The barycentric interpolation collocation method for a class of nonlinear vibration systems

Abstract

A nonlinear vibration system arises in physics. Besides its mathematical model, it is of great importance to have an accurate and reliable solution to the system. Though there are many analytical methods, such as the variational iteration method and the homotopy perturbation method, numerical approaches are rare. This paper suggests the barycentric interpolation collocation method to solve nonlinear oscillators. The Duffing equation is adopted as an example to elucidate the solution process. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.

The piecewise reproducing kernel method for the time variable fractional order advection-reaction-diffusion equations

“…( 14) could not have a good numerical accuracy. In order to solve this problem, we use the piecewise reproducing kernel method [21][22][23].…”

“…In recent years several numerical methods have been proposed, such as the variational iteration method and the homotopy perturbation method [6][7][8][9][10][11], the reproducing kernel method [12][13][14][15], etc. In previous work, the Taylor's formula or Delta function was used to construct the reproducing kernel space [16][17][18][19], which has been proved to be an effective tool to solving various kinds of differential equations [20][21][22][23][24][25]. In this paper, we structure some new reproductive kernel spaces based on Legendre polynomials for numerical approach to time-fractional advection-reaction-diffusion equations.…”

The space spectral interpolation collocation method for reaction-diffusion systems

“…Many approaches are used to solve reaction-diffusion systems. These methods include the finite difference method [10], Galerkin finite element method [11], the best uniform (BURA) rational approximation [12], the variational iteration method (VIM) [13][14] and the homotopy perturbation methods (HPM) [15][16][17], the barycentric interpolation collocation method (BICM) [18][19][20] and the reproducing kernel method (RKM) [21][22][23][24][25], etc.…”

Numerical Simulation of the Lorenz-Type Chaotic System Using Barycentric Lagrange Interpolation Collocation Method

“…With the development of numerical analysis, there are some high-precision methods, such as variational iteration method [11][12][13], BLICM [14][15][16][17][18][19][20][21][22], and so on [23]. J.P. Berrut [24][25][26] introduced barycentric Lagrange interpolation, [27,28] studied numerical stability of barycentric Lagrange interpolation, and [15,16] give algorithm of BLICM.…”

“…J.P. Berrut [24][25][26] introduced barycentric Lagrange interpolation, [27,28] studied numerical stability of barycentric Lagrange interpolation, and [15,16] give algorithm of BLICM. Some authors [14,[17][18][19][20][21][22] have used BLICM to solve all sorts of problems and show the BLICM is a high precision numerical method. This paper suggests the BLICM to solve the Lorenz system.…”