Neumann Functions Research Articles

Infinite integrals involving Bessel functions by contour integration

The approach of contour integration and residue theorem is systematically developed to evaluate infinite integrals involving Bessel functions or closely related ones. Three independent formulations applicable to different types of integrals are presented. The conventional formulation using Hankel functions as auxiliary functions, whose previous development leave considerable room for improvement, is fully developed. The second and third formulations, which appear unnoticed before, uses as auxiliary functions linear combinations of Bessel and Struve functions, and linear combinations of Anger and Weber functions, respectively. These functions play a crucial role as the Hankel functions do in the conventional formulation. Several general formulae are derived, each expressing an infinite integral in terms of residues. The integrands are products of rational functions, powers and Bessel (Neumann, Struve, Anger or Weber) functions, with some relation satisfied between the index of the power and the order of the Bessel or related function. Typical examples are worked out for each type of integral.

Periodic Homogenization of Green and Neumann Functions

For a family of second‐order elliptic operators with rapidly oscillating periodic coefficients, we study the asymptotic behavior of the Green and Neumann functions, using Dirichlet and Neumann correctors. As a result we obtain asymptotic expansions of Poisson kernels and the Dirichlet‐to‐Neumann maps as well as optimal convergence rates in Lp and W1,p for solutions with Dirichlet or Neumann boundary conditions. © 2014 Wiley Periodicals, Inc.

Polyharmonic boundary value problem in a sector

In this article, a polyharmonic Neumann function in a sector with angle θ=π/n(n∈N) is studied by convolution. Especially, the outward normal derivatives at three corner points are defined properly. We give the recursive expressions for the polyharmonic Neumann function, obtaining the solution and the condition of solvability for the related polyharmonic Neumann problem.

Neumann functions for second order elliptic systems with measurable coefficients

We study Neumann functions for divergence form, second order elliptic systems with bounded measurable coefficients in a bounded Lipschitz domain or a Lipschitz graph domain. We establish existence, uniqueness, and various estimates for the Neumann functions under the assumption that weak solutions of the system enjoy interior H\"older continuity. Also, we establish global pointwise bounds for the Neumann functions under the assumption that weak solutions of the system satisfy a certain natural local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solutions of the system is in fact equivalent to the global pointwise bound for the Neumann function. We present a unified approach valid for both the scalar and the vectorial cases.

Conformal mappings of multiply connected domains onto canonical domains using the Green and Neumann functions

In this study, some properties of the Green and Neumann functions associated with boundary value problems for elliptic equations are analysed. The described Green and Neumann functions are used for the construction of conformal mappings of multiply connected domains onto some canonical domains.

Tri-harmonic boundary value problems in a sector

This paper is devoted to the construction of a tri-harmonic Green function and a tri-harmonic Neumann function in a sector with angle explicitly, as well as a tri-harmonic Neumann function in the unit disk. Then the related Dirichlet and Neumann problems for the tri-Poisson equation are discussed, respectively.

Greenʼs function for second order parabolic systems with Neumann boundary condition

We study the Neumann Greenʼs function for second order parabolic systems in divergence form with time-dependent measurable coefficients in a cylindrical domain Q=Ω×(−∞,∞), where Ω⊂Rn is an open connected set such that a multiplicative Sobolev embedding inequality holds there. Such a domain includes, for example, a bounded Sobolev extension domain, a special Lipschitz domain, and an unbounded domain with compact Lipschitz boundary. We construct the Neumann Greenʼs function in Q under the assumption that weak solutions of the systems satisfy an interior Hölder continuity estimate. We also establish global Gaussian bounds for Neumann Greenʼs function under an additional assumption that weak solutions with zero Neumann data satisfy a local boundedness estimate. In the scalar case, such a local boundedness estimate is a consequence of De Giorgi–Moser–Nash theory holds for equations with bounded measurable coefficients in Sobolev extension domains, while in the vectorial case, one may need to impose further regularity assumptions on the coefficients of the system as well as on the domain to obtain such an estimate. We present a unified approach valid for both the scalar and vectorial cases and discuss some applications of our results including the construction of Neumann functions for second order elliptic systems with measurable coefficients in two dimensional domains.

