Question

Find the general solution of an 1D heat equation: ?_t (?, ?) = 4?_xx (?, ?) with the boundary conditions ?(0, ?) = ?(2, ?) = 0. Note that ?(?, ?) denotes the temperature profile along ? of a uniform rod of length 2.

Answer 1

Given: I-D Heat Equation - 4 Toon (2,t) Te (2,+ Conditions : Boundary To,t) T(2,t) = 0 deryth of uniform read let + u Torse (2,+) - X(X) 714). where, It X T' Tax =x"T 2 4 (X"T) XT' 47 d 7 da? x dT dt = IdT I dt k= -d² 4 d²7 dx² let

I dT 4 d²7 2 -d? 7 at dT dt -d²T Х dre dx? 4 dt dt : -X²T de + dx² 4 ons com plemestery feruetrom Equations for above dilferuatrol TE -akt ce x - A les +(t) = c.e las (2x) + B 3 sin (2x) X(x) = Acos cos(que) fax) + Bänlaa * UT (aut) = X(X) Tt) (cennie). (Acos(ag) +86-(3) ( CA = A' (8 = 8' ) - 479,+) = Al cos (12) + B' sin (2x) T(n,+) = -8²t е 4 * ( Alue) ... (13]

from boundary condition م = (رع) من در کوره - ( م ) À was (2x) A+ ہو۔ [A' و + و therefore, و: ام ود (+ ,2) - (نن) - A+ e B' sin ( 2 x (1) (o' din a - %م ہ

B'sina - Quals 2 A los an & B sin/72 Turfor, ²t +(at) = e y Al=0 - (@nnit tla,t) Eē a- (2n-1) B sin /(2n-1) (@u-) n=1 2 (fen-v.)'t en sin (fam.)2X 4 Tln,t) = E e av 17x)] hai