A Legendre Petrov–Galerkin method for fourth-order differential equations

Abstract

a b s t r a c tIn this paper, we present a Legendre Petrov-Galekin method for one-dimensional linear fourth-order differential equations. A Legendre Petrov-Galerkin and Chebyshev collocation method is developed for the nonlinear Kuramoto-Sivashinsky equation. Numerical results are presented to demonstrate the efficiency of the proposed schemes, and optimal rates of convergence in the L 2 -norm are rigorously derived.

Crank–Nicolson/quasi-wavelets method for solving fourth order partial integro-differential equation with a weakly singular kernel

“…A number of elegant numerical methodology have been introduced for the fourth-order problems in the past few decades. These success include finite difference scheme [6,7], finite element methods [4,11], the homotopy perturbation [8], spectral-Galerkin method [21,22], spectral element methods [34], domain decomposition method [10], finite volume method [17]. And in recent years, Mehrdad and Mehdi [16] developed chebyshev cardinal function method for fourth-order integro-differential equations, but the equation does not include the terms of partial derivative of time and the weakly singular kernel.…”

Modelling of Displacement Washing of Pulp Fibers Using the Hermite Collocation Method

“…The diffusion dispersion problems arising from packed bed reactors have always motivated the scientists for the development of new models (Pellet, 1966;Grähs, 1974;Neretnieks, 1974Neretnieks, , 1976Perron & Lebeau, 1977;Al-Jabari et al, 1994;Kukreja et al, 1995;Eriksson et al, 1996;Potůček, 1997;Potůček & Pulcer, 2004;Arora et al, 2006Arora et al, , 2008. It has given rise to the use of a variety of analytical and numerical techniques for the solution of these models such as Laplace transforms (Brenner, 1962;Pellett, 1966;Rasmuson & Neretnieks, 1980;Liao & Shiau, 2000;Aminikhah, 2012), orthogonal collocation method (Villadsen & Stewart, 1967;Michelsen & Villadsen, 1971;Raghvan & Ruthven, 1983;Adomaitis & Lin, 2000;Solsvik & Jakobsen, 2012), orthogonal collocation on finite elements (Carey & Finlayson, 1975;Ma & Guiochon, 1991;Arora et al, 2005Arora et al, , 2006Arora et al, , 2008, Galerkin method (Liu & Bhatia, 2001;Onah, 2002;Bhrawy & El-Soubhy, 2010;Nadukandi et al, 2010;Shen et al, 2011;Zhu et al, 2011;Solsvik & Jakobsen, 2012), Tau method (ElDaou & Al-Matar, 2010;Vanani & Aminataei, 2011;…”

A Petrov-Galerkin spectral method for fourth-order problems

“…Over the past few decades, rapid progress has been made in numerical simulation for fourth‐order problems, and a variety of numerical methods have been developed for these problems, such as the series‐type method , differential quadrature method , and finite element method . Meanwhile, the applications of spectral methods in fourth‐order problems have also attracted much attention because of their high order of accuracy, see for instance, the theoretical work for one‐dimensional fourth‐order equations and for two‐dimensional fourth‐order equations. Moreover, many kinds of efficient algorithms are proposed for numerical solutions of fourth‐order equations in one and two dimensions (cf.…”