In this paper we present the recent development in the numerical integration of the
Schrödinger equation and related systems of ordinary differential equations with oscillatory …

The standard monograph in this area is the book Exponential fitting by Ixaru and Vanden
Berghe (Kluwer, Boston-Dordrecht-London, 2004) but a fresh look on things is necessary …

Second order boundary value problems are solved by means of exponentially-fitted
Numerov methods. These methods, which depend on a parameter, can be constructed …

In this paper we study the connection between:(i) closed Newton-Cotes formulae of high
order,(ii) trigonometrically-fitted and exponentially-fitted differential methods,(iii) symplectic …

Runge–Kutta–Nyström pair of orders 7 (5) using six stages per step have been discovered
very recently. Here we modify four of its weights. The resulting method integrates exactly the …

When a differential equation exhibits chaos, stiffness, damping and/or oscillation in its
solution component, such a differential equation is termed rigid. Over the years, solving such …

One of the well-known schemes for the direct numerical integration of second-order initial-
value problems is due to Falkner. This paper focuses on the construction of a family of …

It is the purpose of this paper to derive two-step hybrid methods for y ″= f (x, y), with
oscillatory or periodic solutions, specially tuned to the behaviour of the solution, through the …

In this paper we introduce a new explicit hybrid Numerov-type method. This method is of
fourth algebraic order and has phase-lag and its first two derivatives equal to zero. We …

In this paper, a new scheme based on the exponential fitting technique is presented for
solving the nonlinear time-fractional Swift-Hohenberg equation, where the first and second …