Gauss-Laguerre Quadrature Formula Research Articles

Energy efficiency optimization in a parallel relay-assisted UWOC system with simultaneous lightwave information and power transfer.

In this paper, we investigate the energy efficiency optimization for a parallel relay-assisted underwater wireless optical communication (UWOC) system with simultaneous lightwave information and power transfer (SLIPT) over an aggregate channel. In this system, relay nodes are equipped with energy harvesting devices, getting energy from the direct current component of the received signal transmitted by the source node. These nodes utilize the harvested energy to transmit the signal to the destination node with the decoding and forwarding strategy. The harvested energy for each relay node is derived by the Gauss-Laguerre quadrature formula and the outage probability is deduced by the Meijer-G function. Then, the system's energy efficiency can be calculated and an energy efficiency maximization problem is built up with respect to the bias current. We propose a three-level-iteration algorithm to solve this problem. In the first level, the Dinkelbach method is used to represent energy efficiency in a parametric subtractive form. In the second level, we use the penalty function method to convert the object function and constraint. In the third level, the objective function is transformed into a quadratic function by using a successive convex approximate method, thereby solving for the bias current. The effects of system parameters on energy efficiency are also analyzed. Theoretical results and Monte Carlo simulations suggest that employing the solved bias current can significantly improve the system's energy efficiency.

Outage probability analysis of a cooperative NOMA UWOC system with multiuser scheduling under weak oceanic turbulence.

A downlink cooperative non-orthogonal multiple access (NOMA) multiuser underwater wireless optical communication (UWOC) system with a greedy scheduling scheme has been proposed for the Internet of Underwater Things. In particular, the near user plays as a relay to assist the far user, and both decode-and-forward and amplify-and-forward relaying protocols are considered. Relying on the Gauss-Laguerre quadrature formula, the analytical expressions for the outage probability of users are derived considering the degrading effects of the underwater channel, namely, absorption, scattering, and turbulence-induced fading. The outage performance is further analyzed systematically under different water types, targeted data rates, the number of users, the receiver aperture size, and the power allocation coefficient. Numerical results demonstrate that the performance of the far user can be improved by the proposed cooperative NOMA technology. Moreover, the proposed cooperative NOMA system performs better compared with both conventional OMA and non-cooperative NOMA systems. Monte Carlo simulation results are presented to confirm the accuracy of derived expressions, which have a tight agreement with analytical results.

Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation

The impulse response of the fractional oscillation equation was investigated, where the damping term was characterized by means of the Riemann–Liouville fractional derivative with the order α satisfying 0≤α≤2. Two different analytical forms of the response were obtained by using the two different methods of inverse Laplace transform. The first analytical form is a series composed of positive powers of t, which converges rapidly for a small t. The second form is a sum of a damped harmonic oscillation with negative exponential amplitude and a decayed function in the form of an infinite integral, where the infinite integral converges rapidly for a large t. Furthermore, the Gauss–Laguerre quadrature formula was used for numerical calculation of the infinite integral to generate an analytical approximation to the response. The asymptotic behaviours for a small t and large t were obtained from the two forms of response. The second form provides more details for the response and is applicable for a larger range of t. The results include that of the integer-order cases, α= 0, 1 and 2.

IMPLEMENTATIONS OF BOUNDARY–DOMAIN INTEGRO-DIFFERENTIAL EQUATION FOR DIRICHLET BVP WITH VARIABLE COEFFICIENT

In this paper, we present the numerical results of the Boundary-Domain Integro-Differential Equation (BDIDE) associated to Dirichlet problem for an elliptic type Partial Differential Equation (PDE) with a variable coefficient. The numerical constructions are based on discretizing the boundary of the problem region by utilizing continuous linear iso-parametric elements while the domain of the problem region is meshed by using iso-parametric quadrilateral bilinear domain elements. We also use a semi-analytic method to handle the integration that exhibits logarithmic singularity instead of using Gauss-Laguare quadrature formula. The numerical results that employed the semi-analytic method give better accuracy as compared to those when we use Gauss-Laguerre quadrature formula. The system of equations that obtained by the discretized BDIDE is solved by an iterative method (Neumann series expansion) as well as a direct method (LU decomposition method). From our numerical experiments on all test domains, the relative errors of the solutions when applying semi-analytic method are smaller than when we use Gauss-Laguerre quadrature formula for the integration with logarithmic singularity. Unlike Dirichlet Boundary Integral Equation (BIE), the spectral properties of the Dirichlet BDIDE is not known. The Neumann iterations will converge to the solution if and only if the spectral radius of matrix operator is less than 1. In our numerical experiment on all the test domains, the Neumann series does converge. It gives some conclusions for the spectral properties of the Dirichlet BDIDE even though more experiments on the general Dirichlet problems need to be carried out.

