The aim of this paper is to design a new family of numerical methods of arbitrarily high order for systems of rst-order diierential equations which are to be termed pseudo two-step Runge-Kutta methods. By using collocation techniques, we can obtain an arbitrarily high-order stable pseudo two-step Runge-Kutta method with any desired number of implicit stages in retaining the two-step nature. In ...

We describe a new method for numerical integration, dubbed bandlimited collocation implicit Runge–Kutta (BLC-IRK), and compare its efficiency in propagating orbits to existing techniques commonly used in Astrodynamics. The BLC-IRK scheme uses generalized Gaussian quadratures for bandlimited functions. This new method allows us to use significantly fewer force function evaluations than explicit ...

In this paper, using a model transformation approach a system of linear delay differential equations (DDEs) with multiple delays is converted to a non-delayed initial value problem. The variational iteration method (VIM) is then applied to obtain the approximate analytical solutions. Numerical results are given for several examples involving scalar and second order systems. Comparisons with the...

*Correspondence: [email protected] Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China Abstract Uncertain differential equations have been widely applied to many fields especially to uncertain finance. Unfortunately, we cannot always get the analytic solution of uncertain differential equations. Early researchers have put up a numerical method based on t...

8. First-Order Equations: Numerical Methods 8.1. Numerical Approximations 2 8.2. Explicit and Implicit Euler Methods 3 8.3. Explicit One-Step Methods Based on Taylor Approximation 4 8.3.1. Explicit Euler Method Revisited 4 8.3.2. Local and Global Errors 4 8.3.3. Higher-Order Taylor-Based Methods (not covered) 5 8.4. Explicit One-Step Methods Based on Quadrature 6 8.4.1. Explicit Euler Method Re...

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 ...

A new diagonally implicit Runge-Kutta-Nyström (RKN) method is developed for the integration of initial-value problems for second-order ordinary differential equations possessing oscillatory solutions. Presented is a method which is three-stage fourth-order with dispersive order six and 'small' principal local truncation error terms and dissipation constant. The analysis of phase-lag, dissipatio...

A Runge–Kutta method takes small time steps, to approximate the solution to an initial value problem. How accurate is this approximation? If the error is asymptotically proportional to hp, where h is the stepsize, the Runge–Kutta method is said to have “order” p. To find p, write the exact solution, after a single time-step, as a Taylor series, and compare with the Taylor series for the approxi...