A non‐commutative Landau‐Zener formula

We study semi‐classical measures of families of solutions to a 2 × 2 Dirac system with 0 mass, which presents bands crossing. We focus on constant electro‐magnetic fields. The fact that these fields are orthogonal or not leads to different geometric situations. In the first case, one reduces to some well‐understood model problem. For studying the second case, we introduce some two‐scale semi‐classical measures associated with symplectic submanifold. These measures are operator‐valued measures and the transfer of energy at the crossing is described by a non‐commutative Landau‐Zener formula for these measures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On semi‐classical d‐orthogonal polynomials

In this paper a general theory of semi‐classical d‐orthogonal polynomials is developed. We define the semi‐classical linear functionals by means of a distributional equation , where Φ and Ψ are matrix polynomials. Several characterizations for these semi‐classical functionals are given in terms of the corresponding d‐orthogonal polynomials sequence. They involve a quasi‐orthogonality property for their derivatives and some finite‐type relations.     

Semi‐transitive graphs

In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi‐transitive and semi‐transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi‐transitive spider‐graphs PS(k,N;r) and MPS(k,N;r). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi‐transitive graph (if it has more than one) is a bi‐transitive graph. We show how the alternets can be used to understand the structure of a semi‐transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi‐transitive graphs of degree 4 begun by Marus?ic? and Praeger. This classification shows that nearly all such graphs are spider‐graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 1–27, 2004     

A Fourier pseudospectral method for a generalized improved Boussinesq equation

In this article, we propose a Fourier pseudospectral method for solving the generalized improved Boussinesq equation. We prove the convergence of the semi‐discrete scheme in the energy space. For various power nonlinearities, we consider three test problems concerning the propagation of a single solitary wave, the interaction of two solitary waves and a solution that blows up in finite time. We compare our numerical results with those given in the literature in terms of numerical accuracy. The numerical comparisons show that the Fourier pseudospectral method provides highly accurate results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 995–1008, 2015     

Convergence analysis for parallel‐in‐time solution of hyperbolic systems

Parallel‐in‐time algorithms have been successfully employed for reducing time‐to‐solution of a variety of partial differential equations, especially for diffusive (parabolic‐type) equations. A major failing of parallel‐in‐time approaches to date, however, is that most methods show instabilities or poor convergence for hyperbolic problems. This paper focuses on the analysis of the convergence behavior of multigrid methods for the parallel‐in‐time solution of hyperbolic problems. Three analysis tools are considered that differ, in particular, in the treatment of the time dimension: (a) space–time local Fourier analysis, using a Fourier ansatz in space and time; (b) semi‐algebraic mode analysis, coupling standard local Fourier analysis approaches in space with algebraic computation in time; and (c) a two‐level reduction analysis, considering error propagation only on the coarse time grid. In this paper, we show how insights from reduction analysis can be used to improve feasibility of the semi‐algebraic mode analysis, resulting in a tool that offers the best features of both analysis techniques. Following validating numerical results, we investigate what insights the combined analysis framework can offer for two model hyperbolic problems, the linear advection equation in one space dimension and linear elasticity in two space dimensions.     

Stationary availability of a semi‐Markov system with random maintenance

We consider a reparable system with a finite state space, evolving in time according to a semi‐Markov process. The system is stopped for it to be preventively maintained at random times for a random duration. Our aim is to find the preventive maintenance policy that optimizes the stationary availability, whenever it exists. The computation of the stationary availability is based on the fact that the above maintained system evolves according to a semi‐regenerative process. As for the optimization, we observe on numerical examples that it is possible to limit the study to the maintenance actions that begin at deterministic times. We demonstrate this result in a particular case and we study the deterministic maintenance policies in that case. In particular, we show that, if the initial system has an increasing failure rate, the maintenance actions improve the stationary availability if and only if they are not too long on the average, compared to the repairs ( a bound for the mean duration of the maintenance actions is provided). On the contrary, if the initial system has a decreasing failure rate, the maintenance policy lowers the stationary availability. A few other cases are studied. Copyright © 2000 John Wiley & Sons, Ltd.     