Biharmonic Green function and biharmonic Neumann function in a sector

In this article, a biharmonic Green function and a biharmonic Neumann function in a sector with angle θ = π/n (n ∈ ℕ) are studied, and then we investigate the related Dirichlet problem and Neumann problem for the bi-Poisson equation, respectively.

Estimating utility functions using generalized maximum entropy

This paper estimates von Neumann and Morgenstern utility functions using the generalized maximum entropy (GME), applied to data obtained by utility elicitation methods. Given the statistical advantages of this approach, we provide a comparison of the performance of the GME estimator with ordinary least square (OLS) in a real data small sample setup. The results confirm the ones obtained for small samples through Monte Carlo simulations. The difference between the two estimators is small and it decreases as the width of the parameter support vector increases. Moreover, the GME estimator is more precise than the OLS one. Overall, the results suggest that GME is an interesting alternative to OLS in the estimation of utility functions when data are generated by utility elicitation methods.

Sharp estimates for the Neumann functions and applications to quantitative photo-acoustic imaging in inhomogeneous media

We obtain sharp Lp and Hölder estimates for the Neumann function of the operator ∇⋅γ∇−ik on a bounded domain. We also obtain quantitative description of its singularity. We then apply these estimates to quantitative photo-acoustic imaging in inhomogeneous media. The problem is to reconstruct the optical absorption coefficient of a diametrically small anomaly from the absorbed energy density.

The Narrow Escape Problem—A Short Review of Recent Results

The narrow escape problem in diffusion theory, which goes back to Lord Rayleigh, is to calculate the mean first passage time, also called the narrow escape time (NET), of a Brownian particle to a small absorbing window on the otherwise reflecting boundary of a bounded domain. The renewed interest in the NET problem is due to its relevance in molecular biology and biophysics. The small window often represents a small target on a cellular membrane, such as a protein channel, which is a target for ions, a receptor for neurotransmitter molecules in a neuronal synapse, a narrow neck in the neuronal spine, which is a target for calcium ions, and so on. The leading order singularity of the Neumann function for a regular domain strongly depends on the geometric properties of the boundary. It can give a smaller contribution than the regular part to the absorption flux through the small window when it is located near a boundary cusp. We review here recent results on the dependence of the absorption flux on the geometric properties of the domain and thus reveal geometrical features that can modulate the flux. This indicates a possible way to code information physiologically.

On the Neumann function and the method of images in spherical and ellipsoidal geometry

The invention of an image system for a boundary value problem adds to a significant understanding of the structure of the problem, both at the mathematical and at the physical level. In this paper, the interior and exterior Neumann functions for the Laplacian in the cases of spherical and ellipsoidal domains are represented in terms of images. Besides isolated images, the presence of the normal derivative in the Neumann condition demands an additional continuous distribution of images, which in the spherical cases, can be restricted to a one‐dimensional manifold, whereas for the ellipsoid, both a one‐dimensional and a two‐dimensional distribution of images is needed. Copyright © 2012 John Wiley & Sons, Ltd.

Modified harmonic Robin function

The harmonic Robin function is redefined as interpolation between the related Green and Neumann functions and explicitly constructed for the unit disc and a circular ring.

Harmonic expressions for the eigen mode fields of step and graded index fibers

For clarifying the mode field characteristics of step and graded index fibers, the Helmholtz's equation and local planar wave approximation are engaged, a novel idea that the eigenfunction of the traveling wave field in radial grade index dielectric is similar to positive or negative half order Hankel function is advanced. Then, the eigenfunction of core layer of graded index fiber is similar to the combination of half order Bessel function and half order Neumann function, the eigenfunction of cladding layer of graded index fiber is similar to the first kind half order imaginary parameter Hankel function, are suggested. The phase shift for total reflection effect at the turning point of grade fiber is proposed. The angular phase shift pi for the ray propagating along the fiber circumference is given. The eigen equation of new improved WKB method for analysing the mode field characteristics of graded index fiber is presented. Moreover, the eigen equation of new improved WKB method for analyzing the mode field characteristics of graded index fiber is held good for analysing the mode field characteristics of step index fiber too, it indicates that, the eigen equation of new improved WKB method is of universal significance.

