In this paper, the Legendre--Petrov--Galerkin method for the Korteweg--de Vries equation
with nonperiodic boundary conditions is analyzed. The nonlinear term is computed with the …

A Legendre--Petrov--Galerkin (LPG) method for the third-order differential equation is
developed. By choosing appropriate base functions, the method can be implemented …

We present an overview of the theory and applications of spectral penalty methods for the
stable solution of initial boundary value problems. Through a number of case studies we …

The stability and convergence of the Galerkin method for differential equations with
symmetric operators have been confirmed with numerical results, while this is not the case …

We present some relations that allow the efficient approximate inversion of linear differential
operators with rational function coefficients. We employ expansions in terms of a large class …

In this paper, a well-conditioned collocation method is constructed for solving general p th
order linear differential equations with various types of boundary conditions. Based on a …

We describe a robust, adaptive algorithm for the solution of singularly perturbed two-point
boundary value problems. Many different phenomena can arise in such problems, including …

Automatic Chebyshev spectral collocation methods for Fredholm and Volterra integral and
integro-differential equations have been implemented as part of the chebfun software …

The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for
computing both Caputo and (modified) Riemann–Liouville (RL) fractional pseudospectral …

We introduce a hybrid Gegenbauer (ultraspherical) integration method (HGIM) for solving
boundary value problems (BVPs), integral and integro-differential equations. The proposed …