D • CONDUCTION OF HEAT 



only two nonlinear problems: (1) The problem of surface melting as a 

 factor in erosion and (2) the problem of heat conduction in a material 

 with thermal properties ik and c) which vary with the temperature. 

 Finally, among the linear problems we shall discuss primarily those which 

 involve only one space coordinate. 



In order to reduce the boundary conditions to homogeneous form we 

 make the transformation 







^ - 1 



Then the linear problems to be discussed are included in the following 

 formulation : 



d_ 



dx 



\ dxj dy \ by) 



pc 



dQ 



dt 



dQ 



+ 



6 = 



mG 



on r 



(2-la) 



dy, 



^ m 



dn ' k 



e = H{x, y) at i = J 



where s is the arc length on V measured from a reference position and 

 k, p, and c are functions only of position. Detailed discussion of the 

 mathematical system represented by Eq. 2-la is given in texts dealing 

 generally with the theory of partial differential equations. Here we merely 

 summarize what may be considered as the most important characteristics 

 of the system, particularly so in relation to analysis of heat conduction in 

 solid mediums. 



1. It is always possible to eliminate the time either by separation of 

 variables or by the Laplace transform [6]. Thus by the method of sepa- 

 ration of variables, if we write 



e = <p{x)m 



then 



pCcp 



d_ 



dx 



dcp 

 dx 



+ 



dy 



(4:) 



1# 



\p dt 



-X2 



X must be independent of x, y, and t, i.e. a constant, since the left- 

 hand side is independent of t while the right-hand side is independent of 

 X and y. Moreover, it can be shown that X^ is positive [8, Vol. 1, p. 252], 

 We obtain firstly 



and secondly the eigenvalue problem 



d_ 



dx 



(4r)+f,H:)+^v.=o i„G 



k ^+ h<p = on r 

 dn 



(2-2) 



< 256 ) 



