Spectral Lanczos’ Tau Method for Systems of Nonlinear Integro-Differential Equations

Abstract

In this paper an extension of the spectral Lanczos' tau method to systems of nonlinear integro-differential equations is proposed. This extension includes (i) linearization coefficients of orthogonal polynomials products issued from nonlinear terms and (ii) recursive relations to implement matrix inversion whenever a polynomial change of basis is required and (iii) orthogonal polynomial evaluations directly on the orthogonal basis. All these improvements ensure numerical stability and accuracy in the approximate solution. Exposed in detail, this novel approach is able to significantly outperform numerical approximations with other methods as well as different tau implementations. Numerical results on a set of problems illustrate the impact of the mathematical techniques introduced.

Almost Exact Computation of Eigenvalues in Approximate Differential Problems

“…To approximate eigenvalues for this problem, we first have to approximate cos(2x) and cosh(2x) by polynomials. Considering, as in the previous example, z 16 ≈ cos(2x) the 16th degree Tau solution of z (x) + 4z(x) = 0, 0 < x < π 2 , z(0) = 1, z( π 2 ) = −1,…”

“…are matrices approximating the operational matrices associated with differential equations (16). For each fixed q, we define matrices Tau T 1 and T 2 , representing Mathieu and modified Mathieu equations, respectively.…”

“…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.…”

Solving Nonholonomic Systems with the Tau Method

“…Using suitable matrices, see for example [4,9,10], the differential problem ( 11) is translated into an algebraic problem. Such matrices must be computed with criteria in order to ensure stable computations [10].…”

Explicit formulae for derivatives and primitives of orthogonal polynomials

“…For high degree approximation the accuracy of the approximate solutions is degraded by the bad conditioning of the matrices involved. In recent works, dealing with the extension of spectral methods to systems of nonlinear integro-differential problems [17] and to problems with non-polynomial coefficients [16], the error propagation when working with operational matrices is referred as a drawback. This fact is of great importance when there is need of a large number of coefficients computed with great precision, as * Research funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT -Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2013.…”