D • CONDUCTION OF HEAT 



(Xn — Ki) (jj PlCl(plm<Plndxdy + // P2C2<P2m<p2ndxdyj 

 Gi Gi 



[[if <Plm d(pin\ , / d(p2m ^ d(p2n \\ , 



— <Plm 



the integrands on the exterior boundary vanishing identically as in Eq. 

 2-3, n being a normal to the boundary. 

 Hence the function set 



$n = Vpc (fn (n = 1, 2, . . .) 

 will be orthogonal if 



, difln J d(p2n ^ -n ro K\ 



<P\n = (fin ki -r— = K2 -^ on Ti (2-5) 



Now, the above conditions represent the physical requirements of con- 

 tinuity of temperature and heat flux through any internal boundary, and 

 therefore the above relations represent the boundary conditions which 

 must be used in internal boundaries separating domains of different 

 mediums. It is seen, therefore, that by the imposition of the physical 

 requirements, the orthogonality property of the function set -y/pc ^„ is 

 always automatically assured. 



A particular solution is then 



Qnix, y, t) = <pn{x, y)e-^'^ 



and, since the system is linear, the complete solution is a linear sum of 

 particular solutions: 



e{x, y,t) = Yj ^"^" = E ^"'^"^~'"' (2-6) 



71= 1 n = 1 



e{x, y,0) = Yj ^n^Pnix, y) = Fix, y) 



n = l 



The amplitudes An still remain to be determined. 



3. Now, a further theorem [8, Vol. 1, p. 319] states that if F{x, y) is a 

 continuous function in the domain G with continuous first and second 

 derivatives and in addition satisfies the boundary conditions of the problem, 

 then F{x, y) can be expanded in an absolutely and uniformly convergent 



< 258 ) 



