Radial Differential Equations Research Articles

Properties of axisymmetric Bernstein modes in an infinite-length non-neutral plasma

We have observed axisymmetric Bernstein modes in an infinite-length particle-in-cell code simulation of a non-neutral plasma. The plasmas considered were in global thermal equilibrium and there were at least 50 Larmor radii within the plasma radius. The density of the plasma in the simulation is parameterized by β, the ratio of the central density to the density at the Brillouin limit. These modes have m = 0 and kz=0, where the eigenfunctions vary as ei(mθ+kzz). The modes exist both near the Coriolis-shifted (by the plasma rotation) upper-hybrid frequency, ωuh=ωc2−ωp2, and near integer multiples (2, 3, etc.) of the Coriolis-shifted cyclotron frequency (called the vortex frequency, ωv=ωc2−2ωp2). The two modes near ωuh and 2ωv are the main subject of this paper. The modes observed are clustered about these two frequencies and are separated in frequency at low plasma density roughly by δω≈10(rL/rp)2ωp2/ωc. The radial velocity field of the modes has a J1(kr) dependence in the region of the plasma where the density is nearly constant. For any given density, there are three classes of modes that exist: (1) The fundamental mode is slightly above the upper-hybrid frequency, (2) the upper branch is above the higher of ωuh and 2ωv, and (3) the lower branch is below the lower of ωuh and 2ωv, with similar values of k for both the upper and the lower frequency branches. The modes are fully kinetic and the resulting pressure tensor has significant anisotropy, including off-diagonal terms. A Vlasov analysis of these modes considering only particle resonances up to 2ωv produces a radial mode differential equation whose solution agrees well with the simulations, except at high density (β greater than about 0.9) where higher-order resonances become important.

Precise response functions in all-electron methods: Generalization to nonspherical perturbations and application to NiO

In a previous publication [Betzinger, Friedrich, G\"orling, and Bl\"ugel, Phys. Rev. B 85, 245124 (2012)] we presented a technique to compute accurate all-electron response functions, e.g., the density response function, within the full-potential linearized augmented-plane-wave (FLAPW) method. Response contributions that are not captured (completely) within the finite Hilbert space spanned by the LAPW basis are taken into account by an incomplete-basis-set correction (IBC). The latter is based on a formal response of the basis functions themselves, which is derived by exploiting their dependence on the effective potential. Its construction requires the solution of radial differential equations, having the form of Sternheimer equations, by numerical integration. The approach includes a formally exact treatment of the response contribution from the core states. While we restricted the formalism to spherical perturbations in the previous work, we here generalize the formalism to nonspherical perturbations. The improvements are demonstrated with exact-exchange optimized-effective-potential (EXX-OEP) calculations of antiferromagnetic NiO. It is shown that with the generalized IBC a basis-set convergence is realized that is as fast as in density-functional theory calculations using standard local or semilocal functionals. The EXX-OEP band gap, magnetic moment, and spectral function of NiO are in substantially better agreement with experiment than results obtained from calculations with local and semilocal functionals.

Numerical modelling of wave interaction with a concentric cylindrical system with an arc-shaped porous outer cylinder

The scaled boundary finite element method (SBFEM) is a novel semi-analytical technique, which combines the advantages of the finite element method and the boundary element method. In this paper, the SBFEM is extended to deal with the short-crested wave interaction with a concentric cylindrical system, where the inner cylinder is solid, while the arc-shaped outer cylinder is thin in thickness and porous to protect the interior cylinder. Both cylinders are surface-piercing and fixed on the bottom. As a key element, a virtual circular cylinder extending the arc-shaped porous outer cylinder with the same centre is introduced so that the entire computational domain is partitioned along the virtual cylinder into a bounded domain and an unbounded domain with common interface. Meanwhile, a variable porous-effect parameter G caused by the complex configuration of the system is introduced for the matching condition of the common interface so that the final SBFEM equation still can be handled in the traditional manner. A new variational principle formulation is used to derive the SBFEM equation in each sub-domain. Hankel functions and Bessel functions are chosen as the basis functions for the solutions of the unbounded and bounded domains, respectively. Then the coefficients for the solutions of the two domains are obtained based on the matching condition. The SBFEM discretizes only the common interface with curved surface finite elements and keeps the radial differential equation solved completely analytically. The results of numerical verification show that the present method yields excellent results with only a few nodes and quick convergence. The effects of the wide range wave parameters and structure configurations including opening angle, position and porous-effect parameter of the outer cylinder as well with the annular spacing on the wave forces, surface elevations and the diffracted wave contours are examined.

