Question

Solving PDE's: du k. u (0,t) 0 (L,t) O u (x,0)- flx) ax

Please answer the problem above (PDE). Provide a solution that is easy to follow. Thank you!

Answer 1

Heat Equuation' Ju at (1) -(2) u(,0) de (o,t) 4 a dution uat) x(x) TIt) Let Pastial dosuva tues a(5) ane Qu xxTt) at x)TCt) paitial denivatus 5) m) Sulstrtuting the we get k x()Tt) x(TCE)ko Sepatating the Natualtes we Tt) gel- x) T(t) T 6) Lot Tt) Firom (G) e got ond k TIt) (8) X(a) k T P

wations (7) amd ae inear ODE iT and X Case Let Tt T) Cie TIt Omd .2 o (Auony eaton) dintind nal tako t/ xx) 2e vk uait e (e 3e Lo p 2 casei -t T(t) Ca e 2 Tit) Aulny eauatim m+-o ox.cogalos Then xiz) cs C + Sin )00 nt)Ce Sm

Tt) (7)Ct Ca ut) the dutions G), d0) and C) a Auitalts luto () is ahundion uhich atst cos the conditicn tha uxit) must donsae as t cases Such dution is uxit)c tr csxtSm ) 2 cae VE and auating to Sut-sttuing wegd rau öz/t0,t) ( ) Cae K o=(0 (0)s ) Cy (8)) ( 0 C4 e Cato, then So, 4t0, uir)o Catnualseluton)

Surstluting 0in 02), 2duces t uxE)=Ca can2 12) u(at) Ca os we get ) Sim C4 CS L, and euatng t 3e0 e get S -Case VE becomes ucxt) 0 eso the (3) CS t0 Sim C Hena Sm C4 to, Cs to 4mik m0,12- Sulstiutim -m m 3), we get w- CaCs e uxit) ux t) e eas

.The gomenal Aeu bon ux.t)2 ane .2 04) m0 (14) a Foutuo Cne kenuos and the Coeriont& an com e detaimed ueranla fx) dz bbhe L utzo) fa fesx dx u0) 12 c urt)