{"id":1647,"date":"2022-04-14T15:53:02","date_gmt":"2022-04-14T07:53:02","guid":{"rendered":"https:\/\/gsclab.ntou.edu.tw\/wordpress\/?p=1647"},"modified":"2022-12-08T09:18:44","modified_gmt":"2022-12-08T01:18:44","slug":"infinitely-smooth-polyharmonic-rbf-collocation-method-for-numerical-solution-of-elliptic-pdes","status":"publish","type":"post","link":"https:\/\/gsclab.ntou.edu.tw\/wordpress\/?p=1647","title":{"rendered":"Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs"},"content":{"rendered":"\n<p>Abstract: In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation<br>method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with<br>natural logarithm is a piecewise smooth function in the conventional radial basis function collocation<br>method for solving governing equations. We converted the piecewise smooth PRBF into an<br>infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial<br>distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly,<br>the PRBF and its derivatives in the governing PDEs were always continuous. The seismic<br>wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater<br>contamination problem were investigated to reveal the robustness of the proposed PRBF.<br>Comparisons of the conventional PRBF with the proposed method were carried out as well. The<br>results illustrate that the proposed approach could provide more accurate solutions for solving<br>PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated<br>that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no<br>longer required.<\/p>\n\n\n\n<p><br>Keywords: partial differential equations; polyharmonic; radial basis function; source point; collocation<br>method<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><a href=\"https:\/\/www.mdpi.com\/2227-7390\/9\/13\/1535\/pdf\">https:\/\/www.mdpi.com\/2227-7390\/9\/13\/1535\/pdf<\/a><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><a href=\"https:\/\/www.mdpi.com\/2227-7390\/9\/13\/1535\">https:\/\/www.mdpi.com\/2227-7390\/9\/13\/1535<\/a><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-style-default\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"649\" src=\"https:\/\/gsclab.ntou.edu.tw\/wordpress\/wp-content\/uploads\/2022\/04\/image-1-1024x649.png\" alt=\"\" class=\"wp-image-1648\" srcset=\"https:\/\/gsclab.ntou.edu.tw\/wordpress\/wp-content\/uploads\/2022\/04\/image-1-1024x649.png 1024w, https:\/\/gsclab.ntou.edu.tw\/wordpress\/wp-content\/uploads\/2022\/04\/image-1-300x190.png 300w, https:\/\/gsclab.ntou.edu.tw\/wordpress\/wp-content\/uploads\/2022\/04\/image-1-768x486.png 768w, https:\/\/gsclab.ntou.edu.tw\/wordpress\/wp-content\/uploads\/2022\/04\/image-1.png 1159w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Abstract: In this article, a novel infinitely smooth po [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":1650,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14],"tags":[],"class_list":["post-1647","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-article"],"_links":{"self":[{"href":"https:\/\/gsclab.ntou.edu.tw\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1647","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gsclab.ntou.edu.tw\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gsclab.ntou.edu.tw\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gsclab.ntou.edu.tw\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gsclab.ntou.edu.tw\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1647"}],"version-history":[{"count":1,"href":"https:\/\/gsclab.ntou.edu.tw\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1647\/revisions"}],"predecessor-version":[{"id":1649,"href":"https:\/\/gsclab.ntou.edu.tw\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1647\/revisions\/1649"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/gsclab.ntou.edu.tw\/wordpress\/index.php?rest_route=\/wp\/v2\/media\/1650"}],"wp:attachment":[{"href":"https:\/\/gsclab.ntou.edu.tw\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1647"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gsclab.ntou.edu.tw\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1647"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gsclab.ntou.edu.tw\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1647"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}