The Zakian Method

The Laplace transform of fit) is denoted as F(s), given in Eq. 9.1, and is rewritten again here for convenience 

we assume that f(t) is integrable (piecewise continuous) and is of exponential order (t, that is, f(t) I Mexpfwor). 

Recalling the Dirac s delta fimction, defined in Eq. 8.10, and if we replace of Eq. 8.10 by unity, we obtain 

if t lies between 0 and T, we can replace the above integration range to [0, r] without changing the value of the integral that is, 

Using the property of the Dirac delta function, we can write 

---

Table 93 Comparison Between the Numerical Inverse Obtained by the Zakian Method and the Analytical Solution for Example 9.16...

To show an example of numerical inversion by the Zakian method, we take the function treated in Example 9.12... 

Table 9.4 shows the comparison between the analytical and numerical inverse. The error increases with an increase in time, suggesting that the Zakian method does not work well with oscillating functions. 

Table 9.5 shows the numerical inverse obtained by the Zakian method, and the performance of the routine is even worse than the previous example. 

In conclusion, the Zakian numerical technique yields good results only when the inverse does not exhibit oscillating behavior or when it increases exponentially with time. The Fourier approximation method presented in the next section will remedy the disadvantages encountered by the Zakian method. 

Taking the Examples 9.16, 9.17, and 9.18, we compute the numerical inverse and show the results in the Tables 9.6, 9.7, and 9.8, where it is seen that the Fourier series approximation is a better approximation than the Zakian method. 

The Fourier series approximation is a better method to handle oscillating functions, but it requires more computation time than the Zakian method. With the advent of high speed personal computers, this is not regarded as a serious disadvantage. 

Previous sections dealt with the analytical development of Laplace transform and the inversion process. The method of residues is popular in the inversion of Laplace transforms for many applications in chemical engineering. However, there are cases where the Laplace transform functions are very complicated and for these cases the inversion of Laplace transforms can be more effectively done via a numerical procedure. This section will deal with two numerical methods of inverting Laplace transforms. One was developed by Zakian (1969), and the other method uses a Fourier series approximation (Crump 1976). Interested readers may also wish to perform transforms using a symbolic algebra language such as Maple (Heck 1993). 

To illustrate problems for transcendental functions in the numerical inversion method of Zakian, we take the function... 

---