Now as k takes on the values 2-» m-1, equation (9) will produce m-2 equations 



in the m unknowns (z,,z . . . z ). In order to solve this system, two additional 

 I 2 m 



equations are needed. These additional equations are obtained through 



utilization of an appropriate set of boundary conditions. There are several 



reasonable boundary conditions which may be selected. Bhattacharyya (1969) 



used the assumption that the second derivative is zero at the points (x,,x ). 

 1 I m 



In the present development we will use the boundary conditions proposed by 

 Pennington (1965). These conditions state that the second derivatives at the 

 two end points are a linear extrapolation of those associated with the adjoining 

 interval. Thus, from equation (1) we have 



Vi = f ( Vi } = \ + v (10) 



For k = 2 this reduces to 



-x + z 2 {t + t) -x =0 ' 



(ii) 



1 x 2 



and for k = m-1 we have 



d m _ 2 - 1 \ d m - 2 d m V d m-l 



Combining equations (11) and (12) with the m-2 equations obtained from (9) for 



k = 2, 3 . . . m-1, yields a set of m equations in m unknowns (z.,z . . .z ). 



1 2. m 



With the exception of one additional element in the first and last row, this set 

 of equations forms a tridiagonal matrix which is easily solved using the standard 

 Gauss reduction method. In matrix form, the set is given by AZ = B where 

 A is mxm, Z is mxl and B is mxl thus, 