Solving partial integro-differential option pricing problems for a wide class of infinite activity Lévy processes

In this paper, numerical analysis of finite difference schemes for partial integro-differential models related to European and American option pricing problems under a wide class of Lévy models is studied. Apart from computational and accuracy issues, qualitative properties such as positivity are treated. Consistency of the proposed numerical scheme and stability in the von Neumann sense are included. Gauss–Laguerre quadrature formula is used for the discretization of the integral part. Numerical examples illustrating the potential advantages of the presented results are included.

Research on Optimization of Perforated Horizontal Wells with Separated Production Scheme Based on Genetic Algorithm

Based on the separated production scheme of perforated horizontal well, the production is closely related to the number, length and location of horizontal intervals. New reservoir / wellbore coupling models were modeled which included two boundaries: closed boundary and constant pressure boundary. For these coupling models, two new optimization models were modeled, which took the production as the objective function and took the length and location of horizontal intervals as the optimization variables. Then the numerical calculation models were proposed by using the Gauss-Laguerre quadrature formula. Applying the genetic algorithm and writing the numerical stimulation software, the optimized schemes with 2 segments, 3 segments and 4 segments for planning intervals of the perforated horizontal wells were presented. By comparing the optimized schemes, we can see that the horizontal well production was not infinite increasing with the length of horizontal intervals, but only in the suitable length of horizontal intervals to maintain high production. And it is not the more segments the higher production. For the perforated horizontal well in closed reservoir, the production was the highest based on the optimized schemes with 3 segments.

Transient Sloshing in Partially Filled Laterally Excited Spherical Vessels

AbstractThree-dimensional transient liquid sloshing in a nondeformable spherical container filled to an arbitrary depth and subjected to an arbitrary lateral external acceleration is investigated. The problem solution is achieved by using the linear potential theory in conjunction with the Mehler-Fock transform of general order in the toroidal coordinate system, resulting in a pair of coupled integro-differential equations, which are discretized using Gauss-Laguerre quadrature formula and then solved by appropriate numerical schemes. The transient sloshing response to a real seismic event is calculated and discussed. Also, a ramp-step function is used to simulate the lateral acceleration excitation during an idealized turning maneuver. The effects of the fill depth on the time histories of the horizontal hydrodynamic forces acting on the spherical container are examined. Limiting cases are considered, and good agreements with available analytic and numerical solutions and experimental data are obtained.

Liquid sloshing in half-full horizontal elliptical tanks

A two dimensional hydrodynamic analysis based on the linear potential theory is introduced to study the natural sloshing frequencies of transverse modes in a half-filled non-deformable horizontal cylindrical container of elliptical cross section, without or with a pair of inflexible horizontal longitudinal side baffles of arbitrary extension positioned at the free liquid surface. Successive conformal coordinate transformations in conjunction with the method of separation of variables and the relevant boundary conditions are employed to obtain standard truncated matrix eigen-value problems which are then solved numerically for the resonance eigen-frequencies. The Gauss–Laguerre quadrature formula is used to approximate the integral eigen-problem obtained in the unbaffled case. Plots of the sloshing frequencies as functions of the container aspect ratio and baffle extension are presented and discussed for the three lowest antisymmetric and symmetric transverse oscillation modes. A convergence study is performed to demonstrate the fast convergence and remarkably small computational cost of the Fourier series approach used for the baffled container, and the effects of tank geometry and baffle length on the convergence are also examined. Limiting cases are considered and good agreements with available analytic and numerical solutions as well as experimental data are obtained, demonstrating the accuracy of proposed models.

