Question

2. For the 1-D heat equation solve uha, t) with Cs and ICs wing seperating BC (0,0) = 0, Lt) = 0 ICs (2,0) = cos 21

Answer 1

Pax-) .2 eill c²a u -I = ди. at an? put U=XT, of 'n a ond where 'x' is the function & 't' is the function of of only Only of ' t Now, & du -x di st at = Tay džu du? da² Substitute in I X aT = c² T.dt c² T. dx e po flage at an² = c²d²x z-P² (say) I dI at T x du? 'peo ej Constant

page- 2 or dT 1 dx x daa =-p? c²7 at -p² = or I dt (²7 dt dT + P22²T=0 dt 2 oti d²x +P²=0 = I d²x X du? -pa da? on solving these , we get -P²²t T=c, e & x = C2 Cospatz simpa But U=XT sinpon) (c, épzet) -1) xt +C3 riiu u=(cz caspx

page-3 As per the Boundary condition, given ə uto,t) - о. ди -pekt и есте (“, Now -C₂ Sinpr xp t CC65Pч (Р) 24 cipele (ressinfe tes cospe Put nao, tat, & du = 0, we get. Әя 5. 2. o=c, perette (-ea sino + Cacoso) - с хо + Сухі ) o=c,capeph ( . - Р°** О= c, Pe - Р°** Pet p. e Since, #о сүс го - 1

Pagey Now U(L,t)=0, lie, a ush, t-t, Uso Substitute these values in II, weget -p²c²t o PL + C₂ Sin PL) c, e 0 = (ca cosp But eph -prezt to - C, C₂ Cospl + C, Cg Simpl = 0 (from II) C, C3=0 But C, Cz COSPL (criven). u(a,0) Cosa san al Now for Substitute in II, we get

Page 5 с e, e cospu + C sinpr) Соли L с с Cospurte, la simpa= cos são 2 L (SuT c, c = 0 ... с. с. с 6s px св. А. Equating coefficients, & cospracossan 2. L 5

Paseo CIC221 & P= ST AL - P²227 U= Cie (Cacos pa tez simpe). -p²eat -P²c²t U= C, Cze cospu + C, Cze simpa Put C, C3=0 f C,C2=1, we get _p²c²t - е. cospa to But P = ST/aL %ct cos(sh u= e ~ ) 2L 07 - 951 et 422 cos sin u e QL g