=   (x  x^{}) on domain 0 < x < L  (4)  
k_{i} + h_{i}G  =  0; i = 1 or 2  (5) 
Case  Range  1D steady GF; units are meters. 
X11  x > x^{}  x^{}(1  x/L) 
x < x^{}  x(1  x^{}/L)  
X12  x > x^{}  x^{} 
x < x^{}  x  
X13  x > x^{}  x^{}[1  B_{2}(x/L)/(1 + B_{2})] 
x < x^{}  x[1  B_{2}(x^{}/L)/(1 + B_{2})]  
X21  x > x^{}  L  x 
x < x^{}  L  x^{}  
X22^{ * }  x > x^{}  ( x^{2} + (x)^{2})/(2L)  x + L/3 
x < x^{}  ( x^{2} + (x^{2})^{2})/(2L)  x^{} + L/3  
X23  x > x^{}  L(1 + 1/B_{2}  x/L) 
x < x^{}  L(1 + 1/B_{2}  x^{}/L)  
X31  x > x^{}  (B_{1}x^{}  B_{1}x^{}x/L + L  x)/(1 + B_{1}) 
x < x^{}  (B_{1}x  B_{1}xx^{}/L + L  x^{})/(1 + B_{1})  
X32  x > x^{}  L(1/B_{1} + x^{}/L) 
x < x^{}  L(1/B_{1} + x/L)  
X33  x > x^{}  B_{1}B_{2}x^{} + B_{1}x^{}  B_{1}B_{2}x^{}x/L  B_{2}x + B_{2}L + L 
÷ (B_{1}B_{2} + B_{1} + B_{2})  
x < x^{}  B_{1}B_{2}x + B_{1}x  B_{1}B_{2}x^{}x/L  B_{2}x^{} + B_{2}L + L  
÷ (B_{1}B_{2} + B_{1} + B_{2}) 
G_{1D}(x, x^{}) =  (6) 
X_{n} ^{' '} (x) + X_{n}(x) = 0.  (7) 
Case  X_{n}(x)  N_{x}^{1}  
X11  sin(x)  2/L  
X12  sin(x)  2/L  
X13  sin(x)  2/L  
X21  cos(x)  2/L  
X22 



X23  cos(x)  2/L  
X31  sin((L  x))  2/L  
X32  cos((L  x))  2/L  
X33  Lcos(x) + (h_{1}L/k)sin(x)  2/L  
note:  = (L)^{2} + (h_{i}L/k)^{2} ÷ (L)^{2} + (h_{i}L/k)^{2} + h_{i}L/k  
= ÷ (L)^{2} + (h_{1}L/k)^{2} + (h_{1}L/k) 
Case  Eigencondition  Eigenvalues 
X11  sin(L) = 0  , n = 1, 2,... 
X12  cos(L) = 0  , n = 1, 2,... 
X13  Lcot(L) =  h_{2}L/k  
X21  cos(L) = 0  , n = 1, 2,... 
X22  sin(L) = 0  , n = 0, 1, 2,... 
X23  Ltan(L) = h_{2}L/k  
X31  Lcot(L) =  h_{1}L/k  
X32  Ltan(L) = h_{1}L/k  
X33  tan(L) = [(h_{1} + h_{2})/k]/[  h_{1}h_{2}k^{2}] 