quadraturesize = posint

This command computes a numerical discrete inverse Laplace transform using a method of Peter den Iseger involving only simple linear algebra and InverseFFT and $\mathrm{O}\left(M\right)$ evaluations of the function f.

For a function given by f, this command outputs a vector of M equally spaced numerical values of the inverse Laplace transform of $f\left(t\right)$, each Delta apart, starting at $t=0$. The algorithm is expected to be accurate for $f$ with a smooth inverse transform.

Intermediate calculations are done using hardware double precision floating-point numbers, so raising the value of $\mathrm{Digits}$ will not increase the precision of the results.

$\mathrm{with}\left(\mathrm{inttrans}\right)\:$

$\mathrm{invlaplace}\left(\frac{s^2}{{\left(s^2+2\right)}^{\frac{3}{2}}}\,16\,0.01\,'\mathrm{numeric}'\right)$

$\begin{array}{c}\left[\begin{array}{c}0.999999999865338\\ 0.999850003031420\\ 0.999400049950828\\ 0.998650253123779\\ 0.997600799979164\\ 0.996251952856572\\ 0.994604048973614\\ 0.992657500370408\\ 0.990412793824270\\ 0.987870490612488\\ ⋮\end{array}\right]\\ \hfill \text{16 element Vector[column]}\end{array}$

$F≔\mathrm{unapply}\left(\mathrm{invlaplace}\left(\frac{s^2}{{\left(s^2+2\right)}^{\frac{3}{2}}}\,s\,t\right)\,t\right)$

$F≔t\mapsto -\sqrt{2}\cdot t\cdot \mathrm{BesselJ}\left(1\,\sqrt{2}\cdot t\right)+\mathrm{BesselJ}\left(0\,\sqrt{2}\cdot t\right)$

$\mathrm{Vector}\left(16\,i\mapsto \mathrm{evalf}\left(F\left(\left(i-1\right)\cdot 0.01\right)\right)\right)$

$\begin{array}{c}\left[\begin{array}{c}1.\\ 0.9998500031\\ 0.9994000500\\ 0.9986502531\\ 0.9976007999\\ 0.9962519528\\ 0.9946040488\\ 0.9926575003\\ 0.9904127936\\ 0.9878704902\\ ⋮\end{array}\right]\\ \hfill \text{16 element Vector[column]}\end{array}$

$\mathrm{invlaplace}\left(s\mapsto \frac{\mathrm{\Gamma }\left(\frac{4}{3}\right)}{s^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\,16\,0.01\,'\mathrm{numeric}'\right)$

$\begin{array}{c}\left[\begin{array}{c}0.0199179741463930\\ 0.215443269930483\\ 0.271441691441147\\ 0.310723206544908\\ 0.341995154356477\\ 0.368403118253306\\ 0.391486733105780\\ 0.412128497784202\\ 0.430886903160949\\ 0.448140435725261\\ ⋮\end{array}\right]\\ \hfill \text{16 element Vector[column]}\end{array}$

den Iseger, Peter. "Numerical Transform Inversion Using Gaussian Quadrature. Probability" in The Engineering and Informational Sciences, Vol. 20, pp. 1-44, 2006.

The inttrans/invlaplace/numeric command was introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.