Liouvillian quasi-normal modes of Kerr-Newman black holes

The radial differential equations associated with separable perturbations of Kerr-Newman black holes are known to admit Liouvillian (closed-form) solutions for constrained frequencies and black hole parameters. In this paper, we show that the parameter constraints are satisfied exactly in the case of no rotation and thereby obtain a countable infinity of exact purely damped quasi-normal modes of fields on a Reissner-Nordstrom background at special values of the black hole charge-mass ratio. We show that with rotation the parameter constraints for Liouvillian quasi-normal modes are satisfied approximately in two distinct physical scenarios, where analytical approximations for angular eigenvalues are known. We arrive at functional expressions for quasi-normal frequencies and wave-functions in the case of near-extremal slow rotation and in a particular case of highly damped scalar modes of Kerr and Kerr-Newman. In the near-extremal case, our formulas extend a recent result of Hod to electromagnetic and gravitational perturbations.

Holonomic systems of Gegenbauer type polynomials of matrix arguments related with Siegel modular forms

Differential operators on Siegel modular forms which behave well under the restriction of the domain are essentially intertwining operators of the tensor product of holomorphic discrete series to its irreducible components. These are characterized by polynomials in the tensor of pluriharmonic polynomials with some invariance properties. We give a concrete study of such polynomials in the case of the restriction from Siegel upper half space of degree 2n to the product of degree n. These generalize the Gegenbauer polynomials which appear for n = 1. We also describe their radial parts parametrization and differential equations which they satisfy, and show that these differential equations give holonomic systems of rank 2n.

Short-crested waves interaction with a concentric cylindrical structure with double-layered perforated walls

This paper aims at extending the semi-analytical scaled boundary finite element method (SBFEM) to deal with the short-crested waves interaction with a surface-piercing concentric cylindrical structure, which consists of a solid inner cylinder and a coaxial double-layered perforated wall. The whole fluid domain is divided into three sub-domains including two bounded and one unbounded domains, and a variational principle formulation is used to derive the SBFEM equation in each sub-domain. Hankel functions and Bessel functions are chosen as the basis function for the solution of the unbounded and bounded domains, respectively. Although the structure adds a porous cylinder compared to that of Tao's work (2009), the SBFEM also needs to discretize only the outermost porous cylinder with curved surface finite-elements and keeps the radial differential equation solved completely analytically. However, the unknown coefficient vectors for solving the SBFEM equations increase compared with that of Tao's work. Thus we re-derive the solving process for the SBFEM equations. The results of numerical verification show that the present method yields excellent results with only a few nodes and quick convergence. The major factors including wave parameters and structure configuration that affect the wave forces, surface elevations and the diffracted wave contours are examined.

Analysis of quadruple corner‐cut ridged elliptical waveguide by scaled boundary finite element method

SUMMARYThe scaled boundary FEM (SBFEM) is a novel semi‐analytical approach, which combines the advantages of the FEM and BEM with appealing features of its own. In this paper, the method is applied to analyse the quadruple corner‐cut ridged elliptical waveguide. A quarter of the waveguide is adopted and divided into a few subdomains. SBFEM only discretizes the surface boundaries of the subdomains in the sense of FEM, and no discretization of side‐face boundaries is needed, leading to great flexibility in mesh generation with very few nodes. It transforms the governing PDEs to ODEs of the radial parameter by the variational principle. The radial differential equation is then solved fully analytically without adoption of any numerical scheme, which brings out the inherent advantage for solving a singularity problem. Based on the SBFEM governing equation and introducing the dynamic stiffness of waveguide, a generalised eigenvalue equation with respect to cut‐off wave number is formulated by a continued fraction solution without introducing an internal mesh. The numerical example demonstrates the simplicity, excellent computation accuracy and efficiency of the present SBFEM approach. Influences of corner‐cut ridge dimensions on the cut‐off wave numbers of modes are examined in detail. Therefore, these results provide an extension to the existing design data for ridged waveguides and are considered helpful in practical applications. Copyright © 2011 John Wiley & Sons, Ltd.