Exact formula for the sum of the squares of the Bessel function and the Neumann function of the same order of half-odd integer

Sum of the squares of the Bessel function and the Neumann function of the same order of half-odd integer has been found to be very useful in addressing a puzzle in nuclear physics. One approximate formula available in the literature is valid for the complex argument whose real part is greater than zero, and the absolute value of error term is undefined for half-odd integers. Another approximate formula which is valid for all complex arguments has been obtained using sophisticated mathematical method called Barnes' method. However, the error in the formula is very difficult to calculate. We have obtained exact formula for the sum of the squares of Bessel and Neumann functions of the same order of half-odd integers which is valid for all complex arguments, and its proof is also given.Keywords: Bessel functions; Neumann functions; Anomalous absorption; Partial waves DOI: 10.4038/josuk.v4i0.2694J Sci.Univ.Kelaniya 4 (2008): 15-20

Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel′s Nonlinear ODEs of the First Kind and Relative Degenerate Equations)

We provide a new mathematical technique leading to the construction of the exact parametric or closed form solutions of the classes of Abel′s nonlinear differential equations (ODEs) of the first kind. These solutions are given implicitly in terms of Bessel functions of the first and the second kind (Neumann functions), as well as of the free member of the considered ODE; the parameter ν being introduced furnishes the order of the above Bessel functions and defines also the desired solutions of the considered ODE as one‐parameter family of surfaces. The nonlinear initial or boundary value problems are also investigated. Finally, introducing a relative mathematical methodology, we construct the exact parametric or closed form solutions for several degenerate Abel′s equation of the first kind.

Asymptotic Formulae of the Modal Acoustic Impedance for the Asymmetric Vibrations of a Clamped Circular Plate

The asymptotic and approximate formulae for the asymmetric modal acoustic selfand mutual-impedance have been presented for a clamped circular plate embedded into a flat rigid baffle. The formulae have been obtained for the wide frequency band covering the low frequencies, the high frequencies and the middle frequencies. The high frequency asymptotics have been achieved using the method of contour integral and the method of stationary phase. The products of the Bessel and Neumann functions have been expressed as the asymptotic expansions. Further, the approximate formulae valid within the low and middle frequencies have been obtained from the high frequency asymptotics using some mathematical manipulations. The formulae presented are valid for both the axisymmetric vibrations and the asymmetric vibrations.

Variation of Neumann and Green functions under homotopies of the boundary

We develop variational formulas for certain Neumann and Green functions of multiply connected planar domains, valid for any smooth homotopy of the boundary. The modified Neumann and Green functions under consideration arise in the study of holomorphic functions with single-valued primitives.

Harmonic boundary value problems in half disc and half ring

The Schwarz problem for the Cauchy-Riemann equation and the Dirichlet and Neumann problems for the Poisson equation are explicitly solved in a half disc and a half ring of the complex plane. The respective Poisson kernels and the Green and Neumann functions are given.

Readily computable Green’s and Neumann functions for symmetry-preserving triangles

Neumann and Green’s functions of the Laplacian operator on 30 30 - 60 60 - 90 ∘ 90^{\circ } and 45 45 - 45 45 - 90 ∘ 90^{\circ } triangles can be generated with appropriately placed multiple sources/sinks in a rectangular domain. Highly accurate and easily computable Neumann and Green’s function formulas already exist for rectangles. The extension to equilateral triangles is illustrated. In applications, closed-form expressions can be constructed for the potential, the streamfunction, or the various spatial derivatives of these properties. The derivation of analytic line integrals of these functions allows the proper handling of singularities and facilitates extended applications to problems on domains with open boundaries. Using a boundary integral method, it is demonstrated how one can construct semi-analytical solutions to problems defined on domains that exhibit spatially-dependent properties (heterogeneous media) or possess irregular boundaries.