A Contour Integral Method for Linear Differential Equations in Complex Plane

This paper presents the contour integral method for solving the linear constant coefficient ordinary differential equations in complex plane,and obtains the uniform expressions of the general solutions.Firstly,by using Residue Theorem,the general form of the contour integral representation for the homogeneous complex differential equation is obtained,which can be degenerated to classical results in real line.As for inhomogeneous complex differential equations with constant coefficients,we construct the integral expression of the particular solution for any continuous forcing term,and give rigorous proof via Residue Theorem.Thus the general solutions of inhomogeneous complex differential equations are also given.The main purpose of this paper is to give a foundation for a complete theory of linear complex differential equations with constant coefficients by a contour integral method.The results can not only solve the inhomogeneous complex differential equation well,but also explain the forms that are difficult to be understood in the classical solutions.

Extraction of Stress Intensity Factors by Using the P-Version Finite Element Method and Contour Integral Method

The stress intensity factors(SIFs)for two-dimensional cracks are extracted using the p-version finite element method(P-FEM)and the contour integral method.Several numerical experiments,e.g.,crack initiating from the edge of a circular hole under an unidirectional uniform tension and two equal-length,unequal-length hole-edge cracks,respectively,at a rectangular plate,an inclined centered crack under uniaxial tension at a square plate and a pipeline crack model,are used to demonstrate the accuracy and effectiveness of the approaches.SIFs are presented for the effects of various crack lengths and length-width ratio.Numerical results are analyzed and compared with reference solutions and results obtained by the Voronoi cell finite element method,boundary element method,high-order extended finite element method(high-order XFEM)and commercial finite element software ABAQUS in the available literature.Numerical results are in good agreement with the benchmark problems and show faster convergence rate,higher accuracy and better numerical stability.

Multivariate form of Hermite sampling series

In this paper,we establish a new multivariate Hermite sampling series involving samples from the function itself and its mixed and non-mixed partial derivatives of arbitrary order.This multivariate form of Hermite sampling will be valid for some classes of multivariate entire functions,satisfying certain growth conditions.We will show that many known results included in Commun Korean Math Soc,2002,17:731-740,Turk J Math,2017,41:387-403 and Filomat,2020,34:3339-3347 are special cases of our results.Moreover,we estimate the truncation error of this sampling based on localized sampling without decay assumption.Illustrative examples are also presented.

FURTHER INVESTIGATION OF THE J_2-INTEGRAL IN BIMATERIAL SOLIDS

The J_2-integral induced from the interface of bimaterial solids(J_2^(interface))is stud- ied by numerical method.First,the effect on the J_2-integral induced from the interface is very significant in bimaterial solids,which is inherently related to that induced from the subinterface cracks.Moreover,it can be concluded that either the first or the second component of the J_k- vector is always equal to zero when the contour encloses both the cracks and the whole interface in bimaterial solids.Secondly,it can also be concluded that the interface does produce significant effect on the J_2-integral induced from the subinterface cracks(J_2^(sub))in bimaterial solids.This effect depends on the geometry of the crack arrangement,which is corresponding to the different interaction effect among the cracks and the interface.Moreover,the interface effect on the J_2^(sub) can be neglected when the distance from the crack center to the interface is large enough,which reveals that the bimaterial solids can be regarded as homogenous solids in fracture analysis when the subinterface crack is far enough from the interface.Three examples are given in this paper.

Improving the Contour Saliency by Tensor Superposition

Contour integration is an important intermediate stage of the object recognition, in which line segments belonging to an object boundary are perceptually linked and segmented from complex backgrounds. This paper puts forward an improved contour integration algorithm. The method computes the vector sum of all neighboring ball voting vectors for an object image pixel to improve the saliency of edges. Four groups of experiments are performed on typical contours to compare the contour in painting effect and the precision. Experimental results show that the contour saliency can be improved by tensor superposition.

