Page 14 - Laplace transform numerical inversion

Bruno Josso & Leif Larsen: Laplace transform numerical inversion - June 2012 - p 12/18
In practice, the series expansion given in(19) is performed on a limited number of terms K . In
theory, the bigger the integer K , the more accurate the inversion process yielding to f (t). In the
current implementation the values retained corresponded to the Stehfest number used in section
(4.1.1 ). On the other hand, when the number of coefficients K becomes too big, numerical
approximation problems occur. Again, an extra implementation effort should be made in order to
get a very accurate reconstruction linked to a big number of weighing coefficients. The situation
is strictly similar here as it was with Gaver-Stehfest algorithm.
4.1.3.1 Legendre polynomials
A description of this algorithm can be found in[1] . For Legendre polynomials, weighing coef-
ficients αk decrease slowly. Therefore, one has to choose K big enough to maximise inversion
accuracy. Comparatively, one recalls that for Gaver-Stehfest algorithm, a standard Stehfest num-
ber value is typically N = 8.
In Figure 10,one can see the reconstruction error obtained when inverting ˆ
F (p), the Laplace
transform of function f (t) = e−µt by using Legendre algorithm. It oscillates, especially at early
time, but error magnitude remains fairly stable. Unfortunately, the way the reconstructed function
converges towards the exact solution still causes problem. Indeed, one can see that at some
point, the exponential function becomes negative.
4.1.3.2 Chebyshev polynomials
A description of this algorithm can be found in[7] . In Figure 11,one can see the reconstruc-
tion error obtained when inverting ˆ
F (p), the Laplace transform of function f (t) = e−µt by using
Chebyshev algorithm. It oscillates and increases with time. Inversion accuracy is one of the
worst among the tested algorithms. One can also see that the reconstructed function eventually
becomes negative.
4.1.3.3 Laguerre polynomials
A description of this algorithm can be found in[7] . In Figure 12,one can see the reconstruction
error obtained when inverting ˆ
F (p), the Laplace transform of function f (t) = e− µt by using Laguerre
algorithm. It oscillates but decreases with time. Unfortunately the inverted function occasionally
becomes negative.
For further investigation, it should be mentioned that Laguerre polynomials algorithm can be
extended by considering a complex Laplace argument p ∈ C, by opposition to the real complex
argument p ∈ R scheme tested here. This latter strategy is the one retained by Weeks[ 8] .
4.1.3.4 Conclusion on polynomials series expansion algorithms
As could be seen from section( 4.1.3.1)to(4.1.3.3 )none of polynomials series expansion
algorithms could give a satisfying inversion result. They do not perform better than the standard
Gaver-Stehfest reference method. Furthermore, inverted test function always become negative
at some points whereas it should always remain strictly positive.