Question

(3) (a) A separated-variable solution to the heat equation is U(T) = cos(lx) cos(my) cos(nz)ept. What is p in terms of the wavevector T = li + mj+nk? (b) In 1 space dimension the heat equation is aU/ət = kd’U/ax?. Try the non-separated- variable form: Utlob(t, x) = t-1/2e-z*/Axt. This represents a localized blob of heat spread- ing out. (c) In 2 space dimensions you can multiply the spreading blob solution in (b) with the same function Utlob(t, y) and get U2d blot(t, x, y) =t-le-(z*+y*)/Axt. Is this also a solution?

Answer 1

(3) (a) A separated-variable solution to the heat equation is U(T) = cos(lx) cos(my) cos(nz)ept. What is p in terms of the wa

solutions Heat equation general form in 30 k per at 22 D x2 + + 23x day . kl doce where, อน 22 aya dza By selaration of variable method v Geit) = T) v6) vy) vz) dez) dekle z dze where 1² = 1 l=etmetne where. Ka Conductivity PC P density of May (₂ skific heate take time part only, 누 Let. kok DT It =-ke ² tp -kle = +P -) at -Przo trial solution at (a-P) e a 20 =) La +P. +Pt .T=e

so. Pa ferm'tu). vololol (t, x) = tolle xY4kt du It #tv3 Yukle -112 -44xt/-1 t e + et 4kt? D2v dx2 at the 2 Le luke the 7 ( Yake st tilla x44kt) e akt dx tilla d²v docz *²/4kt tu - 2x -x%4kt akt 4kt Cocina [l- -1/2 x/4kt อใบ Oxa t x2 e akt akt dro Heate equation =k Now we Doxe we get Dw tilla 4485 ka -k. .e dx2 akt بل akt

-1/2 -x2/4kt = t L . е at 4kt -) Ik a²v Joca du at Proved -1/2 Say v=t solution. is a ② for 2o cafe heat equation. 22 du at + ง V zobel (tixiy) = t'e fc?ty?)/4kt ว่าบา -fety")/4kt - e 7 at ketos) (419) () Et १ 는 -fa2+y^) 14kt It e te 4kt av Dx? a 42/4kt th 2x 22 [exYukt] e "laket e-<44kt) (x.e-*4468) e Dx 4kt . -t: e-9*/4 dx akt -92/4kt _x?[4kt sc?/zkt e akt?

++ +5+ - + х/ _kt +7ъ 25 ду (et --- Cone 7 се с "Чон а р . Ман ) Бе Бъ -y-lkt -t-1 를 е Ee ye 72/4kE 2 ke - ++y+){tt (-1 ч*/4 akt 1 + 2ktu -k+y9144ң อใด у -1 +- E + - + _kt дох st- 2kt ду - Gs-y-)(4kt, — L+ с 2 я" ) Proved а) ) о-г ду OUT = Ot + Ох (9,t) - t-e -(-499) (4ke So, Veobdob solution