Question

An infinitely long copper wire coated insulting material on its surface is assumed perfectly insulated laterally. Thus heat in the wire flows in x-direction only.- In this case the temperature of the wire can be modeled by 1-D heat equation: au au ax at Assume that initially the temperature distribution of the wire is u(x,0)-T for x<a A. Please show that the solution of 1-D heat equation in this case is ulxt)-o [Acos(px)Bsin (px)]eptdP where A and B are the coefficients to be determined by the initial condition, c is the thermal diffusivity B. According to the given initial condition, please find A and B in (A). C. Please write down the entire solution in integral form the satisfying (B)

Answer 1

A. Led ula.) XT) 2 C X X dap 2 2 d2 X Fume" 6 In the above ed leFt hand ide is a tcnly nhereas 0nly o, thay must be aonstant. night hamd ride is a Fune" 2 -2 X 1- d T 2 2 T 구뚜 =-아at 21 2 2 LA=e= Conat7 T= A'EC

XP da tx 0 X= A,eoKpa + 8, inpa .ula,)=x)T) (A,n' coapa t 6,f'sint)e =(AcoapatBainp)efn-R-8 referation eonsd. - We tooh +he De the above nstu is valid Fon any values f ranging betreen o foo. Acon pat Boin 8. (reen ab 7 os pa+ Brin ipa +e 2 - e dp T P +8 2i Nw the term ba So the coeFFicient wil diverge at the ea in the ufper linit. indegnand must be 2eno. i.e + 2 2i B-Ai indegration beco mes The -ipn TPT PA -TA-ia T T Ae B=aT

(f'9 Acos pat Bain pr e T coath+aTinpaled C in T dp =inT dp - inT 2, 2 =e 2