15 research outputs found
Eigenvalues of fourth order Sturm-Liouville problems using Fliess series
AbstractWe shall extend our previous results (Chanane, 1998) on the computation of eigenvalues of second order Sturm-Liouville problems to fourth order ones. The approach is based on iterated integrals and Fliess series
Two-parameter Sturm-Liouville problems
This paper deals with the computation of the eigenvalues of two-parameter
Sturm- Liouville (SL) problems using the Regularized Sampling Method, a method
which has been effective in computing the eigenvalues of broad classes of SL
problems (Singular, Non-Self-Adjoint, Non-Local, Impulsive,...). We have shown,
in this work that it can tackle two-parameter SL problems with equal ease. An
example was provided to illustrate the effectiveness of the method.Comment: 9 page
Application of Galerkin Weighted Residual Method to 2nd, 3rd and 4th order Sturm-Liouville Problems
The aim of this paper is to compute the eigenvalues for a class of linear Sturm-Liouville problems (SLE) with Dirichlet and mixed boundary conditions applying Galerkin Weighted Residual methods. We use Legendre polynomials over [0,1] as trial functions to approximate the solutions of second, third and fourth order SLE problems. We derive rigorous matrix formulations and special attention is given about how the polynomials satisfy the corresponding homogeneous form of Dirichlet boundary conditions of Sturm-Liouville problems. The obtained approximate eigenvalues are compared with the previous computational studies by various methods available in literature. Keywords: Sturm-Liouville problems, eigenvalue, Legendre polynomials, Galerkin method.
Differential quadrature method (DQM) and Boubaker Polynomials Expansion Scheme (BPES) for efficient computation of the eigenvalues of fourth-order Sturm-Liouville problems
The differential quadrature method (DQM) and the Boubaker Polynomials Expansion Scheme (BPES) are applied in order to compute the eigenvalues of some regular fourth-order Sturm-Liouville problems. Generally, these problems include fourth-order ordinary differential equations together with four boundary conditions which are specified at two boundary points. These problems concern mainly applied-physics models like the steady-state Euler-Bernoulli beam equation and mechanicals non-linear systems identification. The approach of directly substituting the boundary conditions into the discrete governing equations is used in order to implement these boundary conditions within DQM calculations. It is demonstrated through numerical examples that accurate results for the first kth eigenvalues of the problem, where k= 1,. 2,. 3,. .... , can be obtained by using minimally 2(k+. 4) mesh points in the computational domain. The results of this work are then compared with some relevant studies. © 2011 Elsevier Inc
The Magnus expansion and some of its applications
Approximate resolution of linear systems of differential equations with
varying coefficients is a recurrent problem shared by a number of scientific
and engineering areas, ranging from Quantum Mechanics to Control Theory. When
formulated in operator or matrix form, the Magnus expansion furnishes an
elegant setting to built up approximate exponential representations of the
solution of the system. It provides a power series expansion for the
corresponding exponent and is sometimes referred to as Time-Dependent
Exponential Perturbation Theory. Every Magnus approximant corresponds in
Perturbation Theory to a partial re-summation of infinite terms with the
important additional property of preserving at any order certain symmetries of
the exact solution. The goal of this review is threefold. First, to collect a
number of developments scattered through half a century of scientific
literature on Magnus expansion. They concern the methods for the generation of
terms in the expansion, estimates of the radius of convergence of the series,
generalizations and related non-perturbative expansions. Second, to provide a
bridge with its implementation as generator of especial purpose numerical
integration methods, a field of intense activity during the last decade. Third,
to illustrate with examples the kind of results one can expect from Magnus
expansion in comparison with those from both perturbative schemes and standard
numerical integrators. We buttress this issue with a revision of the wide range
of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its
applications to several physical problem
Modelling and Adaptive Control; Proceedings of an IIASA Conference, Sopron, Hungary, July 1986
One of the main purposes of the workshop on Modelling and Adaptive Control at Sopron, Hungary, was to give an overview of both traditional and recent approaches to the twin theories of modelling and control which ultimately must incorporate some degree of uncertainty. The broad spectrum of processes for which solutions of some of these problems were proposed was itself a testament to the vitality of research on these fundamental issues. In particular, these proceedings contain new methods for the modelling and control of discrete event systems, linear systems, nonlinear dynamics and stochastic processes