A Note on a Fourier Sine Transform

This is a compilation of definite integrals of the product of the hyperbolic cosecant function and polynomial raised to a general power. In this work, we used our contour integral method to derive a Fourier sine transform in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. A summary table of the results are produced for easy reading. A vast majority of the results are new.

The quantum-mechanical Coulomb propagator in an L2 function representation

The quantum-mechanical Coulomb propagator is represented in a square-integrable basis of Sturmian functions. Herein, the Stieltjes integral containing the Coulomb spectral function as a weight is evaluated. The Coulomb propagator generally consists of two parts. The sum of the discrete part of the spectrum is extrapolated numerically, while three integration procedures are applied to the continuum part of the oscillating integral: the Gauss–Pollaczek quadrature, the Gauss–Legendre quadrature along the real axis, and a transformation into a contour integral in the complex plane with the subsequent Gauss–Legendre quadrature. Using the contour integral, the Coulomb propagator can be calculated very accurately from an L$$^2$$ 2 basis. Using the three-term recursion relation of the Pollaczek polynomials, an effective algorithm is herein presented to reduce the number of integrations. Numerical results are presented and discussed for all procedures.

Pseudospectral Roaming Contour Integral Methods for Convection-Diffusion Equations

We generalize ideas in the recent literature and develop new ones in order to propose a general class of contour integral methods for linear convection–diffusion PDEs and in particular for those arising in finance. These methods aim to provide a numerical approximation of the solution by computing its inverse Laplace transform. The choice of the integration contour is determined by the computation of a few suitably weighted pseudo-spectral level sets of the leading operator of the equation. Parabolic and hyperbolic profiles proposed in the literature are investigated and compared to the elliptic contour originally proposed by Guglielmi, López-Fernández and Nino 2020, see Guglielmi et al. (Math Comput 89:1161–1191, 2020). In summary, the article provides a comparison among three different integration profiles; proposes a new fast pseudospectral roaming method; optimizes the selection of time windows on which one may arbitrarily approximate the solution by no extra computational cost with respect to the case of a fixed time instant; focuses extensively on computational aspects and it is the reference of the MATLAB code [20], where all algorithms described here are implemented.

Direct Evaluation of the Stress Intensity Factors for the Single and Multiple Crack Problems Using the P-Version Finite Element Method and Contour Integral Method

Stress intensity factor (SIF) is one of three important parameters in classical linear elastic fracture mechanics (LEFM). The evaluation of SIFs is of great significance in the field of engineering structural and material damage assessment, such as aerospace engineering and automobile industry, etc. In this paper, the SIFs of a central straight crack plate, a slanted single-edge cracked plate under end shearing, the offset double-edge cracks rectangular plate, a branched crack in an infinite plate and a crucifix crack in a square plate under bi-axial tension are extracted by using the p-version finite element method (P-FEM) and contour integral method (CIM). The above single- and multiple-crack problems were investigated, numerical results were compared and analyzed with results using other numerical methods in the literature such as the numerical manifold method (NMM), improved approach using the finite element method, particular weight function method and exponential matrix method (EMM). The effectiveness and accuracy of the present method are verified.

Integral Representation of Polynomial Xn (x;a,b)

In the present paper we have obtained fine difference formula, contour integral representation, real integral representation, infinite single integral representation, finite single integral representation, finite double integral representation, finite double integral representation of polynomial ࢔ࢄ .(࢈ ,ࢇ ;࢞) Keywords: Finite difference, single integral representation, contour integral representation, simple generating relation, double integral representation

Numerical methods for scattering problems in periodic waveguides

In this paper, we propose new numerical methods for scattering problems in periodic waveguides. Based on [20], the “physically meaningful” solution, which is obtained via the Limiting Absorption Principle (LAP) and is called an LAP solution, is written as an integral of quasi-periodic solutions on a contour. The definition of the contour depends both on the wavenumber and the periodic structure. The contour integral is then written as the combination of finite propagation modes and a contour integral on a small circle. Numerical methods are developed and based on the two representations. Compared with other numerical methods, we do not need the LAP process during numerical approximations, thus a standard error estimation is easily carried out. Based on this method, we also develop a numerical solver for halfguide problems. The method is based on the result that any LAP solution of a halfguide problem can be extended to the LAP solution of a fullguide problem. At the end of this paper, we also give some numerical results to show the efficiency of our numerical methods.

Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function

We apply our simultaneous contour integral method to an infinite sum in Prudnikov et al. and use it to derive the infinite sum of the Incomplete gamma function in terms of the Hurwitz zeta function. We then evaluate this formula to derive new series in terms of special functions and fundamental constants. All the results in this work are new.

A lubrication contact pair model for simulating gear micro-pitting damage characteristics based on contour integral

A new two-dimensional finite element model of a lubricated contact pair, based on a contour integral, is proposed to investigate the formation of micro-pitting on gear tooth surfaces. Meanwhile, the contact properties and elasto-hydrodynamic lubrication (EHL) conditions of the gears are considered in the lubricated contact pair model. Then, the stress intensity factors (SIFs) KI and KII and the propagation angle θ C at the crack tip are analyzed by ABAQUS software. Next, the equivalent SIF Kσ can be calculated according to the maximum tangential stress (MTS) criterion, which is often used as the criterion for crack propagation. Considering the effect of a moving contact, the crack more easily propagates under the load x0/ b = −0.895. Furthermore, the pit shapes and variation of stress intensity factor are determined for various combinations of initial crack length a0 and angle β. The results show that longer germinated cracks propagate in areas that are deeper below the tooth surface. And the total length of final crack increases with the initial length and germination angle. These research results provide theoretical support for contact fatigue life analysis and meshing stiffness calculations of micro-pitting gears.