Rational and semi‐rational solutions of a nonlocal (2 + 1)‐dimensional nonlinear Schrödinger equation

We consider the fully parity‐time (PT) symmetric nonlocal (2 + 1)‐dimensional nonlinear Schrödinger (NLS) equation with respect to x and y. By using Hirota's bilinear method, we derive the N‐soliton solutions of the nonlocal NLS equation. By using the resulting N‐soliton solutions and employing long wave limit method, we derive its nonsingular rational solutions and semi‐rational solutions. The rational solutions act as the line rogue waves. The semi‐rational solutions mean different types of combinations in rogue waves, breathers, and periodic line waves. Furthermore, in order to easily understand the dynamic behaviors of the nonlocal NLS equation, we display some graphics to analyze the characteristics of these solutions.     

The semi‐iterative method applied to the hyper‐power iteration

The semi‐iterative method (SIM) is applied to the hyper‐power (HP) iteration, and necessary and sufficient conditions are given for the convergence of the semi‐iterative–hyper‐power (SIM–HP) iteration. The root convergence rate is computed for both the HP and SIM–HP methods, and the quotient convergence rate is given for the HP iteration. Copyright © 2005 John Wiley & Sons, Ltd.     

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.     

Existence and stability of traveling wave solutions to one‐sided mixed initial‐boundary value problem for first‐order quasilinear hyperbolic systems

When one characteristic of the system is linearly degenerate, under suitable boundary conditions, we get the existence of traveling wave solutions located on the corresponding characteristic trajectory to the one‐sided mixed initial‐boundary value problem. When the system is linearly degenerate, by introducing the semi‐global normalized coordinates, we derive the related formulas of wave decomposition to prove the stability of traveling wave solutions corresponding to all leftward and the rightmost characteristic trajectories. Finally, for the traveling wave solutions corresponding to other rightward characteristic trajectories, some examples show their possible instability. Copyright © 2014 John Wiley & Sons, Ltd.     

Null‐field approach for Laplace problems with circular boundaries using degenerate kernels

In this article, a semi‐analytical method for solving the Laplace problems with circular boundaries using the null‐field integral equation is proposed. The main gain of using the degenerate kernels is to avoid calculating the principal values. To fully utilize the geometry of circular boundary, degenerate kernels for the fundamental solution and Fourier series for boundary densities are incorporated into the null‐field integral equation. An adaptive observer system is considered to fully employ the property of degenerate kernels in the polar coordinates. A linear algebraic system is obtained without boundary discretization. By matching the boundary condition, the unknown coefficients can be determined. The present method can be seen as one kind of semianalytical approaches since error only attributes to the truncated Fourier series. For the eccentric case, vector decomposition technique for the normal and tangential directions is carefully considered in implementing the hypersingular equation in mathematical essence although we transform it to summability to divergent series. The five advantages, well‐posed linear algebraic system, principal value free, elimination of boundary‐layer effect, exponential convergence, and mesh free, are achieved. Several examples involving infinite, half‐plane, and bounded domains with circular boundaries are given to demonstrate the validity of the proposed method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Exact boundary observability for 1‐D quasilinear wave equations

By means of a direct and constructive method based on the theory of semi‐global C² solution, the local exact boundary observability and an implicit duality between the exact boundary controllability and the exact boundary observability are shown for 1‐D quasilinear wave equations with various boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

Fast and stable two‐way algorithm for diagonal plus semi‐separable systems of linear equations

An algorithm for the solution of linear systems of equations where the coefficient matrix is diagonal plus a semi‐separable matrix is considered. The algorithm is stable with linear complexity. Furthermore, it is suitable for an implementation on a system of two processors. Copyright © 2001 John Wiley & Sons, Ltd.     

A note on the complex semi‐definite matrix Procrustes problem

This note outlines an algorithm for solving the complex ‘matrix Procrustes problem’. This is a least‐squares approximation over the cone of positive semi‐definite Hermitian matrices, which has a number of applications in the areas of Optimization, Signal Processing and Control. The work generalizes the method of Allwright (SIAM J. Control Optim. 1988; 26 (3):537–556), who obtained a numerical solution to the real‐valued version of the problem. It is shown that, subject to an appropriate rank assumption, the complex problem can be formulated in a real setting using a matrix‐dilation technique, for which the method of Allwright is applicable. However, this transformation results in an over‐parametrization of the problem and, therefore, convergence to the optimal solution is slow. Here, an alternative algorithm is developed for solving the complex problem, which exploits fully the special structure of the dilated matrix. The advantages of the modified algorithm are demonstrated via a numerical example. Copyright © 2007 John Wiley & Sons, Ltd.     

