{"id":124,"date":"2021-04-13T11:47:19","date_gmt":"2021-04-13T10:47:19","guid":{"rendered":"http:\/\/blogs.cardiff.ac.uk\/mathsugprojects\/?p=124"},"modified":"2022-11-11T10:24:53","modified_gmt":"2022-11-11T10:24:53","slug":"qualitative-theory-of-partial-differential-equations-n-dirr","status":"publish","type":"post","link":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/qualitative-theory-of-partial-differential-equations-n-dirr\/","title":{"rendered":"Qualitative Theory of Partial Differential Equations &#8211; Supervisor: Prof. N. Dirr"},"content":{"rendered":"<p style=\"color: #383735\"><strong>Title of project:<\/strong><br \/>\nQualitative Theory of Partial Differential Equations<\/p>\n<p style=\"color: #383735\"><strong>Code:<\/strong><br \/>\nND2122B<\/p>\n<p style=\"color: #383735\"><strong>Supervisor:\u00a0<\/strong>Prof. N. Dirr<\/p>\n<p style=\"color: #383735\"><strong>Project description:<\/strong><\/p>\n<p>Certain partial differential equations of the form<\/p>\n<p>\u2202tu(x,t)=\u2206u(x,t)+f(x,u(x,t))<\/p>\n<p>can be transformed in an integral equation and solved by a Peano-Iteration in a similar way as it is done for ODEs. The difference is however, that we have to work in infinite dimensional vector spaces instead of R. Combining ideas from ODEs and Functional Analysis, a lot can be said about the qualitative behaviour of such equations (stability, long-time behaviour etc.) similar to the ODE case.<br \/>\nThese equations, called reaction-diffusion equations, have applications in chemistry, biology and physics.<br \/>\nA project could, depending on the interest of the student,\u00a0 focus on numerics and\/or on analysis.<br \/>\nBackground Reading:<br \/>\nL.C. Evans, Partial Differential Equations, AMS Grad. Studies in Math. 19<\/p>\n<p style=\"margin: 0cm 0cm 18.0pt 0cm\"><strong><span style=\"font-family: 'Source Sans Pro',sans-serif;color: #383735\">Project offered as double module, single module, or both:<\/span><\/strong><span style=\"font-family: 'Source Sans Pro',sans-serif;color: #383735\"><br \/>\nDouble<\/span><\/p>\n<p style=\"margin: 0cm 0cm 18.0pt 0cm\"><strong><span style=\"font-family: 'Source Sans Pro',sans-serif;color: #141412\">Prerequisite 2nd year modules:<\/span><\/strong><span style=\"font-family: 'Source Sans Pro',sans-serif;color: #141412\"><br \/>\nReal Analysis, Series and Transforms. Not prerequisite but recommended is Modelling with ODEs or a Year 2 Numerical Analysis Module. <\/span><\/p>\n<p style=\"margin: 0cm 0cm 18.0pt 0cm\"><strong><span style=\"font-family: 'Source Sans Pro',sans-serif;color: #383735\">Recommended 3rd year module for concurrent study:<\/span><\/strong><span style=\"font-family: 'Source Sans Pro',sans-serif;color: #383735\"><br \/>\nOrdinary Differential Equations, Fourier and Functional Analysis<\/span><\/p>\n<p style=\"margin: 0cm 0cm 18.0pt 0cm\"><strong><span style=\"font-family: 'Source Sans Pro',sans-serif;color: #383735\">Number of students who can be supervised on this project:<\/span><\/strong><span style=\"font-family: 'Source Sans Pro',sans-serif;color: #383735\"><br \/>\n1<\/span><\/p>\n<p style=\"margin: 0cm 0cm 18.0pt 0cm\"><span style=\"font-family: 'Source Sans Pro',sans-serif;color: #383735\">\u00a0<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Title of project: Qualitative Theory of Partial Differential Equations Code: ND2122B Supervisor:\u00a0Prof. N. Dirr Project description: Certain partial differential equations of the form \u2202tu(x,t)=\u2206u(x,t)+f(x,u(x,t)) can be transformed in an integral equation and solved by a Peano-Iteration in a similar way as it is done for ODEs. The difference is however, that we have to work &hellip; <a href=\"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/qualitative-theory-of-partial-differential-equations-n-dirr\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Qualitative Theory of Partial Differential Equations &#8211; Supervisor: Prof. N. Dirr<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[2,4,7],"tags":[],"class_list":["post-124","post","type-post","status-publish","format-standard","hentry","category-2021-2022","category-double","category-yr-3-descriptions-2021-2022"],"aioseo_notices":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/posts\/124","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/comments?post=124"}],"version-history":[{"count":8,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/posts\/124\/revisions"}],"predecessor-version":[{"id":2061,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/posts\/124\/revisions\/2061"}],"wp:attachment":[{"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/media?parent=124"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/categories?post=124"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/tags?post=124"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}