Flow pattern analysis of linear gradient flow distribution

This paper uses the Oseen transformation to solve the differential equations governing motion of the vertical linear gradient flow distribution close to a wall surface. The Navier-Stokes equations are used to consider the inertia term along the flow direction. A novel contour integral method is used to solve the complex Airy function. The boundary conditions of linear gradient flow distribution for finite problems are determined. The vorticity function, the pressure function, and the turbulent velocity profiles are provided, and the stability of particle trajectories is studied. An Lx-function form of the third derivative circulation is used to to simplify the solution. Theoretical results are compared with the experimental measurements with satisfactory agreement.

A New Numerical Method for Calculation of Complex Dielectric Permittivity

In this paper, a new numerical method-Kuhn algorithm with contour integral-is intrroduced and used to calculate the complex dielectric permittivities of materials from the short-circuitedwaveguide measurements. Some numerical results are given to verify the validity and efficiency ofthe new numerical method and the complex dielectric permittivities of materials calculated by thenew numerical method are quite identical with the results given by other works.

TORSIONAL OSCILLATION OF RIGID DISK IN INFINITE TRANSVERSELY ISOTROPIC ELASTIC CYLINDER

In the present analysis torsional oscillation of a rigid disk in an infinite transversely isotropic elastic cylinder is considered. The effects of anisotropy in the stress intensity factor are shown graphically.

RECURSIVE INTEGRAL METHOD FOR THE NONLINEAR NON-SELFADJOINT TRANSMISSION EIGENVALUE PROBLEM

The transmission eigenvalue problem is an eigenvalue problem that arises in the scatter- ing of time-harmonic waves by an inhomogeneous medium of compact support. Based on a fourth order formulation, the transmission eigenvalue problem is discretized by the Mor- ley element. For the resulting quadratic eigenvalue problem, a recursive integral method is used to compute real and complex eigenvalues in prescribed regions in the complex plane. Numerical examples are presented to demonstrate the effectiveness of the proposed method.

A Meshless and Parallelizable Method for Differential Equations with Time-Delay

Numerical computation plays an important role in the study of differential equations with time-delay,because a simple and explicit analytic solution is usually un available.Time-stepping methods based on discretizing the temporal derivative with some step-size∆t are the main tools for this task.To get accurate numerical solutions,in many cases it is necessary to require∆t<τand this will be a rather unwelcome restriction whenτ,the quantity of time-delay,is small.In this paper,we propose a method for a class of time-delay problems,which is completely meshless.The idea lies in representing the solution by its Laplace inverse transform along a carefully de-signed contour in the complex plane and then approximating the contour integral by the Filon-Clenshaw-Curtis(FCC)quadrature in a few fast growing subintervals.The computations of the solution for all time points of interest are naturally parallelizable and for each time point the implementations of the FCC quadrature in all subintervals are also parallelizable.For each time point and each subinterval,the FCC quadrature can be implemented by fast Fourier transform.Numerical results are given to check the efficiency of the proposed method.

Explicit Expressions for the Ruin Probabilities of Erlang Risk Processes with Pareto Individual Claim Distributions

In this paper we first consider a risk process in which claim inter-arrival times and the time until the first claim have an Erlang (2) distribution. An explicit solution is derived for the probability of ultimate ruin, given an initial reserve of u when the claim size follows a Pareto distribution. Follow Ramsay[8], Laplace transforms and exponential integrals are used to derive the solution, which involves a single integral of real valued functions along the positive real line, and the integrand is not of an oscillating kind. Then we show that the ultimate ruin probability can be expressed as the sum of expected values of functions of two different Gamma random variables. Finally, the results are extended to the Erlang(n) case. Numerical examples are given to illustrate the main results.

Asymptotic of Enumerative Invariants in ℂP^(2)

In this paper,we give the asymptotic expansion of n_(0,d )and n_(1,d),where(3d−1+g)!n_(g,d )counts the number of genus g curves in ℂP^(2) through 3d−1+g points in general position and can be identified with certain Gromov-Witten invariants.

Application of the Residue Theorem to Trigonometric Sum Identities

By evaluating a contour integral with the Cauchy residue theorem, we prove a general summation formula on trigonometric sum, which contains several interesting trigonometric identities as special cases.