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation,II

This paper is a continuation of our recent paper 8 . We will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove global existence (in time) of small data energy solutions assuming suitable regularity on the L² scale with additional L¹ regularity for the data. In order to deal with this L² regularity in the non‐linear part, we will develop and employ some tools from Harmonic Analysis.     

A reliability semi‐Markov model involving geometric processes

We consider a semi‐Markov process that models the repair and maintenance of a repairable system in steady state. The operating and repair times are independent random variables with general distributions. Failures can be caused by an external source or by an internal source. Some failures are repairable and others are not. After a repairable failure, the system is not as good as new and our model reflects that. At a non‐repairable failure, the system is replaced by a new one. We assume that external failures occur according to a Poisson process. Moreover, there is an upper limit N of repairs, it is replaced by a new system at the next failure, regardless of its type. Operational and repair times are affected by multiplicative rates, so they follow geometric processes. For this system, the stationary distribution and performance measures as well as the availability and the rate of occurrence of different types of failures in stationary state are calculated. Copyright © 2002 John Wiley & Sons, Ltd.     

Semi‐implicit schemes for transient Navier–Stokes equations and eddy viscosity models

This study presents two computational schemes for the numerical approximation of solutions to eddy viscosity models as well as transient Navier–Stokes equations. The eddy viscosity model is one example of a class of Large Eddy Simulation models, which are used to simulate turbulent flow. The first approximation scheme is a first order single step method that treats the nonlinear term using a semi‐implicit discretization. The second scheme employs a two step approach that applies a Crank–Nicolson method for the nonlinear term while also retaining the semi‐implicit treatment used in the first scheme. A finite element approximation is used in the spatial discretization of the partial differential equations. The convergence analysis for both schemes is discussed in detail, and numerical results are given for two test problems one of which is the two dimensional flow around a cylinder. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A fourth‐order H1‐Galerkin mixed finite element method for Kuramoto–Sivashinsky equation

A H¹‐Galerkin mixed finite element method is applied to the Kuramoto–Sivashinsky equation by using a splitting technique, which results in a coupled system. The method described in this article may also be considered as a Petrov–Galerkin method with cubic spline space as trial space and piecewise linear space as test space, since the second derivative of a cubic spline is a linear spline. Optimal‐order error estimates are obtained without any restriction on the mesh for both semi‐discrete and fully discrete schemes. The advantage of this method over that presented in Manickam et al., Comput. Math. Appl. vol. 35(6) (1998) pp. 5–25; for the same problem is that the size (i.e., (n + 1) × (n + 1)) of each resulting linear system is less than half of the size of the linear system of the earlier method, where n is the number of subintervals in the partition. Further, there is a requirement of less regularity on exact solution in this method. The results are validated with numerical examples. Finally, instability behavior of the solution is numerically captured with this method.     

Integrable couplings,bi‐integrable couplings and their Hamiltonian structures of the Giachetti–Johnson soliton hierarchy

On the basis of zero curvature equations from semi‐direct sums of Lie algebras, we construct integrable couplings of the Giachetti–Johnson hierarchy of soliton equations. We also establish Hamiltonian structures of the resulting integrable couplings by the variational identity. Moreover, we obtain bi‐integrable couplings of the Giachetti–Johnson hierarchy and their Hamiltonian structures by applying a class of non‐semisimple matrix loop algebras consisting of triangular block matrices. Copyright © 2014 John Wiley & Sons, Ltd.     

A parareal approach of semi‐linear parabolic equations based on general waveform relaxation

We present a parareal approach of semi‐linear parabolic equations based on general waveform relaxation (WR) at the partial differential equation (PDE) level. An algorithm for initial‐boundary value problem and two algorithms for time‐periodic boundary value problem are constructed. The convergence analysis of three algorithms are provided. The results show that the algorithm for initial‐boundary value problem is superlinearly convergent while both algorithms for the time‐periodic boundary value problem linearly converge to the exact solutions at most. Numerical experiments show that the parareal algorithms based on general WR at the PDE level, compared with the parareal algorithm based on the classical WR at the ordinary differential equations (ODEs) level (the PDEs is discretized into ODEs), require much fewer number of iterations to converge.