Solving Vlasov–Maxwell equations in singular geometries

This paper is devoted to the solution of the time-dependent Vlasov–Maxwell equations in singular geometries, i.e. when the boundary includes reentrant corners or edges. Indeed, computing the electromagnetic fields in this case is a challenge per se, as these geometrical singularities generate very strong solutions in their neighborhood. Moreover, they have also an influence over the solution of the Vlasov equation, through the coupling. We propose here a method to solve this problem, illustrated by numerical examples.     

Rejewski’s equations: Solving for the entry permutation

In 1932, Marian Rejewski, who was a young mathematician working at the Polish Cipher Bureau, brilliantly recovered the internal wiring of the military Enigma. His initial efforts were unsuccessful because he assumed that the entry permutation was the same as in the commercial machine. Luckily he tried the identity permutation as an alternative and that proved to be correct. This note describes how Rejewski’s equations may be used to deduce the entry permutation without any guesswork, a technique that was later rediscovered by Alan Turing and by Lieutenant Andrew Gleason.     

Goal-oriented error estimation and h-adaptivity for Maxwell’s equations

Goal-oriented error estimates, based on dual solutions, are derived for the S-parameters of a waveguide cavity resonator and a computationally cheap method is proposed to compute these. Numerical results show that the error estimators are relatively accurate, usually well within a factor 2 from the exact errors. Adaptive mesh-refinement based on these estimates recovers optimal convergence rates for complete and incomplete, first and second order, curl-conforming finite elements of Nédélec type, despite singularities at reentrant corners. It is also shown numerically that the convergence rate is independent of the element-order when singularities are present and the mesh is refined uniformly.     

Solving nonlinear Volterra–Fredholm integro-differential equations using He's variational iteration method

In this paper, a nonlinear Volterra–Fredholm integro-differential equation is solved by using He's variational iteration method. The approximate solution of this equation is calculated in the form of a sequence where its components are computed easily. The accuracy of the proposed numerical scheme is examined by comparing with the modified Adomian decomposition method. The existence and uniqueness of the solution and convergence of the proposed method are proved.     

A Legendre Pseudospectral Penalty Scheme for Solving Time-Domain Maxwell’s Equations

In this paper we present a Legendre pseudospectral algorithm based on a tensor product formulation for solving the time-domain Maxwell equations. Our approach starts by conducting an analysis for finding well-posed boundary operators for the Maxwell equations. We then discuss equivalent characteristic boundary conditions for common physical boundary constraints. These theoretical results are then employed to construct a pseudospectral penalty scheme which is asymptotically stable at the semidiscrete level. Numerical computations based on the proposed scheme are also provided for different cases where exact solutions exist. By measuring the differences between the computed and exact solutions, we observe the expected convergence patterns of the scheme. This work is supported by National Science Council grant No. NSC 95-2120-M-001-003.     

Solving a class of nonlinear optimal control problems via he’s variational iteration method

This paper presents an analytical approximate solution for a class of nonlinear quadratic optimal control problems. The proposed method consists of a Variational Iteration Method (VIM) together with a shooting method like procedure, for solving the extreme conditions obtained from the Pontryagin’s Maximum Principle (PMP). This method is applicable for a large class of nonlinear quadratic optimal control problems. In order to use the proposed method, a control design algorithm with low computational complexity is presented. Through the finite iterations of algorithm, a suboptimal control law is obtained for the nonlinear optimal control problem. Two illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method.     

Sensitivity analysis of shock wave Burgers’ equation via a novel algorithm based on scale-3 Haar wavelets

In this paper, a novel technique is being formulated for the numerical solutions of Shock wave Burgers' equations for planar and non-planar geometry. It is well known that Burgers' equation is sensitive to the perturbations in the diffusion term. Thus we use robustness of wavelets generated by dilation and translation of Haar wavelets on third scale to capture the sensitivity information. The present approach is an improved form of the scale-2 Haar wavelet method. The scheme is based on the forward finite difference scheme for time integration, scale-3 Haar wavelets for space integration and the nonlinearity has been tackled via quasilinearzation technique. Through scale-3 Haar wavelet analysis once the wavelet coefficient is calculated then we can compute the solutions at near the perturbation point. The computation cost of the present scheme is negligible. The proposed method is tested on six test problems to check its computational efficiency where the convergence analysis of scale-3 Haar wavelet method is the proof of our computational arguments.     

Solving Kirkman’s Schoolgirl Problem in a Few Seconds

The Social Golfer Problem has been extensively used by the constraint community in recent years as an example of a highly symmetric problem. It is an excellent problem for benchmarking symmetry breaking mechanisms such as SBDS or SBDD and for demonstrating the importance of the choice of the right model for one problem. We address in this paper a specific instance of the Golfer Problem well known as Kirkmans Schoolgirl Problem and list a collection of techniques and tricks to find efficiently all its unique solutions. In particular, we propose SBDD+, a generic improvement over SBDD which allows a deep pruning when a symmetry is detected during the search. Our implementation of the presented techniques improves previously published results by an order of magnitude for CPU time as well as for number of backtracks. It computes the seven unique solutions of Kirkmans problem in a few seconds.     

