Barcelona, Spain"},{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]},{"given":"Ali Saleh","family":"Alshomrani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]}],"member":"311","reference":[{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijthermalsci.2009.01.007"},{"key":"5","year":"2006"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/0017-9310(92)90131-B"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1115\/1.2911377"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1115\/1.2822329"},{"key":"9","year":"1997"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijheatmasstransfer.2008.08.029"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(92)90324-R"},{"key":"11","year":"1994"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-7278-1"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijthermalsci.2009.08.007"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1002\/nme.2473"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1002\/num.20531"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1080\/10407790.2010.489878"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1007\/s10827-010-0289-5"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2010.07.017"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2013.03.026"},{"key":"1","year":"2012"}],"container-title":["Advances in Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/archive\/2016\/8376061.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/archive\/2016\/8376061.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/archive\/2016\/8376061.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,9]],"date-time":"2020-12-09T00:39:51Z","timestamp":1607474391000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ana\/2016\/8376061\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,5,16]]},"references-count":18,"alternative-id":["8376061","8376061"],"URL":"https:\/\/doi.org\/10.1155\/2016\/8376061","relation":{},"ISSN":["1687-9562","1687-9570"],"issn-type":[{"value":"1687-9562","type":"print"},{"value":"1687-9570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,5,16]]}}}