Francisco R. Villatoro

Padé numerical method for the Rosenau-Hyman compacton equation

Three implicit finite difference methods based on Pade approximations in space are developed for the Rosenau-Hyman K(n,n) equation. The analytical solutions and their invariants are used to assess the accuracy of these methods. Shocks which develop after the interaction of compactons are shown to be independent of the numerical method and its parameters indicating that their origin may not be numerical. The accuracy in long-time integrations of high-order Pade methods is shown.

Self-similar radiation from numerical Rosenau-Hyman compactons

The numerical simulation of compactons, solitary waves with compact support, is characterized by the presence of spurious phenomena, as numerically induced radiation, which is illustrated here using four numerical methods applied to the Rosenau-Hyman K(p,p) equation. Both forward and backward radiations are emitted from the compacton presenting a self-similar shape which has been illustrated graphically by the proper scaling. A grid refinement study shows that the amplitude of the radiations decreases as the grid size does, confirming its numerical origin. The front velocity and the amplitude of both radiations have been studied as a function of both the compacton and the numerical parameters. The amplitude of the radiations decreases exponentially in time, being characterized by a nearly constant scaling exponent. An ansatz for both the backward and forward radiations corresponding to a self-similar function characterized by the scaling exponent is suggested by the present numerical results.

On the method of modified equations. I: Asymptotic analysis of the Euler forward difference method

The method of modified equations is studied as a technique for the analysis of finite difference equations. The non-uniqueness of the modified equation of a difference method is stressed and three kinds of modified equations are introduced. The first modified or equivalent equation is the natural pseudo-differential operator associated to the original numerical method. Linear and nonlinear combinations of the equivalent equation and their derivatives yield the second modified or second equivalent equation and the third modified or (simply) modified equation, respectively. For linear problems with constant coefficients, the three kinds of modified equations are equivalent among them and to the original difference scheme. For nonlinear problems, the three kinds of modified equations are asymptotically equivalent in the sense that an asymptotic analysis of these equations with the time step as small parameter yields exactly the same results. In this paper, both regular and multiple scales asymptotic techniques are used for the analysis of the Euler forward difference method, and the resulting asymptotic expansions are verified for several nonlinear, autonomous, ordinary differential equations. It is shown that, when the resulting asymptotic expansion is uniformly valid, the asymptotic method yields very accurate results if the solution of the leading order equation is smooth and does not blow up, even for large step sizes.

A repository of equations with cosine/sine compactons

Nonlinear evolution equations with cosine/sine compacton solutions are reviewed, including the Rosenau-Hyman equation and generalizations of Korteweg-de Vries, Camassa-Holm, Boussinesq, Benjamin-Bona-Mahony, Klein-Gordon and other equations. Each equation is generalized to three dimensions and the conditions for its cosine solitary waves to be either a compacton or a soliton are determined. Several equations claimed in the literature to be different among them are found to be equivalent.

Quantum fractal superlattices

Fractal superlattices consist of a series of thin layers of two semiconductor materials alternately deposited on each other with widths corresponding to the rules of construction of a fractal set. The scattering of electrons in superlattices is obtained using the transfer matrix method for generalized Cantor fractal potentials that are characterized by a lacunarity parameter. The numerical results show the self-similarity of the reflection coefficient and the appearance of lacunarity-independent energies with perfectly transparent tunneling due to the bound states of the particle in each of the individual potential wells.

Numerical methods based on modified equations for nonlinear evolution equations with compactons

Abstract Compactons are traveling wave solutions with compact support resulting from the balance of both nonlinearity and nonlinear dispersion. Numerical methods with second-, fourth-, sixth-, and eighth-order approximations to the spatial derivatives obtained by means of the method of modified equations applied to the Ismail–Taha finite difference scheme for the Rosenau–Hyman equation are developed. The whole set of methods is compared among them in accuracy, invariant conservation, and in compacton collisions. The best method, among those studied, in terms of the tradeoff between accuracy and computational cost is determined.

A transfer matrix method for the analysis of fractal quantum potentials

The scattering properties of quantum particles on a sequence of potentials converging towards a fractal one are obtained by means of the transfer matrix method. The reflection coefficients for both the fractal potential and finite periodic potential are calculated and compared. It is shown that the reflection coefficient for the fractal potential has a self-similar structure associated with the fractal distribution of the potential whose degree of self-similarity has been quantified by means of the correlation function.

Outlier detection in audit logs for application systems

Abstract An outlier is defined as an observation that is significantly different from the other data in its set. An auditor will employ many techniques, processes and tools to identify these entries, and data mining is one such medium through which the auditor can analyze information. The enormous amount of information contained within transactional processing systems׳ logs means that auditors must employ automated systems for anomalous data detection. Several data mining algorithms have been tested, especially those that deal specifically with classification and outlier detection. A group of these previously described algorithms was selected for use in designing and developing a process to assist the auditor in anomalous data detection within audit logs. We have been successful in creating and ratifying an outlier detection process that works in the alphanumeric fields of the audit logs from an information system, thus constituting a useful tool for system auditors performing data analysis tasks.

Tunneling in quantum superlattices with variable lacunarity

Fractal superlattices are composite, aperiodic structures comprised of alternating layers of two semiconductors following the rules of a fractal set. The scattering properties of polyadic Cantor fractal superlattices with variable lacunarity are determined. The reflection coefficient as a function of the particle energy and the lacunarity parameter present tunneling curves, which may be classified as vertical, arc, and striation nulls. Approximate analytical formulae for such curves are derived using the transfer matrix method. Comparison with numerical results shows good accuracy. The new results may be useful in the development of band-pass energy filters for electrons, semiconductor solar cells, and solid-state radiation sources up to THz frequencies.

Adiabatic perturbations for compactons under dissipation and numerically-induced dissipation

Compacton propagation under dissipation shows amplitude damping and the generation of tails. The numerical simulation of compactons by means of dissipative schemes also show the same behaviors. The truncation error terms of a numerical method can be considered as a perturbation of the original partial differential equation and perturbation methods can be applied to its analysis. For dissipative schemes, or when artificial dissipation is added, the adiabatic perturbation method yields evolution equations for the amplitude loss in the numerical solution and the amplitude of the numerically-induced tails. In this paper, such methods are applied to the K(2,2) Rosenau-Hyman equation, showing a very good agreement between perturbative and numerical results.