A lattice Boltzmann model for Maxwell’s equations

In this paper, a lattice Boltzmann model for the Maxwell’s equations is proposed by taking separate sets of distribution functions for the electric and magnetic fields, and a lattice Boltzmann model for the Maxwell vorticity equations with third order truncation error is proposed by using the higher-order moment method. At the same time, the expressions of the equilibrium distribution function and the stability conditions for this model are given. As numerical examples, some classical electromagnetic phenomena, such as the electric and magnetic fields around a line current source, the electric field and equipotential lines around an electrostatic dipole, the electric and magnetic fields around oscillating dipoles are given. These numerical results agree well with classical ones.     

Estimating the influence zone of a vertical magnetic dipole in aerial prospecting

We consider the direct problem of aerial electric sounding for a layered medium with a vertical cylindrical anomaly. We determine the minimum size of the anomaly when it is indistinguishable from an infinite layer of the same conductivity. The integral equation is solved by the integral current method. The concept of apparent conductivity is introduced for sounding problems using a magnetic dipole. The calculations support the conjecture of the locality of aerial sounding and prove the high efficiency of the integral current method for such problems.     

Conformal transformation method as applied to finding the current density distribution and induced magnetic field in a strip conductor with a rectangular cut

Given a conductor with a rectangular cut, the problem of determining the direct current distribution and the induced magnetic field in the conductor is solved with applications to the testing of chip conductors. An analytical expression for the current distribution is found assuming that the corners are rounded. The magnetic field induced by the given current is calculated.     

Neutral Fermion Having Electric and Magnetic Moments in an External Electromagnetic Field

We show that in 2+1 dimensions, the Dirac equation for a neutral fermion possessing electric and magnetic dipole moments in an external electromagnetic field reduces to the Dirac equation for a charged fermion in a external field characterized by a certain 3-pseudo-vector potential. The effective charge of the neutral fermion is determined by its dipole moments. The effects of coupling electric and magnetic moments of the neutral fermion to the external electromagnetic field seem to be inseparable in physical experiments of any type. We find an exact solution of the Dirac equation for a massive neutral fermion with electric and magnetic dipole moments in a external plane-wave electromagnetic field. We derive expressions for the fermionic vacuum current induced by neutral fermions in the presence of external electromagnetic fields.     

A uniqueness theorem in the inverse problem for the integrodifferential electrodynamics equations

Under study is the problem of finding the kernel and the index of dielectric permeability for the system of integrodifferential electrodynamics equations with wave dispersion. We consider a direct problem in which the external pulse current is a dipole located at a point y on the boundary ?B of the unit ball B. The point y runs over the whole boundary and is a parameter of the problem. The information available about the solution to the direct problem is the trace on ?B of the solution to the Cauchy problem given for the times close to the time when a wave from the dipole source arrives at a point x. The main result of the article consists in obtaining some theorems related to the uniqueness problems for a solution to the inverse problem.     

On the Inverse Source Problem for the Newtonian Potential

The paper deals with the inverse source problems for the Newtonian potential in the sense of distribution (generalized functions). The inverse source problem is defined as follows: A domain G is given. To find is a distribution creating the potential which is known outside of the closed domain G. New results can be applied to electronical fields, because the electrical charge is distributed only on the surface of conductor.     

Electromagnetic interaction of a conducting cylinder with a magnetic dipole caused by steady rotation

The motion of a conductor in a magnetic field induces eddy currents whose interaction with the field produces Lorentz forces opposing the motion. One can determine the velocity of the conductor from the force on the magnet system since the latter is equal but opposite to the Lorentz force on the conductor. This contactless method is known as Lorentz force velocimetry (LFV). We study an idealized configuration of LFV, i.e. a rotating solid cylinder interacting with a point dipole. The understanding of parameter influences in this setup can be helpful for more realistic configurations. We use a purely kinematic approach appropriate for low magnetic Reynolds numbers. Numerical results for small and large distances between dipole and cylinder have been obtained with the commercial software COMSOL Multiphysics. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An algorithm to determine a dipole current in a sphere

Suppose a spherical conductor has a single dipole. If it is assumed that the direction of current is perpendicular to the location of dipole, then we prove that the dipole can be obtained as an exact form from the induced magnetic field and provide an easy algorithm to locate the dipole from the samples of a magnetic field outside the sphere without using any series approximation. A numerical implementation is also presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Solution of the Cauchy Problem for the Three-Dimensional Telegraph Equation and Exact Solutions of Maxwell’s Equations in a Homogeneous Isotropic Conductor with a Given Exterior Current Source

For the solution of the Cauchy problem for the linear telegraph equation in three-dimensional space, we derive a formula similar to the Kirchhoff one for the linear wave equation (and turning into the latter at zero conductivity). Additionally, the problem of determining the field of a given exterior current source in an infinite homogeneous isotropic conductor is reduced to a generalized Cauchy problem for the three-dimensional telegraph equation. The derived formula enables us to reduce this problem to quadratures and, in some cases, to obtain exact three-dimensional solutions with a propagating front, which are of great applied importance for testing numerical methods for solving Maxwell’s equations. As an example, we construct the exact solution of the field from a Hertzian dipole with an arbitrary time dependence of the current in an infinite homogeneous isotropic conductor.     

Solving an electrostatics-like problem with a current dipole source by means of the duality method

The aim of this note is to give a simple and direct proof of the existence and uniqueness of the solution to an electrostatics-like problem when the source (namely, the applied current density) is a current dipole. The result is obtained by using the classical duality method.     

Inverse problems in magneto-electroscanning (in encephalography, for magnetic microscopes, etc.)

