677 research outputs found
Variable-step finite difference schemes for the solution of Sturm-Liouville problems
We discuss the solution of regular and singular Sturm-Liouville problems by
means of High Order Finite Difference Schemes. We describe a code to define a
discrete problem and its numerical solution by means of linear algebra
techniques. Different test problems are proposed to emphasize the behaviour of
the proposed algorithm
Classical and vector sturm—liouville problems: recent advances in singular-point analysis and shooting-type algorithms
AbstractSignificant advances have been made in the last year or two in algorithms and theory for Sturm—Liouville problems (SLPs). For the classical regular or singular SLP −(p(x)u′)′ + q(x)u = λw(x)u, a < x < b, we outline the algorithmic approaches of the recent library codes and what they can now routinely achieve.For a library code, automatic treatment of singular problems is a must. New results are presented which clarify the effect of various numerical methods of handling a singular endpoint.For the vector generalization −(P(x)u′)′+Q(x)u = λW(x)u where now u is a vector function of x, and P, Q, W are matrices, and for the corresponding higher-order vector self-adjoint problem, we outline the equally impressive advances in algorithms and theory
Applying numerical continuation to the parameter dependence of solutions of the Schr\"odinger equation
In molecular reactions at the microscopic level the appearance of resonances
has an important influence on the reactivity. It is important to predict when a
bound state transitions into a resonance and how these transitions depend on
various system parameters such as internuclear distances. The dynamics of such
systems are described by the time-independent Schr\"odinger equation and the
resonances are modeled by poles of the S-matrix. Using numerical continuation
methods and bifurcation theory, techniques which find their roots in the study
of dynamical systems, we are able to develop efficient and robust methods to
study the transitions of bound states into resonances. By applying Keller's
Pseudo-Arclength continuation, we can minimize the numerical complexity of our
algorithm. As continuation methods generally assume smooth and well-behaving
functions and the S-matrix is neither, special care has been taken to ensure
accurate results. We have successfully applied our approach in a number of
model problems involving the radial Schr\"odinger equation
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
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Numerical Solution of Linear Ordinary Differential Equations and Differential-Algebraic Equations by Spectral Methods
This thesis involves the implementation of spectral methods, for numerical solution of linear Ordinary Differential Equations (ODEs) and linear Differential-Algebraic Equations (DAEs). First we consider ODEs with some ordinary problems, and then, focus on those problems in which the solution function or some coefficient functions have singularities. Then, by expressing weak and strong aspects of spectral methods to solve these kinds of problems, a modified pseudo-spectral method which is more efficient than other spectral methods is suggested and tested on some examples.
We extend the pseudo-spectral method to solve a system of linear ODEs and linear DAEs and compare this method with other methods such as Backward Difference Formulae (BDF), and implicit Runge-Kutta (RK) methods using some numerical examples. Furthermore, by using appropriate choice of Gauss-Chebyshev-Radau points, we will show that this method can be used to solve a linear DAE whenever some of coefficient functions have singularities by providing some examples. We also used some problems that have already been considered by some authors by finite difference methods, and compare their results with ours.
Finally, we present a short survey of properties and numerical methods for solving DAE problems and then we extend the pseudo-spectral method to solve DAE problems with variable coefficient functions. Our numerical experience shows that spectral and pseudo-spectral methods and their modified versions are very promising for linear ODE and linear DAE problems with solution or coefficient functions having singularities.
In section 3.2, a modified method for solving an ODE is introduced which is new work. Furthermore, an extension of this method for solving a DAE or system of ODEs which has been explained in section 4.6 of chapter four is also a new idea and has not been done by anyone previously.
In all chapters, wherever we talk about ODE or DAE we mean linear
Study of special algorithms for solving Sturm-Liouville and Schrodinger equations
In dit proefschrift beschrijven we een specifieke klasse van numerieke methoden voor het oplossen van Sturm-Liouville en Schrodinger vergelijkingen. Ook de Matlab-implementatie van de methoden wordt besproken
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