Function Of Second Kind Research Articles

Vibration analysis of submerged thin FGM cylindrical shells

This study gives a brief work on vibration characteristics of cylindrical shells submerged in an incompressible fluid. The shell is presumed to be structured from functionally graded material. The effect of the fluid is introduced by using the acoustic wave equation. Love’s first order thin shell theory is utilized in the shell dynamical equations. The problem is framed by combining shell dynamical equations with the acoustic wave equation. Fluid-loaded terms are associated with Hankel function of second kind. Wave propagation approach is employed to solve the shell problem. Some comparisons of numerical results are performed for the natural frequencies of simply supported-simply supported, clamped-clamped and clamped-simply supported boundary conditions of isotropic as well as functionally graded cylindrical shells to check the validity of the present approach. The influence of fluid on the submerged functionally graded cylindrical shells is noticed to be very pronounced.

Diffraction of Elastic Waves in Fluid-Layered Solid Interfaces by an Integral Formulation

In the present communication, scattering of elastic waves in fluid-layered solid interfaces is studied. The indirect boundary element method is used to deal with this wave propagation phenomenon in 2D fluid-layered solid models. The source is represented by Hankel’s function of second kind and this is always applied in the fluid. Our method is an approximate boundary integral technique which is based upon an integral representation for scattered elastic waves using single-layer boundary sources. This approach is typically called indirect because the sources’ strengths are calculated as an intermediate step. In addition, this formulation is regarded as a realization of Huygens’ principle. The results are presented in frequency and time domains. Various aspects related to the different wave types that emerge from this kind of problems are emphasized. A near interface pulse generates changes in the pressure field and can be registered by receivers located in the fluid. In order to show the accuracy of our method, we validated the results with those obtained by the discrete wave number applied to a fluid-solid interface joining two half-spaces, one fluid and the other an elastic solid.

Scholte waves on fluid-solid interfaces by means of an integral formulation

The present work shows the propagation of Scholte interface waves at the boundary of a fluid in contact with an elastic solid, for a broad range of solid materials. It has been demonstrated that by an analysis of diffracted waves in a fluid it is possible to infer the mechanical properties of the elastic solid medium, specifically, its propagation velocities. For this purpose, the diffracted wave field of pressures and displacements, due to an initial wave of pressure in the fluid, are expressed using boundary integral representations, which satisfy the equation of motion. The source in the fluid is represented by a Hankel’s function of second kind and zero order. The solution to this wave propagation problem is obtained by means of the Indirect Boundary Element Method, which is equivalent to the well-known Somigliana representation theorem. The validation of the results is carried out by using the Discrete Wave Number Method and the Spectral Element Method. Firstly, we show spectra of pressures that illustrate the behavior of the fluid for each solid material considered, then, we apply the Fast Fourier Transform to show results in time domain. Snapshots to exemplify the emergence of Scholte’s waves are also included.

Calibration of stochastic models for interest rate derivatives

For the pricing of interest rate derivatives various stochastic interest rate models are used. The shape of such a model can take very different forms, such as direct modelling of the probability distribution (e.g. a generalized beta function of second kind), a short-rate model (e.g. a Hull–White model) or a forward rate model (e.g. a LIBOR market model). This article describes the general structure of optimization in the context of interest rate derivatives. Optimization in finance finds its particular application within the context of calibration problems. In this case, calibration of the (vector-valued) state of a given stochastic model to some target state, which is determined by available relevant market data, implies a continuous optimization of the model parameters such that a global minimum of the distance between the target state and the model state is achieved. In this article, a novel numerical algorithm for the optimization of parameters of stochastic interest rate models is presented. The optimization algorithm operates within the model parameter space on an adaptive lattice with a number of lattice points per dimension which is both low and fixed. In this way, a considerable performance gain is achieved, as compared to algorithms working with non-adaptive lattices requiring increasing and/or large numbers of lattice points. As compared to standard algorithms, e.g. those of the Levenberg–Marquardt type, the presented adaptive lattice algorithm also reduces the danger of getting trapped near a wrong local minimum. As a numerical example, its application is demonstrated by optimizing volatility and mean reversion parameters of the Hull–White model, such that the latter becomes calibrated to the swaption volatility market relevant for a given OTC (over-the-counter, i.e. not exchange traded) bond option.

Plancherel–Rotach asymptotics for certain basic hypergeometric series

In this work we study the chaotic and periodic asymptotics for the confluent basic hypergeometric series. For a fixed q ∈ ( 0 , 1 ) , the asymptotics for Euler's q-exponential, q-Gamma function Γ q ( x ) , q-Airy function of K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, Ramanujan function ( q-Airy function), Jackson's q-Bessel function of second kind, Ismail–Masson orthogonal polynomials ( q −1 -Hermite polynomials), Stieltjes–Wigert polynomials, q-Laguerre polynomials could be derived as special cases.

The Kontorovich–Lebedev integral transformation with a Hankel function kernel in a space of generalized functions of doubly exponential descent

A version of the Kontorovich–Lebedev transformation with the Hankel function of second kind in the kernel is investigated in a space of distributions of doubly exponential descent. The inversion theorem is rigorously established making use in some steps of the proof of a relation of this transform with the Laplace one. Finally, the theory developed is illustrated in solving certain type of partial differential equations.

