This paper presents a new class of fractional order Runge–Kutta (FORK) methods for numerically approximating the solution differential equations (FDEs). We construct explicit and implicit FORK FDEs by using Caputo generalized Taylor series formula. Due to dependence derivatives on fixed base point, in proposed method, we had modify right-hand side given equation all steps methods. Some coeffici...

The construction of s-stage explicit twoand three-step peer methods of order p = 2s and p = 3s is considered for the solution of non-stiff second order initial value problems where the right-hand side does not depend on y. By additional conditions superconvergence of order p + 1 can be achieved. Further criteria for good methods are large stability intervals and small error constants. Numerical...

In this paper, a general and detailed study of linear stability of Runge–Kutta–Nyström (RKN) methods is given. In the case that arbitrarily stiff problems are integrated, we establish a condition that RKN methods must satisfy so that a uniform bound for stability can be achieved. This condition is not satisfied by any method in the literature. Therefore, a stable method is constructed and some ...

Article history: Received 31 January 2008 Received in revised form 18 June 2008 Accepted 23 July 2008 Available online 19 August 2008

High order energy-preserving methods for Hamiltonian systems are presented. For this aim, an energy-preserving condition of continuous stage Runge–Kutta methods is proved. Order conditions are simplified and parallelizable conditions are also given. The computational cost of our high order methods is comparable to that of the average vector field method of order two.

In this paper we introduce and study rational slice monogenic functions. After proving a decomposition theorem for such functions, we are able to prove the Runge approximation theorem for slice monogenic functions. We then show how a similar argument can be used to obtain an analogue of the Runge approximation theorem in the slice regular setting.