A new composite Runge–Kutta (RK) method is proposed for semilinear partial differential equations such as Korteweg–de Vries, nonlinear Schrödinger, Kadomtsev– Petviashvili (KP), Kuramoto–Sivashinsky (KS), Cahn–Hilliard, and others having high-order derivatives in the linear term. The method uses Fourier collocation and the classical fourth-order RK method, except for the stiff linear modes, whi...

Derivative-free iterative methods are useful to approximate the numerical solutions when given function lacks explicit derivative information or derivatives too expensive compute. Exploring convergence properties of such is crucial in their development. The behavior approaches and determining practical applicability require conducting local as well semi-local analysis. In this study, we explore...

The main di culty in the implementation of most standard implicit Runge-Kutta (IRK) methods applied to (sti ) ordinary di erential equations (ODE's) is to e ciently solve the nonlinear system of equations. In this article we propose the use of a preconditioner whose decomposition cost for a parallel implementation is equivalent to the cost for the implicit Euler method. The preconditioner is ba...

We present the equations of condition up to sixth order for Runge-Kutta (RK) methods, when integrating scalar autonomous problems. Two RK pairs of orders 5(4) are derived. The first at a cost of only five stages per step, while the other having an extremely small principal truncation error. Numerical tests show the superiority of the new pairs over traditional ones.

For a graph G, the k-colour Ramsey number Rk(G) is least integer N such that every k-colouring of edges complete KN contains monochromatic copy G. Let Cn denote cycle on n vertices. We show for fixed k?2 and odd sufficiently large,Rk(Cn)=2k?1(n?1)+1. This resolves conjecture Bondy Erd?s large n. The proof analytic in nature, first step which to use regularity method relate this problem theory o...