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

Authors:
Ting-Ting Shen;Kang-Zheng Xing;He-Ping Ma
Affiliations:
-;-;-
Venue:
Computers & Mathematics with Applications
Year:
2011

Citing 11
Cited 1

Efficient spectral-Galerkin method I: direct solvers of second-and fourth-order equations using Legendre polynomials

SIAM Journal on Scientific Computing
Efficient spectral-Galerkin method II.: direct solvers of second- and fourth-order equations using Chebyshev polynomials

SIAM Journal on Scientific Computing
Chebyshev--Legendre Spectral Viscosity Method for Nonlinear Conservation Laws

SIAM Journal on Numerical Analysis
A Legendre Spectral Galerkin Method for the Biharmonic Dirichlet Problem

SIAM Journal on Scientific Computing
Optimal Error Estimates of the Legendre--Petrov--Galerkin Method for the Korteweg--de Vries Equation

SIAM Journal on Numerical Analysis
A Legendre--Petrov--Galerkin and Chebyshev Collocation Method for Third-Order Differential Equations

SIAM Journal on Numerical Analysis
Convergence Analysis of Spectral Collocation Methods for a Singular Differential Equation

SIAM Journal on Numerical Analysis
Efficient Chebyshev---Petrov---Galerkin Method for Solving Second-Order Equations

Journal of Scientific Computing
Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials

Applied Numerical Mathematics
Optimal Error Estimates of the Legendre Tau Method for Second-Order Differential Equations

Journal of Scientific Computing
Sparse Spectral Approximations of High-Dimensional Problems Based on Hyperbolic Cross

SIAM Journal on Numerical Analysis

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

Journal of Computational Physics

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Abstract

In 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.