Abstract
Among the most popular methods for the solution of the Initial Value Problem are the Runge–Kutta (RK) pairs. These methods can be derived by solving a system of nonlinear equations after admitting various simplifying assumptions. The more simplifying assumptions we consider the more we narrow the solution space. In [1] Tsitouras presented an algorithm for the construction of Runge–Kutta pairs of orders 5 and 4 satisfying only the so called “first column simplifying assumption”. In [2] Famelis and Tsitouras have studied the ability of Differential Evolution techniques to find solutions satisfying all the order conditions needed for the derivation of orders 5 and 4 pairs. In this work we propose an modification on the Differential Evolution strategy for the same problem. The current study will be a guide to the construction of other classes of RK that have not been presented in the literature.
Highlights
We consider the numerical solution of the non-stiff initial value problem, y = f (x, y), y(x0) = y0 ∈ IRm, x ≥ x0 (1)where the function f : IR × IRm → IRm is assumed to be as smooth as necessary
[2] Famelis and Tsitouras have studied the ability of Differential Evolution techniques to find solutions satisfying all the order conditions needed for the derivation of orders 5 and 4 pairs
In this work we propose an modification on the Differential Evolution strategy for the same problem
Summary
Yn+1 = yn + hn bj f j and yn+1 = yn + hn b j f j, j=1 j=1 where i−1 fi = f (xn + cihn, yn + hn ai j f j), i = 1, 2, · · · , s As in both formulae ci, ai j are common the coefficients bj, b j define the (p−1)−th and p−th order approximations respectively. The coefficients of such a pair of methods can be presented in a matrix-array form using the following Butcher Tableau [3, 4]. Instead nonlinear optimizers based to stochastic direct search seem to work very efficiently
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