Second-order wave diffraction by a circular cylinder using scaled boundary finite element method

The scaled boundary finite element method (SBFEM) has achieved remarkable success in structural mechanics and fluid mechanics, combing the advantage of both FEM and BEM. Most of the previous works focus on linear problems, in which superposition principle is applicable. However, many physical problems in the real world are nonlinear and are described by nonlinear equations, challenging the application of the existing SBFEM model. A popular idea to solve a nonlinear problem is decomposing the nonlinear equation to a number of linear equations, and then solves them individually. In this paper, second-order wave diffraction by a circular cylinder is solved by SBFEM. By splitting the forcing term into two parts, the physical problem is described as two second-order boundary-value problems with different asymptotic behaviour at infinity. Expressing the velocity potentials as a series of depth-eigenfunctions, both of the 3D boundary-value problems are decomposed to a number of 2D boundary-value sub-problems, which are solved semi-analytically by SBFEM. Only the cylinder boundary is discretised with 1D curved finite-elements on the circumference of the cylinder, while the radial differential equation is solved completely analytically. The method can be extended to solve more complex wave-structure interaction problems resulting in direct engineering applications.

Coulomb spheroidal representation in the hyperspherical approach to the three-body problem

The construction of the most simple representation coordinated with the physical asymptotic conditions is considered for the Coulomb three-body scattering wave function at energies below the threshold of the three-particle breakup. It is shown that such a representation can be constructed with the use of the well-known solutions of the separable Coulomb two-centre (CTC) problem if the role of the intercentre distance is given to the hyperradius. For the simplest version of the approach (Abramov, J Phys B 41:175201, 2008) the corresponding system of radial differential equations and the statement of the three-body scattering problem for the total angular momentum J ≥ 0 are presented.

Scaled Boundary FEM Model for Interaction of Short-Crested Waves with a Concentric Porous Cylindrical Structure

This paper describes the development of an efficient scaled boundary finite-element model (FEM) for the simulation of short-crested wave interaction with a concentric porous cylindrical structure. By weakening the governing differential equation in the circumferential direction, the SBFEM is able to solve analytically the weakened equation in the radial direction. Only the cylinder boundary on the circumference of the exterior porous cylinder is discretized with curved surface finite elements, while a complete analytical representation is obtained for the radial differential equation. Comparisons of the numerical results on wave diffraction forces and surface wave elevations at the cylinder to available analytical solutions demonstrate that excellent accuracy can be achieved by the SBFEM with a very small number of surface finite elements. The influence of varying the wave parameters as well as the system configuration on the system hydrodynamics, including the wave force, wave run-up, and diffracted wave contour is examined and extensive results on them are presented. This parametric study will help determine the various hydrodynamic effects of a concentric porous cylindrical structure.

A family of Runge-Kutta methods with zero phase-lag and derivatives for the numerical solution of the Schrödinger equation and related problems

We construct a family of two new optimized explicit Runge-Kutta methods with zero phase-lag and derivatives for the numerical solution of the time-independent radial Schrodinger equation and related ordinary differential equations with oscillating solutions. The numerical results show the superiority of the new technique of nullifying both the phase-lag and its derivatives.

Linear tearing mode stability equations for a low collisionality toroidal plasma

Tearing mode stability is normally analysed using MHD or two-fluid Braginskii plasma models. However for present, or future, large hot tokamaks like JET or ITER the collisionality is such as to place them in the banana regime. Here we develop a linear stability theory for the resonant layer physics appropriate to such a regime. The outcome is a set of ‘fluid’ equations whose coefficients encapsulate all neoclassical physics: the neoclassical Ohm's law, enhanced ion inertia, cross-field transport of particles, heat and momentum all play a role. While earlier treatments have also addressed this type of neoclassical physics we differ in incorporating the more physically relevant ‘semi-collisional fluid’ regime previously considered in cylindrical geometry; semi-collisional effects tend to screen the resonant surface from the perturbed magnetic field, preventing reconnection. Furthermore we also include thermal physics, which may modify the results. While this electron description is of wide relevance and validity, the fluid treatment of the ions requires the ion banana orbit width to be less than the semi-collisional electron layer. This limits the application of the present theory to low magnetic shear—however, this is highly relevant to the sawtooth instability—or to colder ions. The outcome of the calculation is a set of one-dimensional radial differential equations of rather high order. However, various simplifications that reduce the computational task of solving these are discussed. In the collisional regime, when the set reduces to a single second-order differential equation, the theory extends previous work by Hahm et al (1988 Phys. Fluids 31 3709) to include diamagnetic-type effects arising from plasma gradients, both in Ohm's law and the ion inertia term of the vorticity equation. The more relevant semi-collisional regime pertaining to JET or ITER, is described by a pair of second-order differential equations, extending the cylindrical equations of Drake et al (1983 Phys. Fluids 26 2509) to toroidal geometry.

