{"id":1972,"date":"2021-04-13T11:24:00","date_gmt":"2021-04-13T10:24:00","guid":{"rendered":"http:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/?p=1972"},"modified":"2022-11-11T10:24:55","modified_gmt":"2022-11-11T10:24:55","slug":"qualitative-theory-of-partial-differential-equations-supervisor-professor-nicholas-dirr","status":"publish","type":"post","link":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/qualitative-theory-of-partial-differential-equations-supervisor-professor-nicholas-dirr\/","title":{"rendered":"Qualitative Theory of Partial Differential Equations &#8211; Supervisor: Professor Nicholas Dirr"},"content":{"rendered":"\n<p><strong>Title\nof project:<\/strong><br>\nQualitative Theory of Partial Differential Equations<\/p>\n\n\n\n<p><strong>Code:<\/strong><br> ND2021B<\/p>\n\n\n\n<p><strong>Supervisor:&nbsp;<\/strong>Prof. N. Dirr<\/p>\n\n\n\n<p><strong>Project\ndescription:<\/strong><\/p>\n\n\n\n<p>Certain\npartial differential equations of the form<\/p>\n\n\n\n<p>\u2202tu(x,t)=\u2206u(x,t)+f(x,u(x,t))\n<br>\ncan be transformed in an integral equation and solved by a Peano-Iteration in a\nsimilar way as it is done for ODEs. The difference is however, that we have to\nwork in infinite dimensional vector spaces instead of Rn. Combining ideas from\nODEs and Functional Analysis, a lot can be said about the qualitative behaviour\nof such equations (stability, long-time behaviour etc.) similar to the ODE\ncase.<br>\nThese equations, called reaction-diffusion equations, have applications in\nchemistry, biology and physics.<br>\nA project should always contain some proofs, but, depending on the interest of\nthe student, could then focus on numerics or on analysis.<br>\nBackground Reading:<br>\nD. Henry, Geometric Theory of Semilineat Parabolic Differential Equations\nSpringer Lecture Notes in Mathematics 840<br>\nL.C. Evans, Partial Differential Equations, AMS Grad. Studies in Math. 19<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><strong>Project\noffered as double module, single module, or both:<\/strong><br>\nDouble<\/p>\n\n\n\n<p><strong>Prerequisite\n3rd year modules:<\/strong><br>\nMeasure Theory, Ordinary Differential <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Title of project: Qualitative Theory of Partial Differential Equations Code: ND2021B Supervisor:&nbsp;Prof. 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-supervisor-professor-nicholas-dirr\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Qualitative Theory of Partial Differential Equations &#8211; Supervisor: Professor Nicholas 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":[2,4,8],"tags":[],"class_list":["post-1972","post","type-post","status-publish","format-standard","hentry","category-2021-2022","category-double","category-yr-4-description-2021-2022"],"aioseo_notices":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/posts\/1972","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=1972"}],"version-history":[{"count":3,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/posts\/1972\/revisions"}],"predecessor-version":[{"id":2059,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/posts\/1972\/revisions\/2059"}],"wp:attachment":[{"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/media?parent=1972"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/categories?post=1972"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/tags?post=1972"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}