Question

6. A bar of length I siting on the x axis with on end at the origin is initially (t = 0) at a temperature u(3,0) = 0° From t= 0 on the ends are held at 20° (.eu (0,t) = u(lt) = 20°), find the temperature, u(a,0), for t > 0.

Answer 1

- Thermal conductivity dt = a de da2 2p2t The temperature function (n.) Satisfies the I-D heat equation where a ²-k 50 (Density) (specific heat) We have 3 possible solutions for uCat) (1) u(x, t) = CA, ePx +AePx) Az each (2) V(x, t) = (Ay Cospu + As sinpx) Abela (3) U(x, t) = ( A7 + As ) Ag Now at t-o u(a,0) ive when the temperature u depends only ona but not ont. c) AG & Q?p? dx2 The general solution is u=anth Given when a=0 u=200 ie (6=20 when x=l u=20 e 2o=al+20° So, la=0" Now I can write u(x,0)=20. which is initial temperature distribulion. increased When the temperature at A and B are reduced to 20°c the state is not more steady state. For this transient state the boundary conditions are. ci) u (2,0)=20° = f(n)= 20 , OLAL2O (ii) u (ot) = 20° , & to (iii) u (e,t)= 20 , & t o

Here we have have non zero boundary Values. In such cases the temp. tunction u(x, t) = u(x) + 44(x, t). Where us (x) is a solution of (I) involving only and satisfying boundary conditions (ii) and (ii). To find us Cx) the general solution is a xt bi Condition (1) when a=0 u=20 so, b = 20 Condition (in) when x=1 u=20 So, a l + 20 20 ore, a, o Us (x) = 20 To find u (r, t) = ulo,t)- Us (x). ut (x=0, t) = u(o,t) - Us (0) = 20-20 = 0 ut (x=1, t) = ull, t-Us (l) = 20-20=0 Ut(x, t0) = f(x) = u(x,0) - Us (x) = 20-20=0 Now, we have f(x=0, t) = 0 Suitable solution which satisfies boundary condition is 44(x, t) = (A cospx + Bsinpx) e a22+ So, Aē a?p?t=0 ) ea?p?t to A0 Thus Ut Cr, t) = B sin prea? p? Again ut (x = 1, t) = 0 So Brinpl é a 2p26=0 Here ea? p?t to Bto : Simpl= 0 = sinna 2 2 Now u(x, t) = B sin nax e a ?n??t

By superposition principle the most general solution is ut (ait) = { By sin nax ea Now Ut (m, t =0) = f(x) = 20,0 22 2 2 = 20 B sin pas = f(x) = 20 To find Bo, expand f(x) in a hal range fourier sine Series in the Interval (0, l=20). where bn = 2 / f(x) sin ny do We have Bn= bn. .. Bp = 2 FC_) Sin ngx ax 21 20 sin pas de com 4ro cont tout] Bn - 40 (1-(-10%); when n is even Bn = 0 ; when n is odd Br=80/17 ... The required most general solution is 4(vt) = & Teh (-1)") } sin pas a?q Hence u(x, t)= 20 + sin naye apat so, u(,0) = 207 3 tao sin noy (for t70). Пл N=1,3,5-- n. 1,3,5 A