Exponentially Fitted Dissipative Numerov-type Methods

4 Exponentially Fitted Dissipative Numerov-type Methods. - Simos and Williams have developed an exponentially fitted fourth algebraic order method which has the form  

In order that the method (101) integrates exactly any linear combination of the fimctions x, x, x , x, exp( vx) the following system of equations is obtained  

For the trigonometrically fitted case a replacement of w by iw in the third equation of the system of equations (102) must be done. Taking real and imaginary parts gives two relations for z. The real part gives  

If we eliminate z between the above relations we obtain a linear relation between the c, values  

The values of the coefficients co, Ci and C2 are obtained by the solution of the system of the two equations of the system of equations (102) and (103). After the determination of the coefficients cq, ci and C2 the eoeffieient z ean be defined from the equations given before equation (103). The finite differenee method will now integrate exaetly any eombination of the fimction set 1, X, x, x x , x eosvx, sinux. For small values of v the above formulae are subject to heavy caneellations. In this ease Taylor series expansions must be used. The local truncation error is O(h ). 

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