Invariant solutions of integro-differential Vlasov–Maxwell equations in Lagrangian variables by Lie group analysis

The Lie point symmetries of the Vlasov–Maxwell system in Lagrangian variables are investigated by using a direct method for symmetry group analysis of integro-differential equations, with emphasis on solving nonlocal determining equations. All similarity reduction forms for the system are obtained by using different approaches and some analytical and numerical solutions are presented.     

Unified Analysis of Leap-Frog Methods for Solving Time-Domain Maxwell’s Equations in Dispersive Media

In this paper, we consider the time dependent Maxwell’s equations resulting from dispersive medium models. First, the stability and Gauss’s law are proved for all three most popular dispersive medium models: the isotropic cold plasma, the one-pole Debye medium and the two-pole Lorentz medium. Then leap-frog mixed finite element methods are developed for these three models. Optimal error estimates are proved for all three models solved by the lowest-order Raviart-Thomas-Nédélec spaces. Extensions to multiple pole dispersive media are presented also. Numerical results confirming the analysis are presented.     

An overdetermined problem for a weighted Poisson’s equation

In this paper, we discuss an overdetermined problem for a weighted Poisson’s equation. We prove that if there exists a solution of the weighted Poisson’s equation $△u+?logw??u=?1$ on a smooth bounded domain $\Omega$ with both Dirichlet and Neumann constant boundary condition, and the weight function $w$ satisfies some conditions in $\Omega$, then $\Omega$ is a ball. We also study some applications of the overdetermined problems and some overdetermined problems with nonconstant Neumann boundary condition.     

The reconstruction of a time-dependent source from a surface measurement for full Maxwell’s equations by means of the potential field method

This paper is devoted to the study of an inverse source problem governed by full Maxwell’s equations by means of the potential field method (the $\mathit{A}$-$?$ method). The source term is assumed to be separable in time and space, in which the unknown part is solely time-dependent and is recovered from a surface measurement. We prove that the solution to the inverse problem based on the $\mathit{A}$-$?$ formulation is existing and unique. We suggest a constructive scheme for approximating the solution and discuss its convergence. Finally, a few examples are presented to verify the theoretical results.     

Non-Reflecting Boundary Conditions for Maxwell’s Equations

A new discrete non-reflecting boundary condition for the time-dependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented. It is derived from a virtual spatial finite difference discretization of the problem on the unbounded domain. Fourier transforms are used to decouple transversal modes. A judicious combination of edge based nodal values permits us to recover a simple structure in the Laplace domain. Using this, it is possible to approximate the convolution in time by a similar fast convolution algorithm as for the standard wave equation.     

Solving a leader–follower facility problem via parallel evolutionary approaches

A leader–follower facility problem is considered in this paper. The objective is to maximize the profit obtained by a chain (the leader) knowing that a competitor (the follower) will react by locating another single facility after the leader locates its own facility. A subpopulation-based evolutionary algorithm called TLUEGO was recently proposed to cope with this hard-to-solve global optimization problem. However, it requires high computational effort, even to manage small-size problems. In this work, three parallelizations of TLUEGO are proposed, a distributed memory programming algorithm, a shared memory programming algorithm, and a hybrid of the two previous algorithms, which not only allow us to obtain the solution faster, but also to solve larger instances.     

Stability radius of a vector investment problem with Savage’s minimax risk criteria

Based on the classical Markowitz model, we formulate a vector (multicriteria) Boolean problem of portfolio optimization with bottleneck criteria under risk. We obtain the lower and upper attainable bounds for the quantitative characteristics of the type of stability of the problem, which is a discrete analog of the Hausdorff upper semicontinuity of the multivalued mapping that defines the Pareto optimality.     

An illusion efect of Maxwell’s fsh-eye lens

We show that Maxwell’s fsh-eye lens can make a semi-circular perfect electric conductor look like a circular one.Such an efect can also be achieved(not perfectly)by using negative index metamaterials,but only within a single frequency.Maxwell’s fsh-eye lens,however,can work for a set of eigenfrequencies.Numerical simulations are performed to verify the efect.     

An illusion effect of Maxwell’s fish-eye lens

We show that Maxwell＇s fish-eye lens can make a semi-circular perfect electric conductor look like a circular one. Such an effect can also be achieved （not perfectly） by using negative index metamaterials, but only within a single frequency. Maxwell＇s fish-eye lens, however, can work for a set of eigenfrequencies. Numerical simulations are performed to verify the effect.