dx s 2 Q \ s s+Y/ 



(B7) 



Solving Equation B6 together with Equations B4 and B7 yields 



y = - l a - 



1 Q B \ e" qX 1 Q B e" qX 



o 2 Q / qs 2 Q q(s + y) 



(B8) 



i. 2 s 

 where q = — 



E 



3. The inverse Laplace transform of the first term in Equation 

 found to be (Appendix A) 



is 



1 / 1 Q b\/„ fit -x 2 /4Et . x 

 y = - I a - o" 7T" I I 2 V — e ~ x erf c 



(B9) 



The second term is evaluated by applying Duhamel's theorem (Carslaw and Jaeger 

 1959, p. 301) which reads 



( T )f 2 (t - x)d T 



\ = L{f L (t)} L{f 2 (t)} 



(BIO) 



in which L{} represents the Laplace transform operation. The second term of 

 Equation B8 yields, after some rearranging, 



2 

 y = 



n i — & 



e 2 2 I, r 2 



YC -x /4eC 



dC 



(Bll) 



Accordingly, the complete solution is 



B2 



