Abstract: In this paper, the fifth order Improved Runge-Kutta method (IRK5) that uses just five function evaluations per step is developed. The method proposed here are derived with only five stages which results in lower number of function evaluations. Therefore, IRK5 has a lower computational cost than the classical fifth order Runge-Kutta method (RK5). Here, the order conditions of the metho...

 In this paper, we investigate the evolution of supercontinuum and femtosecond optical pulses generation through square lattice elliptical-core photonic crystal fiber (PCF) at 1550 nm by using both full-vector multipole method (M.P.M) and novel concrete algorithms: symmetric  split-step Fourier (SSF) and  fourth order Runge Kutta (RK4) which is an accurate method to solve the general  nonlinear...

In this paper we have been discussed the numerical technical solutions for some undamped force vibrations of a spring problem using Runge-Kutta fourth order classical method and Eulers Modified Method. These two methods are very well known in physical mathematical sciences.

A new implementation of boundary condition based on the half-covolume and bounce-back rule for the non-equilibrium distribution function for the finite volume LBM is proposed here. The numerical simulation results for the expansion channel flow and driven cavity problem indicate that this method is workable for arbitrary meshes. In addition, the fourth order Runge–Kutta scheme is found to be a ...

We present a model for an SIR epidemic in a population consisting of two components—locals and migrants. We identify three equilibrium points and we analyse the stability of the disease free equilibrium. Then we apply optimal control theory to find an optimal vaccination strategy for this 2-group population in a very simple form. Finally we support our analysis by numerical simulation using the...

A system of ordinary differential equations, which describes the interaction of HIV and T -cells in the immune system is utilized, and optimal controls representing drug treatment strategies of this model are explored. Two types of treatments are used, and existence and uniqueness results for the optimal control pair are established. The optimality system is derived and then solved numerically ...

In the present paper, the modified Runge-Kutta method is constructed, and it is proved that the modified Runge-Kutta method preserves the order of accuracy of the original one. The necessary and sufficient conditions under which the modified Runge-Kutta methods with the variable mesh are asymptotically stable are given. As a result, the θ-methods with 1 2 ≤ θ ≤ 1, the odd stage Gauss-Legendre m...

This part is concerned with the numerical solution of initial value problems for systems of ordinary differential equations. We will introduce the most basic one-step methods, beginning with the most basic Euler scheme, and working up to the extremely popular Runge–Kutta fourth order method that can be successfully employed in most situations. We end with a brief discussion of stiff differentia...

In this work, the optimization for a radiative-convective annular fin of arbitrary profile with base wall thermal resistances is considered. A fourth order Runge-Kutta method is used to solve the associated non-linear governing equations. Further differentiations yield the optimum heat transfer and the optimum fin dimensions. To facilitate the thermal design, design charts for optimum dimension...

The flow of a viscous incompressible fluid between two parallel plates due to the normal motion of the plates is investigated. The unsteady Navier-Stokes equations are reduced to a nonlinear fourthorder differential equation by using similarity solutions. Homotopy analysis method HAM is used to solve this nonlinear equation analytically. The convergence of the obtained series solution is carefu...