ABSTRACT
In recent years, spectral methods have become a valuable tool for solving differential equations. Such methods allow the numerical solution of differential equations, many of which would result untreatable with an analytical method. Moreover, such methods do this with good accuracy and a high level of convergence; depending on the nature of the solution [1-3]. On the other hand, in the modeling of physical problems there are several models in which higher order equations are involved, and their study becomes fundamental. In spectral methods one seeks the solution of a differential equation in terms of a series of smooth and known orthogonal functions. Among the different kinds of spectral methods are the pseudospectral or collocation method, the Galerkin method, and the tau approximation method. The first-mentioned method is the one used in this work for assessing the study of third-order differential operators. Third-order operators are present in important physic equations, such as in the KDV equation [4] or in thin-layer equations in fluid physics [5]. Greguš [6] work can be used as a reference for such differential operators. Although there are a large number of articles on spectral methods in second-order differential operators [7] and fourth-order differential operators, this is not the case for third-order operators. Furthermore, in some of the latter open questions remain. Such questions become interesting from the numerical point of view, and this chapter deals with part of them. In the algorithm implementation for the pseudospectral differentiation and the execution of experiments for thiswork, we usedMathematica [8], which is a common tool for scientific calculation and also has programming capabilities. There are publications that use Matlab as programming tool, in particular the works of Trefethen [9] and Weidemann and Reddy [10]. One advantage shown in Mathematica in this research was that of first being able to compute expressions in algebraic form, then simplifying them and in the end obtaining the numerical value. In addition, the algorithm implementation to generate the pseudospectral differentiation matrices is done almost directly.