Zeros of Jacobi functions of second kind

The number of zeros in ( - 1 , 1 ) of the Jacobi function of second kind Q n ( α , β ) ( x ) , α , β > - 1 , i.e. the second solution of the differential equation ( 1 - x 2 ) y ″ ( x ) + ( β - α - ( α + β + 2 ) x ) y ′ ( x ) + n ( n + α + β + 1 ) y ( x ) = 0 , is determined for every n ∈ N and for all values of the parameters α > - 1 and β > - 1 . It turns out that this number depends essentially on α and β as well as on the specific normalization of the function Q n ( α , β ) ( x ) . Interlacing properties of the zeros are also obtained. As a consequence of the main result, we determine the number of zeros of Laguerre's and Hermite's functions of second kind.

Non-Fuchsian Singularities in Quantum Mechanics

A modification of the spiked harmonic oscillator is studied in the case for which the perturbation potential contains both an inverse power and a linear term. It is then possible to evaluate trial functions by solving an integral equation due to the occurrence of the linear term. The general form of such integral equation is obtained by using a Green-function method, and adding a modified Bessel function of second kind which solves an homogeneous problem with Dirichlet boundary condition at the origin.

Obliquely Incident Poincaré Waves on a Sloping Continental Shelf

Abstract An analytical theory of barotropic tides propagating onto a sloping continental shelf from the deep ocean is developed. The plane Poincare waves incident from the deep ocean are obliquely angled, and a full matching of shelf and ocean solutions is implemented. Allowance for a nonzero water depth at the coast requires an additional term, the Bessel function of second kind, in the solution. The full solution is examined for response characteristics for both frictionless tides and for tides affected by a linear bottom friction, and energy dissipation rates are evaluated. Results for narrow continental shelves indicate that a small but nonzero coastal wall depth, in conjunction with the angle of incidence, can play a significant role in modifying the response, while for wider continental shelves both of these features greatly modify the response at resonance.

An inequality for Legendre polynomials

The following inequality is established: ‖Pn(cos ϑ)‖< [√1+(π4/16)(n+1/2)4 sin4 ϑ]−1, 0<ϑ<π, n=1,2,..., where Pn(x) denotes the Legendre polynomial of degree n. The relation P2n(cos ϑ) + (4/π2)× Q2n(cos ϑ) < [√1+(π4/16)(n+1/2)4 sin4 ϑ]−1, n=1,2,..., on [θn1,θn,n+1], is proven where Qn(x) denotes the Legendre function of second kind, cos θn1 the largest zero of Qn(x), and cos θn,n+1=−cos θn1. Similarly we obtain the inequalities ‖J0(x)‖ < [√1+(π4/16)x4]−1, x≠0, and J20(x) + Y20(x)< [√1+(π4/16)x4]−1, x≥y1, where y1=0.893577... is the first positive zero of Y0(x), and J0(x), Y0(x) denote the Bessel functions of the first and second kind, respectively. The results of the present paper arise out of some problems of nuclear and particle physics.

On The Best Estimators of Coefficients of Variation, skewness and Kurtosis of Pareto Distribution and Their Variances

Here we develop the uniformly minimum variance unbiased (best) estimators of coefficient of variation, Pearson's coefficients of skewness and kurtosis of two-parameter Pareto distribution, the variance of these best estimators and the best estimators of those variances. The best estimators are in terms of Kummer's function 1 F1 and Humbert's function ø2 ; and their variances are in terms of F2 , the Appel function of second kind and K 12 , a quadruple hypergeornetric function invented by Exton during the early 1970's.

The Best Estimators of Quantiles and the Three Means of the Pareto Distribution

In this paper we develop the uniformly minimum variance unbiased (best) estimators of the quantiles, mean, geometric mean and harmonic mean of the Pareto distribuion in the case when both the shape parameter a and the scale parameter k are unknown and in cases when one of them alone is unknown. The best estimates are in terms of the Bessel Function o F1 and Kummer's function 1 F1. The variance of the best estimator are found out, which are in terms of F2 , the Appell function of second kind and ψ2 , a confluent hypergeometric function of two variables. Further we prove that every best estimator of this paper is strongly consistent.

An asymptotic relation for the zeros of Bessel functions

We define the function j νκ for all real κ > 0 as follows: for κ = 1, 2, … the j νκ denotes the kth positive zero of the Bessel function J ν ( z) of first kind and for k − 1 < κ < k, j νκ denotes the kth positive zero of the cylinder Bessel function C ν ( z) = cos α J ν ( z) − sin α Y ν ( Z) with α = ( k − ν) π (see [2]), where Y ν ( x) is the Bessel function of second kind. We introduce the function ι( x) for x > − 1, l(x)= lim κ→∞ j κ,x,κ κ . and we prove, among other things, the inequality j νκ < κι( ν κ ) . Moreover, we find the first three terms of the asymptotic expansion of ι( x), for large values of x and other properties of this function.

Combining the Probabilities from Significance Tests

THE statistic Q proposed by van Valen1 is such that, under the null hypothesis, 2 log 1/2Q is distributed as the modulus of the difference of two independent χ2 variables each of 2N degrees of freedom (where N = n + m in van Valen's notation). It follows that the use of Q as ‘the combined probability desired’ is not correct without a functional transformation: namely, using the result of David et al.2 for the distribution of the difference of two χ2 variables, the value: should be used, where: (Km(z) being a standard notation for the modified Bessel function of second kind with imaginary argument).