Neumann Functions Research Articles

Neumann problem for generalized n-Poisson equation

Using a hierarchy of integral operators having higher-order Neumann functions and their derivatives as kernels, the Neumann problem for a 2 n th order linear partial complex differential equation is discussed. The solvability of the problem is obtained.

Traveling wave solutions for some factorized nonlinear PDEs

In this work, some new special traveling wave solutions of the convective Fisher equation, the time-delayed Burgers–Fisher equation, the Burgers–Fisher equation and a nonlinear dispersive–dissipative equation (Kakutani and Kawahara 1970 J. Phys. Soc. Japan 29 1068) are obtained through the factorization technique. All of them share the same type of factorization scheme, which reduces the original equation to a Riccati equation of the same kind, whose general solution is given in terms of Bessel and Neumann functions. In addition, some novel particular solutions of the nonlinear dispersive–dissipative equation are provided.

Unique determination of a perfectly conducting ball by a finite number of electric far field data

In this paper,we prove uniqueness in determining a perfectly conducting ball in the inverse electromagnetic scattering problem by a finite number of electric far field patterns with a single incident direction and polarization. It is emphasized that we use only one electric far field pattern datum to uniquely determine the radius of a ball if it is centered at the origin with radius R < 2 / k . Furthermore, if its center was not given as a prior information, three more measurement data must be added to uniquely determine its center. The main tool used here is some new results on zeros of spherical Bessel and spherical Neumann functions.

Study on field enhancement of a normal-gated field emission nanowire cold cathode

The field enhancement is one of the important factors that indicate the performance of field emission cold cathode devices. It is intimately related to the field emission current density and the threshold voltage of the device. In our paper, the field enhancement factor of a normal-gated field emission nanowire cold cathode model was analytically deduced on the basis of classical electrostatic theory, and it is given by the equation. β=k1{N2·(L-d1)2+［1/k1+(L-d1)］2}1/2. The effect of geometrical parameters of the device on the field enhancement factor was explored. The theoretical analysis showed that the larger the length (L-d1) of nanowire above the gate and the gate hole radius, the larger the enhancement factor is; but the larger the nanowire radius, the smaller the enhancement factor is. When the L is much larger thand1, the enhancement factor satisfies the relation. β∝Ｌ／r0, for which N=N1(k1r0)/N0(k1r0), N0(k1r0) and N1(k1r0) are both Neumann functions and k1=0.8936/R. R, L, r0 and d1 are the gate hole radius, the nanowire length, the nanowire radius and the gate-cathode distance, respectively.

Application of modal wavelets to PIV measurements by selecting basis function

A modal wavelet transform, which overcomes the intrinsic data number limitation of power of two to conventional wavelet transform, has been applied to analysis of axial and eddy pseudo velocity fields, standard PIV velocity field and experimental PIV measurement. The modal wavelet transform is compared with the discrete wavelet transform in order to select the optimum basis function among Neumann, Dirichlet and Green function types basis functions. Consequently, it is verified that Neumann type function is the best basis because the correlation of Neumann type basis function is higher and the root mean square is lower than the other basis functions. Also, the decomposition vector patterns by Neumann type are similar to that by conventional Daubechies basis function of 4th order.

Variable Coefficient Transmission Problems and Singular Integral Operators on Non-Smooth Manifolds

In this paper we discuss a new approach and an extension of the results in [11] regarding transmission boundary value problems and spectral theory for singular integral operators on Lipschitz domains. The main novelty here is the consideration of variable coefficient operators and systems which, in turns, requires a change in the strategy employed in [11]. In that paper, an approach based on the Serrin-Weinberger asymptotic theory, akin to the influential work of B. Dahlberg and C. Kenig [9], has been used. By further building on the work in [11], [20], [44], [34], here we develop an alternative approach, based on the regularity of the Neumann function, which is capable of handling variable coefficient operators of Schrodinger type on Lipschitz subdomains of Riemannian manifolds. One key feature of this approach is that it avoids the discussion of the asymptotic behavior at infinity for solutions of elliptic PDE’s with bounded, measurable coefficients. In order to be more specific we shall now introduce some notation, starting with the geometric setting we have in mind. Assume that M is a compact Riemannian manifold, of real dimension n := dimM≥ 2, equipped with a Lipschitz metric tensor g := ∑ gjkdxj ⊗ dxk. Throughout the paper we let dV := gdx1...dxn, where g := det (gjk), be the volume element on M, and denote by ∆ u := g−1/2 ∑

