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Numerical Methods: Concepts
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
In some cases, the order of the method, and hence m in equation (179.1), will be unknown. The Richardson extrapolation method may still be used, by either estimating m numerically, or by using the Shanks transformation. The Shanks transformation uses three successive terms of the form An=A∞+αhn to estimate A∞ via A∞=An+1An−1−An2An+1+An−1−2An.
Fitting models to data
Published in Victor A. Bloomfield, Using R for Numerical Analysis in Science and Engineering, 2018
The Shanks transformation is used to accelerate the convergence of a slowly convergent series. In many common cases, the true value of the series, S, can be represented as the nth partial sum plus an “error” term decreasing geometrically with n: S = S +Cλn. Manipulation of this equation shows that
Generalized decomposition method: Applications to nonlinear oscillator and MHD fluid flow past cone/wedge geometries
Published in Numerical Heat Transfer, Part B: Fundamentals, 2020
Yasir Nawaz, Muhammad Shoaib Arif
HAM is based on homotopy in topology which includes auxiliary parameter that provides a convenient way to adjust and control the convergence region of solution series. One of the advantages for the use of HAM is that it provides great freedom to choose so-called auxiliary linear operator. In ref. [13], Ariel provided the extended solution of the problem that arises in steady, incompressible, viscous fluid past a radially stretching sheet. The considered problem was boundary value problem that has been reduced from the Navier Stoke equation which models the mentioned flow problem. The solution was found by HPM that was extended in the work. It was shown that the solution can be accelerated by applying Shanks’ transformation. In ref. [14], the problem of fluid flow past a wedge and cone was formed and modeled by partial differential equations and Cattaneo-Christov heat flux was considered. Also it has been shown in ref. [14] that the thermal and momentum boundary layers are different for a cone and a wedge. The effects of dimensionless parameters on velocity and temperature profiles were shown by drawing graphs. It has been shown that the rate of heat transfer in the flow past a cone was more than the rate of heat transfer that was considered to past a wedge. Further work on flow problems can be seen in refs. [14–19]. Some of the work on Chebyshev finite difference method can be seen in ref. [20] and some of the work on supersonic boundary layers can be seen in refs. [21, 22].
An Improved Restarted Adomian-based Solution for the Minimum Reflux Ratio of Multicomponent Distillation Columns
Published in Indian Chemical Engineer, 2018
M. Danish, M. Mubashshir, S. Zaidi
The Shanks transform is used to accelerate the convergence of a slowly convergent series [41]. The Shanks transformation of a sequence {U1, U2, … , Un} is defined as follows:To further accelerate the convergence, the Shanks transformation may be applied repeatedly, that is, and so on.