Volume 9, Issue 3

Real positive definite Hankel matrices arise in many important applications. They have spectral condition numbers which exponentially increase with their orders. We give a structural algorithm for finding positive definite Hankel matrices using the Cholesky factorization, compute it for orders less than or equal to 30, and compare our result with earlier results.

In this paper, we introduce a new spectral method based on ultraspherical polynomials for solving systems of initial value differential algebraic equations. Moreover, the suggested method is applicable for a wide range of differential equations. The method is based on a new investigation of the ultraspherical spectral differentiation matrix to approximate the differential expressions in equations. The produced equations lead to algebraic systems and are converted to nonlinear programming. Numerical examples illustrate the robustness, accuracy, and efficiency of the proposed method.

We have developed a new hybrid model for an heterogeneous traffic flow, based on a coupling of the Lighthill — Whitham and Richards (LWR) macroscopic model and the kinetic model. On the highways of a road network, we consider the macroscopic description of the traffic flow and switch to the kinetic model to compute the mass flux through a junction. This new model reproduces the capacity drop phenomenon at a merge junction, for instance, without imposing any priority rule. We present some numerical simulations in which we compare the results of the hybrid model with those given by the fully macroscopic model. Furthermore, we illustrate the consequences of the velocity distribution on the flow through a merging junction.

We have proved the difference analogue of a Bihari-type inequality. Using this inequality, we study the stability in C and monotonicity of the difference schemes approximating initial-boundary value problems for nonlinear conservation laws and multi-dimensional parabolic equations. It has been shown that in the nonlinear case the stability and monotonicity are determined not only by the behavior of the approximate solution but also by its difference derivatives appearing in the nonlinear terms of the equation. The stability estimates are obtained without any assumptions about the properties of the solution and nonlinear coefficients of the differential problem. Here we use restrictions only on input data (initial and boundary conditions and the right-hand side). The sufficient conditions of the shock wave generation is formulated for input data. For the Riemann problem two exact and stable difference schemes are analyzed.

A convection-dominated singularly perturbed two-point boundary problem is considered. For the numerical analysis of such problems, it is necessary to prove certain a priori bounds on the derivatives of its solution. This paper provides a survey of the ways in which such bounds can be proved, while assessing the feasibility of extending such proofs to convection-dominated partial differential equations, and also introduces a new proof based on a classical finite-difference argument of Brandt.

This paper presents a new technique for numerical treatments of Volterra delay integro-differential equations that have many applications in biological and physical sciences. The technique is based on the mono-implicit Runge — Kutta method for treating the differential part and the collocation method (using Boole’s quadrature rule) for treating the integral part. The efficiency and stability properties of this technique have been studied. Numerical results are presented to demonstrate the effectiveness of the methodology.

We consider image denoising as the problem of removing spurious oscillations due to noise while preserving edges in the images. We will suggest here how to directly make infinitesimal adjustment to standard variational methods of image denoising, to enhance desirable target assumption of the noiseless image. The standard regularization method is used to define a suitable energy functional to penalize the data fidelity and the smoothness of the solution. This energy functional is tailored so that the region with small gradient is isotropically smoothed whereas in a neighborhood of an edge presented by a large gradient smoothing is allowed only along the edge contour. The regularized solution that arises in this fashion is then the solution of a variational principle. To this end the associated Euler — Lagrange equation needs to be solved numerically and the half-quadratic minimization is generally used to linearize the equation and to derive an iterative scheme. We describe here a method to modify Euler — Largrange equation from commonly used energy functionals, in a way to enhance certain desirable preconceived assumptions of the image, such as edge preservation. From an algorithmic point of view, we may deem this algorithm as a smoothing by a local average with an adaptive gradient-based weight. However, this algorithm may result in noisy edges although the edge is preserved and noise is suppressed in the low-gradient regions of the image. The main focus here is to present an edge-preserving regularization in the aforementioned view point, and to provide an alternative and simple way to modify the existing algorithm to mitigate the phenomena of noisy edges without explicitly defining step where we specify an energy functional to be minimized.