Iterated Neumann problem for the higher order Poisson equation

AbstractRewriting the higher order Poisson equation Δn u = f in a plane domain as a system of Poisson equations it is immediately clear what boundary conditions may be prescribed in order to get (unique) solutions. Neumann conditions for the Poisson equation lead to higher‐order Neumann (Neumann‐n ) problems for Δn u = f . Extending the concept of Neumann functions for the Laplacian to Neumann functions for powers of the Laplacian leads to an explicit representation of the solution to the Neumann‐n problem for Δn u = f . The representation formula provides the tool to treat more general partial differential equations with leading term Δn u in reducing them into some singular integral equations. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Schlömilch series and grating sums

We consider sums over the set of positive integers relevant to construction of periodic Green's functions for diffraction gratings and similar problems, and provide a general formula for a combination of Bessel functions of complex order and complex powers of distance from the origin. This general formula is investigated in a number of particular cases, and in particular we provide expressions which enable sums of functions with Neumann series to be re-expressed as combinations of hypergeometric series. We also investigate sums of Neumann functions of integer order, using analytic continuation techniques to provide formulae for their evaluation which we demonstrate are accurate and efficient in both the high and low frequency regions. We also exhibit sums which may be evaluated analytically, and recurrence formulae linking sums.

Neumann series and lattice sums

We consider sums over the square lattice which depend only on radial distance, and provide formulas which enable sums of functions with Neumann series to be reexpressed as combinations of hypergeometric series. We illustrate the procedure using trigonometric sums previously studied by Borwein and Borwein, sums combining logarithms, Bessel functions Jλ, and powers of distance, and sums of Neumann functions. We also exhibit sums which may be evaluated analytically and recurrence formulas linking sums.

Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives

Alternative expressions for calculating the prolate spheroidal radial functions of the second kind R m l ( 2 ) ( c , ξ ) R_{ml}^{\left ( 2 \right )}\left ( c, \xi \right ) and their first derivatives with respect to ξ \xi are shown to provide accurate values over wide parameter ranges where the traditional expressions fail to do so. The first alternative expression is obtained from the expansion of the product of R m l ( 2 ) ( c , ξ ) R_{ml}^{\left ( 2 \right )}(c, \xi ) and the prolate spheroidal angular function of the first kind S m l ( 1 ) ( c , η ) S_{ml}^{\left ( 1 \right )}\left ( c, \eta \right ) in a series of products of the corresponding spherical functions. A similar expression for the radial functions of the first kind was shown previously to provide accurate values for the prolate spheroidal radial functions of the first kind and their first derivatives over all parameter ranges. The second alternative expression for R m l ( 2 ) ( c , ξ ) R_{ml}^{\left ( 2 \right )}\left ( c, \xi \right ) involves an integral of the product of S m l ( 1 ) ( c , η ) S_{ml}^{\left ( 1 \right )}\left ( c, \eta \right ) and a spherical Neumann function kernel. It provides accurate values when ξ \xi is near unity and l − m l - m is not too large, even when c c becomes large and traditional expressions fail. The improvement in accuracy using the alternative expressions is quantified and discussed.

On the optimization of non-uniform acoustic liners on annular nozzles

The solution of the convected wave equation, with uniform axial flow, in cylindrical co-ordinates, is used together with non-uniform impedance wall boundary conditions, to specify the acoustic modes in a cylindrical or annular nozzle. The radial eigenfunctions in this case are Bessel functions, and the method applies equally well to sheared and swirling mean flows, provided that the appropriate eigenfunctions are used. The eigenvalues or radial wavenumbers are determined by: (i) the roots of a linear combination of Bessel functions for a cylindrical nozzle with uniform wall impedance; (ii) the roots of a 2×2 determinant whose terms are linear combinations of Bessel and Neumann functions, for an annular nozzle with uniform but distinct impedances at each wall; (iii) the roots of an infinite determinant for a cylindrical nozzle with circumferentially non-uniform wall impedance; (iv) the roots of the determinant of a 2×2 block of infinite matrices for an annular nozzle with distinct, non-uniform impedance distributions at the two walls. The case of circumferentially non-uniform but axially uniform wall impedance, allows the existence of an axial wavenumber for each frequency and each eigenvalue or radial wavenumber. The acoustic liner may be optimized by maximizing the decay of a particular wave mode, e.g., the slowest decaying, or a combination of them, e.g., the total acoustic energy.

Estimation of the electromagnetic field created at the earth’s surface by an overhead line current

The electromagnetic field that an overhead infinitely line current produces at the surface of the earth can be expressed as inverse Fourier integrals over a horizontal wave number in terms of the Neumann and Struve functions. These functions have known mathematical properties, including the series expansions. The latter are utilized in this work to derive expressions for the electromagnetic field. The integrals of the electromagnetic field are solved and the results are represented graphically.

