In this paper, we analyze the stability of the fourth order Runge-Kutta method for integrating semi-discrete approximations of time-dependent partial differential equations. Our study focuses on linear problems and covers general semi-bounded spatial discretizations. A counter example is given to show that the classical four-stage fourth order Runge-Kutta method can not preserve the one-step st...

This study utilised positive Lyapunov exponents’ criteria to develop chaos diagram on the parameters space of 4-dimensional harmonically excited vibration absorber control Duffing’s Oscillator. Relevant simulations were effected by choice combination of constant step Runge-Kutta methods and Grahm Schmidt Orthogonal rules. Simulations of 4-dimensional hyper-chaotic models of modified Lorenz and ...

We investigate conservative properties of Runge-Kutta methods for Hamiltonian PDEs. It is shown that multi-symplecitic Runge-Kutta methods preserve precisely norm square conservation law. Based on the study of accuracy of Runge-Kutta methods applied to ordinary and partial differential equations, we present some results on the numerical accuracy of conservation laws of energy and momentum for H...

The time-dependent diffusion equation is studied, where the coefficient itself depends simultaneously on space and time. First, a family of novel, nontrivial analytical solutions constructed in one dimension with classical self-similar Ansatz. Then, solution for two different sets parameters reproduced by 18 explicit numerical methods. Fourteen these time integrators are recent unconditionally ...

Strong stability preserving (SSP) high order Runge–Kutta time discretizations were developed for use with semi-discrete method of lines approximations of hyperbolic partial differential equations, and have proven useful in many other applications. These high order time discretization methods preserve the strong stability properties of first order explicit Euler time stepping. In this paper we a...

In this paper, based on the classical Fourth-Order Runge-Kutta method, the modified FourthOrder Runge-Kutta method is presented for solving nonlinear vibration of axially travelling string system, that is to solve time varying and nonlinear differential equations. The classical Fourth-Order Runge-Kutta method can only be used to solve first-order linear differential equations. Its main idea is ...

Enormous amount of real time robot arm research work is still being carried out in different aspects, especially on dynamics of robotic motion and their governing equations. Taha [5] discussed the dynamics of robot arm problems. Research in this field is still on-going and its applications are massive. This is due to its nature of extending accuracy in order to determine approximate solutions a...

In this work we consider Symplectic Runge Kutta Nyström methods with five stages. A new fourth algebraic order method with phase-lag order eight is presented. Also the symplectic Runge Kutta Nyström of Calvo and Sanz Serna with five stages and fourth order is modified to produce a phase-fitted method. We apply the new methods on several Hamiltonian systems and on the computation of the eigenval...

The cubic spline interpolation method, the Runge–Kutta method, and the Newton–Raphson method are extended to dual versions (developed in the context of dual numbers). This extension allows the calculation of the derivatives of complicated compositions of functions which are not necessarily defined by a closed form expression. The code for the algorithms has been written in Fortran and some exam...