Approximating the Kohlrausch function by sums of exponentials (Q2865145)

For a fixed \(\beta\in (0,1)\), the Kohlrausch function \(\exp(-t^{\beta})\) has the Laplace transform representation NEWLINE\[NEWLINE \exp(-t^{\beta}) = \int_0^{\infty}\phi(\beta, p)\exp(-pt)\,dp, \quad t\geq 0, NEWLINE\]NEWLINE where \(\phi(\beta, p)\) can be written using some integral representations. In this paper, a constructive procedure for the approximation of the Kohlrausch function by sums of exponentials is proposed. Nonuniform grids on \((0, \infty)\) are considered for reducing the number of exponentials, and midpoint estimates for the mean value points are utilized. Several numerical examples for different values of \(\beta\) are given.