String field theory vertices for fermions of integral weight

We construct Witten-type string field theory vertices for a fermionic first order system with conformal weights (0,1) in the operator formulation using delta-function overlap conditions as well as the Neumann function method. The identity, the reflector and the interaction vertex are treated in detail paying attention to the zero mode conditions and the U(1) charge anomaly. The Neumann coefficients for the interaction vertex are shown to be intimately connected with the coefficients for bosons allowing a simple proof that the reparametrization anomaly of the fermionic first order system cancels the contribution of two real bosons. This agrees with their contribution c=-2 to the central charge. The overlap equations for the interaction vertex are shown to hold. Our results have applications in N=2 string field theory, Berkovits' hybrid formalism for superstring field theory, the \eta\xi-system and the twisted bc-system used in bosonic vacuum string field theory.

On the derivation of asymptotic expansions for special functions from the corresponding differential equations

Asymptotic expansions of special functions are usually obtained from integral representations e.g. by Watson's lemma. In the present paper a method of derivation of asymptotic expansions is presented, which does not use integral representations, and relies on solutions of the differential equation satisfied by the special functions in question. The method is illustrated by the determination of the asymptotic expansions for Bessel, Neumann and Hankel functions and has wider application

Bessel and Neumann fitted methods for the numerical solution of the Schrödinger equation

The Bessel and Neumann fitted methods for the numerical solution of the Schrödinger equation is the subject of this paper. An eighth-algebraic-order method for the numerical solution of the Schrödinger equation is developed in this paper. The new method has free parameters which are defined such that the method is fitted to spherical Bessel and Neumann functions. A variable-step procedure is obtained based on the newly developed method and the method of Simos [1]. The results produced based on the numerical solution of the radial Schrödinger equation and of coupled differential equations arising from the Schrödinger equation indicate that this new approach is more efficient than other well-known methods.

Highly accurate approximations of Green's and Neumann functions on rectangular domains

Green9s and Neumann functions of —Δ, where Δ is the Laplacian operator, on a rectangular domain are approximated to any desired degree of accuracy by finite series. Many applications require only a modest number of terms. Upper bounds for the errors in these approximations are also derived. The approximating functions reveal the structural similarities and differences in Green9s and Neumann functions.

Global Solutions of Reaction–Diffusion Systems with a Balance Law and Nonlinearities of Exponential Growth

We consider the initial boundary value problem of the form ut−aΔu=−f(u, v), vt−cΔu−dΔv=+f(u, v), x∈Ω∈RN, N⩾1, t∈R+ where f(u, v)⩾0, f(0, v)=0, v∈R; f(u, v)⩽Kφ(u)eσv, K and σ are positive constants, φ(.) is any continuous, nonnegative, locally Lipschitzian function on R such that φ(0)=0, d>a, and c<d−a with bounded continuous nonegative initial data. This system contains in particular the Frank–Kamenetskii approximation to an nth-order exothermic chemical reaction of Arrhenius type and non-systemically autocatalysed reaction–diffusion systems. We prove the existence of global classical solutions and study their large time behaviour. Our main tools are estimates of the Neumann function for the heat equation and local Lp a priori estimates independent of time.

SIMPLE AND ACCURATE EXPLICIT BESSEL AND NEUMANN FITTED METHODS FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION

Explicit second and fourth algebraic order methods for the numerical solution of the Schrödinger equation are developed in this paper. The new methods have free parameters defined so that the methods are fitted to spherical Bessel and Neumann functions. Based on these new methods we obtained a variable-step algorithm. The results produced based on the numerical solution of the radial Schrödinger equation and the coupled differential equations arising from the Schrödinger equation indicate that this new approach is more efficient than other well known ones.

A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation

An explicit eighth algebraic order Bessel and Neumann fitted method is developed in this paper for the numerical solution of the Schrodinger equation. The new method has free parameters which are defined in order the method is fitted to spherical Bessel and Neumann functions. A variable-step procedure is obtained based on the newly developed method and the method of Simos [17]. Numerical illustrations based on the numerical solution of the radial Schrodinger equation and of coupled differential equations arising from the Schrodinger equation indicate that this new approach is more efficient than other well known methods.

Global existence and large time behavior of positive solutions to a reaction diffusion system

We consider a reaction diffusion system whit a triangular matrix of diffusion coefficients satisfying a balance law on a bounded domain with no-flux boundary condition. We demonstrate that globally bounded solutions exist for any reaction term provided a condition on the diffusion coefficients is satisfied. The proof makes use of some properties of the Neumann function for the heat equation posed in a bounded domain recently obtained in [12]. When the spatial domain is {\rm \bf R}$^N$, the proof relies on well-known properties of the fundamental solution of the heat equation.