Abstract
Differential eigenvalue problems arise in many fields of Mathematics and Physics, often arriving, as auxiliary problems, when solving partial differential equations. In this work, we present a method for eigenvalues computation following the Tau method philosophy and using Tau Toolbox tools. This Matlab toolbox was recently presented and here we explore its potential use and suitability for this problem. The first step is to translate the eigenvalue differential problem into an algebraic approximated eigenvalues problem. In a second step, making use of symbolic computations, we arrive at the exact polynomial expression of the determinant of the algebraic problem matrix, allowing us to get high accuracy approximations of differential eigenvalues.
Highlights
Finding eigenfunctions of differential problems can be a hard task, at least for some classical problems
We can find literature in Sturm–Liouville problems, in Mathieu problems or in Orr–Sommerfeld problems describing the difficulties involved in the resolution of those problems [1,2,3,4,5,6,7,8]
We present a procedure based on the Ortiz and Samara’s operational approach to the Tau method described in [9], where the differential problem is translated into an algebraic problem
Summary
Finding eigenfunctions of differential problems can be a hard task, at least for some classical problems. We present a procedure based on the Ortiz and Samara’s operational approach to the Tau method described in [9], where the differential problem is translated into an algebraic problem. This is achieved using the called operational matrices that represent the action of differential operators in a function. Our main purpose is to use the Tau Toolbox, a Matlab numerical library that is being developed by our research group [14,15,16] This library allows a stable implementation of the Tau method for the construction of accurate approximate solutions for integro-differential problems. We present some examples showing that, using this technique in the Tau Toolbox, we are able to obtain results comparable with those reported in the literature and sometimes even better
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