The main purpose of this paper is to review the work on Runge-Kutta methods at the University of Toronto during the period 1963 to the present (1996). To provide some background, brief mention is also made of related work on the numerical solution of ordinary diierential equations, but, with just a few exceptions, speciic references are given only if the referenced material has a direct bearing...

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...

A new selection is made of the most practical of the many explicit Runge-Kutta formulas of order 4 which have been proposed. A new formula is considered, formulas are modified to improve their quality and efficiency in agreement with improved understanding of the issues, and formulas are derived which permit interpolation. It is possible to do a lot better than the pair of Fehlberg currently re...

mhd boundary layer flow of two phase model nanofluid over a vertical plate is investigated both analytically and numerically. a system of governing nonlinear partial differential equations is converted into a set of nonlinear ordinary differential equations by suitable similarity transformations and then solved analytically using homotopy analysis method and numerically by the fourth order rung...

in this paper, the chebyshev spectral collocation method(cscm) for one-dimensional linear hyperbolic telegraph equation is presented. chebyshev spectral collocation method have become very useful in providing highly accurate solutions to partial differential equations. a straightforward implementation of these methods involves the use of spectral differentiation matrices. firstly, we transform ...

Classical and new numerical schemes are generated using evolutionary computing. Differential Evolution is used to find the coefficients of finite difference approximations of function derivatives, and of single and multi‐ step integration methods. The coefficients are reverse engineered based on samples from a target function and its derivative used for training. The Runge‐Kutta schemes are tra...

THEORY AND IMPLEMENTATION OF NUMERICAL METHODS BASED ON RUNGE-KUTTA INTEGRATION FOR SOLVING OPTIMAL CONTROL PROBLEMS

We present new symmetric fourth and sixth-order symplectic Partitioned Runge{ Kutta and Runge{Kutta{Nystrr om methods. We studied compositions using several extra stages, optimising the eeciency. An eeective error, E f , is deened and an extensive search is carried out using the extra parameters. The new methods have smaller values of E f than other methods found in the literature. When applied...