The RK5GL3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on a combination of a fifth-order Runge-Kutta method and 3-point Gauss-Legendre quadrature. In this paper we describe an effective local error control algorithm for RK5GL3, which uses local extrapolation with an eighth-order Runge-Kutta method in tandem with RK5GL3, and a ...

By systematically altering the in-plane flight path and velocity of an aircraft and the rate at which the cable is deployed, the tip of a cable being towed behind an aircraft is shown to rendezvous with a ground-based surface location in minimum time, with minimum control effort required. This is achieved through optimal control, formulated as a nonlinear programming problem using the Legendre-...

Corresponding Author: Simon Liao The University of Winnipeg, Winnipeg, Manitoba, Canada, R3B 2E9, Canada Email: [email protected] Abstract: In this research, a numerical integration method is proposed to improve the computational accuracy of Legendre moments. To clarify the improved computation scheme, image reconstructions from higher order of Legendre moments, up to 240, are conducted. Wi...

We give the explicit time description of three following problems: dynamics and optimal dynamics for some important electromechanical system and Galerkin approximation for beam equation. All these problems are reduced to the problem of the solving of the systems of diierential equations with polynomial nonlinearities and with or without some constraints. The rst main part of our construction is...

We give the explicit time description of four the following problems: dynamics and optimal dynamics for some important electromechanical system, Galerkin approximation for beam equation, computations of Melnikov function for perturbed Hamiltonian systems. All these problems are reduced to the problem of the solving of the systems of diierential equations with polynomial nonlinearities and with ...

Gauss–Legendre quadrature rules are of considerable theoretical and practical interest because of their role in numerical integration and interpolation. In this paper, a series expansion for the zeros of the Legendre polynomials is constructed. In addition, a series expansion useful for the computation of the Gauss–Legendre weights is derived. Together, these two expansions provide a practical ...

The main aim of this article is to generalize the Legendre operational matrix to the fractional derivatives and implemented it to solve the nonlinear multi-order fractional differential equations. In this approach, a truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used. The main characteristic behind the approach using this technique is that...

in recent years, there has been greater attempt to find numerical solutions of differential equations using wavelet's methods. the following method is based on vector forms of haar-wavelet functions. in this paper, we will introduce one dimensional haar-wavelet functions and the haar-wavelet operational matrices of the fractional order integration. also the haar-wavelet operational matrice...

In this article, a shifted Legendre tau method is introduced to get a direct solution technique for solving multi-order fractional differential equations (FDEs) with constant coefficients subject to multi-point boundary conditions. The fractional derivative is described in the Caputo sense. Also, this article reports a systematic quadrature tau method for numerically solving multi-point boundar...

In this paper, we develop a framework to obtain approximate numerical solutions to ordi‌nary differential equations (ODEs) involving fractional order derivatives using Legendre wavelets approximations. The continues Legendre wavelets constructed on [0, 1] are uti‌lized as a basis in collocation method. Illustrative examples are included to demonstrate the validity and applicability of the techn...