Time-resolved Dyson equations in the context of time-dependent density-functional theory: Extension to solid systems

It is shown that under certain conditions an instantaneous resolvent $G(z;t)$ of a time-dependent Kohn-Sham or Kohn-Sham-Dirac operator $\mathcal{H}(t)$ can be defined from which the particle and the magnetization density can be evaluated directly using contour integrations. The corresponding Green's function can be evaluated within a multiple-scattering scheme by solving at each given time $t$ coupled radial differential equations for the single-site problem. This scheme is in particular intriguing since within the adiabatic time-dependent density functional the configurational part of a time-dependent perturbation can be viewed as a contribution to an effective (full) potential. First applications are shown for an Fe atom excited by a femtosecond laser pulse.

Control design and stabilization analysis of non-linear systems using angular representations

The stabilization of non-linear systems using the angular method are considered in this paper. Using the angular method, the system is converted into two subsystems. One is a non-linear equation on a sphere and the other is a radial differential equation. The control is designed by considering the radial subsystem. The radial control can be continuous or discontinuous depending on the structure of the input map. Whenever the input map of the radial subsystem is zero, the radial control is not accessible. In this paper methods for removing this obstacle are presented.

Analytical Solutions of the Gravitational Field Equations in de Sitter and Anti-de Sitter Spacetimes

The generalized Laplace partial differential equation, describing gravitational fields, is investigated in de Sitter spacetime from several metric approaches—such as the Riemann, Beltrami, Borner-Durr, and Prasad metrics—and analytical solutions of the derived Riccati radial differential equations are explicitly obtained. All angular differential equations trivially have solutions given by the spherical harmonics and all radial differential equations can be written as Riccati ordinary differential equations, which analytical solutions involve hypergeometric and Bessel functions. In particular, the radial differential equations predict the behavior of the gravitational field in de Sitter and anti-de Sitter spacetimes, and can shed new light on the investigations of quasinormal modes of perturbations of electromagnetic and gravitational fields in black hole neighborhood. The discussion concerning the geometry of de Sitter and anti-de Sitter spacetimes is not complete without mentioning how the wave equation behaves on such a background. It will prove convenient to begin with a discussion of the Laplace equation on hyperbolic space, partly since this is of interest in itself and also because the wave equation can be investigated by means of an analytic continuation from the hyperbolic space. We also solve the Laplace equation associated to the Prasad metric. After introducing the so called internal and external spaces—corresponding to the symmetry groups SO(3,2) and SO(4,1) respectively—we show that both radial differential equations can be led to Riccati ordinary differential equations, which solutions are given in terms of associated Legendre functions. For the Prasad metric with the radius of the universe independent of the parametrization, the internal and external metrics are shown to be of AdS-Schwarzschild-like type, and also the radial field equations arising are shown to be equivalent to Riccati equations whose solutions can be written in terms of generalized Laguerre polynomials and hypergeometric confluent functions.

ON AXISYMMETRIC TRAVELING WAVES AND RADIAL SOLUTIONS OF SEMI-LINEAR ELLIPTIC EQUATIONS

ABSTRACT. Combining analytical techniques from perturbation methods and dynamical systems theory, we present an elementaryapproach to the detailed construction of axisymmetric diffusive interfaces in semi-linear elliptic equations. Solutions of the resulting non-autonomous radial differential equations can be expressed in terms of a slowlyvarying phase plane system. Special analytical results for the phase plane system are used to produce closed-form solutions for the asymptotic forms of the curved front solutions. These axisym-metric solutions are fundamental examples of more general curved fronts that arise in a wide variety of scientific fields, and we extensivelydiscuss a number of them, with a particular emphasis on connections to geometric models for the motion of interfaces. Related classical results for traveling waves in one-dimensional problems are also reviewed briefly. Manyof the results contained in this article are known, and in presenting known results, it is intended that this article be expositoryin nature, providing elementarydemonstrations of some of the central dynamical phenomena and mathematical techniques. It is hoped that the article serves as one possible avenue of entree to the literature on radiallysymmetric solutions of semilinear elliptic problems, especiallyto those articles in which more advanced mathematical theoryis developed.

