Question

Let u be the solution to the initial boundary value problem for the Heat Equation дли(t, 2) — 4 әғи(t, 2), te (0, o0), те (0,1); with initial condition , u(0, a)f() and with boundary conditions 0. u(t, 0)0 u(t, 1) Find the solution u using the expansion и(t, г) "(2)"т (?)"а " n 1 with the normalization conditions 1 Vn (0) 1, wn 2n a. (3/10) Find the functions wn, with index n> 1. Wn b. (3/10) Find the functions v,n, with index n> 1. Un c. (4/10) Find the coefficients , with indexn> 1 Сп

Answer 1

Given Initial boundarn valu pnblem is (ki)ne CRi?) ne with im tial condihon u(ox) Omd with boundry conditon ue io)=0 u(tis) = 0 So By the method of separation ef Variabte weget U = X(7). T(t). Hhen weet Say separation ecnstond 47 get je T+4m To Solving weget, +BGin m z T=ce4mt NOw Aecosm2 +B'e sinmz)ét. 4mt ( Aeosmz + Bsin m z)e apply ing Boundaray condihons weget NOw u(tro)o O=(Ax1 +Bx0).e4m t A =O 4mt Bsin m2e O Bsinm.e m nn n2143 in m sin n l7

So Hhe Solutions b ecomes, umyt n Ctz)Bsin(nn e Now the mot general Solutton is given by the Suprpo si hon pinciple.ire uCtix= Cn e Sinnn Where en is an rloritrarr n(t)eumt & wnlsin uax ionstont amd Un (0)- e OKh -e- 1 tere wm(m 1 Sin mn.l sin 1 S0, sin n nn a) tha femehon 4t fimetion n= the nitial Condihon ulon) apply Cn Now Ent egeating weget Cn sinnn with Sinnnn after mulhplying xp XULMIS (K 2sinnicht n 2 wpxnuis.o

n sinnandk en= sinnn |-2 Cn eos nnx 1 en