{"id":124294,"date":"2018-02-20T23:50:30","date_gmt":"2018-02-20T23:50:30","guid":{"rendered":"https:\/\/writemyessayfree.com\/partial-differential-equation-problem"},"modified":"2017-08-16T05:57:35","modified_gmt":"2017-08-16T05:57:35","slug":"partial-differential-equation-problem","status":"publish","type":"post","link":"https:\/\/www.benedictsol.com\/blogs\/partial-differential-equation-problem\/","title":{"rendered":"Partial Differential Equation Problem"},"content":{"rendered":"<div class=\"the_content_wrapper\">\n<p>Partial Differential Equation Problem<br \/> 1. The steady concentration of a reactant ?(r, ?) outside a disk of radius 1<br \/> (r &gt; 1) obeys the following equation:<br \/> 0 = ?2? =<br \/> 1<br \/> r<br \/> ?<br \/> ?r (r<br \/> ??<br \/> ?r ) + 1<br \/> r<br \/> 2<br \/> ?<br \/> 2?<br \/> ??2<br \/> The following boundary conditions apply:<br \/> \u02c6r \u00b7 ??(1, ?) = f(?)(1 + a?(1, ?))<br \/> where f(?) is an even function whose Fourier expansion is<br \/> \u02c6f(?) = 1<br \/> 2<br \/> F0 +<br \/> X8<br \/> i=1<br \/> Fn cos(n?)<br \/> (a) Write down an expression for the general solution (before applying boundary<br \/> conditions) for a steady solution as<br \/> ?(r, ?) = X8<br \/> i=-8<br \/> gn(r)(An cos(n?) + Bn sin(n?))<br \/> Then apply the boundary conditions and deduce which boundary conditions<br \/> would vanish.<br \/> (b) For a = 0, show that a steady solution ?(r, ?) exists only if F0 = 0 and<br \/> find the steady solution in this case.<br \/> (c) For a 6= 0, show that the Fourier coefficients An for a steady solution<br \/> ?(r, ?) satisfy the equation<br \/> X8<br \/> n=0<br \/> MmnAn = -Fm<br \/> where Mmn is the m \u00d7 n terms of a m \u00d7 n matrix that depends on the<br \/> coefficients Fm\u2019s.<br \/> Given the solution, write down the nonlinear criterion that F0 must satisfy<br \/> if a steady solution is to exist. (The criterion requires a matrix inversion, which<br \/> you cannot do explicitly since the matrix is infinite; just write it symbolically<br \/> as a matrix inverse)<br \/> (d) For a = 1 and Fn = dn,1, numerically approximate F0 to five digits.<br \/> dn,1 = 1 for n = 1 and 0 otherwise.<br \/> 1<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Partial Differential Equation Problem 1. The steady concentration of a reactant ?(r, ?) outside a disk of radius 1 (r &gt; 1) obeys the following equation: 0 = ?2? = 1 r ? ?r (r ?? ?r ) + 1 <a href=\"https:\/\/www.benedictsol.com\/blogs\/partial-differential-equation-problem\/\" class=\"read-more\">Read More &#8230;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"class_list":["post-124294","post","type-post","status-publish","format-standard","hentry"],"_links":{"self":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/124294","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/comments?post=124294"}],"version-history":[{"count":0,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/124294\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/media?parent=124294"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/categories?post=124294"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/tags?post=124294"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}