Question

Consider the one-dimensional heat equation for nonconstant thermal properties debelo - (Kolon with the initial condition u(x, 0) = f(x). [Hint: Suppose it is known that if u(x, t) = ??(x ) h(t), then 1 dh 1 d do Ko(c) = -1 h dt c(x)p(x)o dx dx You may assume the eigenfuctions are known. Briefly discuss limt- u(x, t). Solve the initial value problem: (a) with boundary conditions u(0, t) = 0 and u(L, t) = 0 au *(b) with boundary conditions ди = ax coefficients using orthogonality.] Ox (L, t) 0,, Derive

Answer 1

mom constant Sal Given from that data. 1- dianensional heat equation for thermal properties c [C] P(c) au - 2c [kolo) are CP ко at = 2² да2 Ut=22uka where 22 = ko СР as Consider the solution is of the fooms u(x,x) = 0 (c) h(t) Them, ut=ph' - ukoc=th Ut=2 UKC -> on=220"h > separation of variables_0" ah Them d'+20=0 In'tq?nn=0 m2+7=0 con +Q21=0 m=trai co=221 Then the general solution of dhc) - am costoc at bn sim Samec - Applying the boundary conditions uxloot) = $'(o) h(b) = 0 => '(o)=0 usa (Lt)=01(b) h (t)=0 = $(4)=0 din (c) = -am dam sim Sana+bm cosſ Ana oo (o)= 0 =) bm ſam=0 = bn=0

Them, Pon (oc) = ancos Sanx dm (L)=0 = 2 Jam =nX= AM = [mx]² Then the solution of Polc)= agoas (masch 2 where, an -=[23] with values of Am, holt)=co and hm (Z) = Core-[ant] I, h=1,2 Thus u(x,6)=2 ton (oc) hm (t) cot zné [ant] n= cos para n=1
Note:plzzz don't give dislike....plzzz comment if you have any problem i will try to solve your problem....plzzz give thumbs up i am in need........