Convergence acceleration of the Gauss-Laguerre quadrature formula

Asymptotic expansions have been obtained for the error of the Gauss-Laguerre quadrature formula for a large class of analytic integrands. Then, a theorem of convergence acceleration is given. Numerical examples are also given.

Numerical evaluation of exponential integral: Theis well function approximation

This paper reviews various approximation methods for evaluating one form of the exponential integral known as the Theis well function W(u) and presents an algorithm that can be easily implemented to one's desired accuracy for 0 < u < ∞. The algorithm is based on a combination of a fast-converging series representation for small u and an easy-implementing Gauss-Laguerre quadrature formula when u becomes large. The partition point for u varies with the desired program accuracy and efficiency. The error analyses suggested that a partition point in the interval of 1 ≤ u ≤ 4 can yield sufficiently accurate results for practical applications of groundwater problems with minimum computational effort. A comparison of the results revealed that the popular Stehfest inverse Laplace transform technique generally yielded a relative error several orders of magnitude greater than that obtained by the proposed algorithm. For large u, the approximation accuracy increased rapidly using the quadrature formula while the results of the Laplace inversion technique showed numerical oscillation between some positive and negative values.

A quadrature formula for correlation integrals

A convenient Gauss–Laguerre quadrature formula is presented for integrands which depend on the radial coordinates r1 and r2 of two bodies as well as on their relative distance r12. This formula generalizes the analytic method by Calais and Löwdin [J. Mol. Spectrosc. 8, 203 (1962)] to cases when the analytical evaluation is not possible and defaults to an exact method when it is. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 513–518, 1997

Abscissae and weights for the Gauss-Laguerre quadrature formula

A program is described which calculates the abscissae and weights for the Gauss-Laguerre quadrature formula for integrals of the form ∫∈e -x x α f( x)d x very rapidly and with high accuracy even in the case of many abscissae. The abscissae are given by the zeros of the Laguerre polynomials, which are found by the Newton-Raphson method with suitable initial approximations. The program is useful for the problem in the case of an integrand f( x) oscillating rapidly.

Addition of Points to Gauss–Laguerre Quadrature Formulas

The Gauss–Laguerre quadrature formula is defined by \[ If = \int_0^\infty {e^{ - x} f(x)dx \simeq \sum_{i = 1}^n {\alpha _i^{(n)} f\left( {\xi _i^{(n)} } \right),} } \] where the numbers ${\alpha _i^{(n)} }$ and ${\xi _i^{(n)} }$ are weights and nodes. A common method of estimating the error of this rule is to evaluate the quadrature rule for two different values of n and to then compare the difference in the answers. Unfortunately, none of the nodes are in common for the two different quadrature rules, and so the function must be evaluated at each separate node. We investigate in this paper the addition of points to the Gauss–Laguerre rule such that the new points are real, lie in the interval of integration, and the associated weights are positive. Such rules enable one to estimate economically the error of quadrature, because the function values at the Gauss–Laguerre abscissae are reused. The weights and nodes for some suitable low-order formulae are given in Table 2.

Tables for the Gaussian computation of ∫₀^{∞}𝑥^{𝛼}𝑒^{-𝑥}𝑓(𝑥)𝑑𝑥 for values of 𝛼 varying continuously between -1 and +1

Tables of coefficients of Chebyshev expansions for computing to 11 places the abscissas and weight factors of the 12-point generalized Gauss-Laguerre quadrature formula are presented.

The errors of first and second kinds in welch-aspln's solution of the behrens-fisher problem

Welch's test for the Behrens-Fisher problem, which tests the equality of means of two normal populations with unknown (possibly unequal) variances, is considered in this paper. The tables of critical regions for Welch's test are of an approximate nature. The errors of the first and second kinds are calculated on computer for this test, using the Gauss-Laguerre quadrature formula. The effects of the discretization and round-off errors in the computation have been eliminated- The values of α and β, the two kinds of errors for Welch's test are tabulated. These tables give a broad picture of the behaviour of Welch's test for different values of the parameters involved.