Adiabatic representation for a hydrogen atom photoionization in a uniform magnetic field

A new effective method of calculating wave functions of discrete and continuous spectra of a hydrogen atom in a strong magnetic field is developed on the basis of the adiabatic approach to parametric eigenvalue problems in spherical coordinates. The two-dimensional spectral problem for the Schrodinger equation at a fixed magnetic quantum number and parity is reduced to a spectral parametric problem for a one-dimensional angular equation and a finite set of ordinary second-order radial differential equations. The results are in good agreement with the photoionization calculations by other authors and have a true threshold behavior.

POTHMF: A program for computing potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field

A FORTRAN 77 program is presented which calculates with the relative machine precision potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field. The potential curves are eigenvalues corresponding to the angular oblate spheroidal functions that compose adiabatic basis which depends on the radial variable as a parameter. The matrix elements of radial coupling are integrals in angular variables of the following two types: product of angular functions and the first derivative of angular functions in parameter, and product of the first derivatives of angular functions in parameter, respectively. The program calculates also the angular part of the dipole transition matrix elements (in the length form) expressed as integrals in angular variables involving product of a dipole operator and angular functions. Moreover, the program calculates asymptotic regular and irregular matrix solutions of the coupled adiabatic radial equations at the end of interval in radial variable needed for solving a multi-channel scattering problem by the generalized R-matrix method. Potential curves and radial matrix elements computed by the POTHMF program can be used for solving the bound state and multi-channel scattering problems. As a test desk, the program is applied to the calculation of the energy values, a short-range reaction matrix and corresponding wave functions with the help of the KANTBP program. Benchmark calculations for the known photoionization cross-sections are presented.

Scaled boundary FEM solution of short-crested wave diffraction by a vertical cylinder

The scaled boundary finite-element method (SBFEM) is a novel semi-analytical method developed in the elasto-statics and elasto-dynamics areas that has the advantages of combining the finite-element method with the boundary-element method. The SBFEM method weakens the governing differential equation in the circumferential direction and solves the weakened equation analytically in the radial direction. It has the inherent advantage of solving the unbounded fluid dynamic problem. In this paper, the boundary-value problem composed of short-crested waves diffracted by a vertical circular cylinder is solved by SBFEM. Only the cylinder boundary is discretized with curved surface finite-elements on the circumference of the cylinder, while the radial differential equation is solved completely analytically. The computation of the diffraction force based on the present SBFEM solution demonstrates a high accuracy achieved with a small number of surface finite elements. The method can be extended to solve more complex wave–structure interaction problems resulting in direct engineering applications.

Measuring brain variability by extrapolating sparse tensor fields measured on sulcal lines

Modeling and understanding the variability of brain structures is a fundamental problem in neurosciences. Improved mathematical representations of structural brain variation are needed to help detect and understand genetic or disease related sources of abnormality, as well as to improve statistical power when integrating functional brain mapping data across subjects. In this paper, we develop a new mathematical model of normal brain variation based on a large set of cortical sulcal landmarks (72 per brain) delineated in each of 98 healthy human subjects scanned with 3D MRI (age: 51.8 ± 6.2 years). We propose an original method to compute an average representation of the sulcal curves, which constitutes the mean anatomy. After affine alignment of the individual data across subjects, the second order moment distribution of the sulcal position is modeled as a sparse field of covariance tensors (symmetric, positive definite matrices). To extrapolate this information to the full brain, one has to overcome the limitations of the standard Euclidean matrix calculus. We propose an affine-invariant Riemannian framework to perform computations with tensors. In particular, we generalize radial basis function (RBF) interpolation and harmonic diffusion partial differential equations (PDEs) to tensor fields. As a result, we obtain a dense 3D variability map that agrees well with prior results on smaller subject samples. Moreover, “leave one (sulcus) out” tests show that our model is globally able to recover the missing information on brain variation when there is a consistent neighboring pattern of variability. Finally, we propose an innovative method to analyze the asymmetry of brain variability. As expected, the greatest asymmetries are found in regions that includes the primary language areas. Interestingly, any such asymmetries in anatomical variance, if it remains after anatomical normalization, could explain why there may be greater power to detect group activation in one hemisphere versus the other in fMRI studies.