How to solve this Laplace integral for an averaging function in an iterative numerical (finite difference) model?
The variable of the weighted mean value zm would convey a more expressive power if denoted as zmean. The RHS integral in formula (14) of your reference is a &quot;g(x)-weighted average of f(x) on the interval [0, t]&quot;, see http://www.sosmath.com/calculus/integ/integ04/integ04.html. In formula (14), f(s) is z2(s) and g(s) = sn. The integral in (14) is not a Laplace transform, but the extended g(x)-weighted mean value, and s is not a dual variable of the Laplace transform, but a dummy variable of integration.
A variable n in (14) is not an index but the exponent of the power function g(s) = sn, and t is also not indexed, but raised to a power n+1. From the fragment you presented in your question, and if they indeed use Newark's method, it does not follow that the authors seek an iterative solution for zm; rather, they use the weighted means as per formula (14) to find the solution for x, x_dotted.