Contrary to the prevailing opinion about the incorrectness of the inverse MEEG-problem, we prove its unique solvability in the framework of the system of Maxwell''s equations [3]. The solution of this problem is the distribution of ${\bf y} \mapsto {\bf q}({\bf y})$ current dipoles of brain neurons that occupies the region $Y \subset \mathbb{R}^3 $. It is uniquely determined by the non-invasive measurements of the electric and magnetic fields induced by the current dipoles of neurons on the patient"s head. The solution can be represented in the form ${\bf q}={\bf q}_0+{\bf p}_0\delta\Big|_{\partial Y}$, where ${\bf q}_0$ is the usual function defined in $Y,$ and ${\bf p}_0\delta\Big|_{\partial Y} $ is a $\delta$-function on the boundary of the domain $Y$ with a certain density ${\bf p}_0$. It is essential that ${\bf p}_0$ and ${\bf q}_0$ are interrelated. This ensures the correctness of the inverse MEEG-problem. However, the components of the required 3-dimensional distribution $ {\bf q} $ must turn out to be linearly dependent if only the magnetic field ${\bf B}$ is taken into account. This question is considered in detail in a flat model of the situation.     

Numerical study of the influence of diffusion of magnetic particles on equilibrium shapes of a free magnetic fluid surface

A mathematical model and a computational method for studying the influence of the particle diffusion on equilibrium shapes of a magnetic liquid is developed. It is then applied on the ferrohydrostatic problem of doubly connected equilibrium shapes of a magnetic fluid located on a horizontal plate around a vertical cylindrical conductor with a direct current. Numerical simulations show the limits of the uniform concentration approximation.     

Unique solvability of the inverse problem for the heat equation with nonlocal boundary conditions

We consider the inverse problem of source identification for the heat conduction problem. The neoclassical formulation of the direct problem with integral boundary condition is used. Conditions for unique solvability of the inverse problem are obtained. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 36–50, 2006.     

Uniqueness for an inverse problem for a nonlinear parabolic system with an integral term by one-point Dirichlet data

We consider an inverse problem arising in laser-induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues is destroyed by coagulation. For the dosage planning quantitatively reliable numerical simulation are indispensable. To this end the identification of the thermal growth kinetics of the coagulated zone is of crucial importance. Mathematically, this problem is a nonlinear and nonlocal parabolic inverse heat source problem. We show in this paper that the temperature dependent thermal growth parameter can be identified uniquely from a one-point measurement.     

Direct and inverse problems for electromagnetic scattering by a doubly periodic structure with a partially coated dielectric

Consider the problem of scattering of electromagnetic waves by a doubly periodic Lipschitz structure. The medium above the structure is assumed to be homogenous and lossless with a positive dielectric coefficient. Below the structure there is a perfect conductor with a partially coated dielectric boundary. We first establish the well‐posedness of the direct problem in a proper function space and then obtain a uniqueness result for the inverse problem by extending Isakov's method. Copyright © 2009 John Wiley & Sons, Ltd.     

Parallel optimization applied to magnetoencephalography

This paper studies a new numerical scheme applicable to magnetoencephalography (MEG), that is, clustering. This method is based on a new theory to the under-determined ill-posed problem, called parallel optimization, and clusters several electric current elements distributed in a volume conductor by one point in time data, without prescribing the number of dipoles. Numerical experiments and optional algorithms are also included.     

A procedure for determining a spacewise dependent heat source and the initial temperature

The inverse problem of determining a spacewise dependent heat source, together with the initial temperature for the parabolic heat equation, using the usual conditions of the direct problem and information from two supplementary temperature measurements at different instants of time is studied. These spacewise dependent temperature measurements ensure that this inverse problem has a unique solution, despite the solution being unstable, hence the problem is ill-posed. We propose an iterative algorithm for the stable reconstruction of both the initial data and the source based on a sequence of well-posed direct problems for the parabolic heat equation, which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for a typical benchmark test example, which has the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure gives accurate numerical approximations in relatively few iterations.     

一维半线性双曲型方程未知源的反问题

We concern the inverse problem of determination of unknown source term for one-dimensional hyperbolic half-linear equation. Approach form for inverse problem is given by using correlative problem of assistant. We concern more ordinary problem than this paper, which is turned into integral equation with the method of characteristic line. We prove the existence and uniqueness of part solution for inverse problem, and unknown source can be solved bv successive approximation.     

A New Energy Inversion for Parameter Identification in Saddle Point Problems with an Application to the Elasticity Imaging Inverse Problem of Predicting Tumor Location

The primary objective of this work is a detailed theoretical and computational study of the elasticity imaging inverse problem for tumor identification within the human body. Apart from this inverse problem's important and interesting application, it also poses noteworthy mathematical challenges since the underlying mathematical model is a system of elasticity involving incompressibility. This gives rise to the “locking” effect and special treatment is necessary for both the direct and inverse problems. To study the inverse problem in an optimization framework, we introduce a general computational scheme for handling parameter identification in saddle point problems along with the introduction and analysis of a new energy output least-squares objective functionals. We also present a treatment of the identification of discontinuous elasticity coefficients using the total variation regularization method. General formulas for the computation of the coefficient-to-solution map and a complete convergence analysis are given for the continuous problem as well as for its discrete analogue. Discrete formulas and implementation issues are discussed in detail and numerical examples for smooth and discontinuous coefficients are given.     

Determination of a spacewise dependent heat source

This paper investigates the inverse problem of determining a spacewise dependent heat source in the parabolic heat equation using the usual conditions of the direct problem and information from a supplementary temperature measurement at a given single instant of time. The spacewise dependent temperature measurement ensures that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. For this inverse problem, we propose an iterative algorithm based on a sequence of well-posed direct problems which are solved at each iteration step using the boundary element method (BEM). The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for various typical benchmark test examples which have the input measured data perturbed by increasing amounts of random noise.