Evaluation of the Wave-Resistance Green Function: Part 2—The Single Integral on the Centerplane

1987 ◽  
Vol 31 (03) ◽  
pp. 145-150 ◽  
Author(s):  
J. N. Newman
Keyword(s):  
Green Function ◽  
Arbitrary Order ◽  
Asymptotic Series ◽  
Neumann Series ◽  
Absolute Accuracy ◽  
Double Integral ◽  
Vertical Coordinate ◽  
Recursive Scheme ◽  
Thin Region ◽  
Single Integral

Effective series expansions are derived for the evaluation of the single integral in the potential of a submerged source which moves with constant velocity, when the source and field point are in the same longitudinal centerplane. In conjunction with the polynomial approximations for the double integral component which have been derived in Part 1 of this work, the present results facilitate the computation of the source potential or Green function. Three complementary domains of the centerplane are considered, with different expansions developed for use in each domain. The principal expansion is based on a Neumann series which is effective for small or moderate distances from the origin, except in a thin region near the free surface. To deal with the latter domain an asymptotic expansion is derived in ascending powers of the vertical coordinate. Both of these expansions are refined by subtracting a simpler component with the same behavior at the origin, and relating this component to Dawson's integral. Special algorithms for the evaluation of the latter function are presented in the Appendix. The third and final expansion, based upon the method of steepest descents, is effective at large distances from the origin. This asymptotic series is derived by a systematic recursive scheme to permit an arbitrary order of the approximation. Used in conjunction with the first two expansions, this permits the single integral to be evaluated with an absolute accuracy of six decimals throughout the centerplane.

scholarly journals Polyanalytic boundary value problems for planar domains with harmonic Green function

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Heinrich Begehr ◽  
Bibinur Shupeyeva
Keyword(s):  
Green Function ◽  
Boundary Value Problems ◽  
Arbitrary Order ◽  
Boundary Value ◽  
Planar Domains ◽  
Schwarz Problem ◽  
Bianalytic Functions ◽  
Neumann Type ◽  
The Neumann Problem ◽  
Well Posed

AbstractThere are three basic boundary value problems for the inhomogeneous polyanalytic equation in planar domains, the well-posed iterated Schwarz problem, and further two over-determined iterated problems of Dirichlet and Neumann type. These problems are investigated in planar domains having a harmonic Green function. For the Schwarz problem, treated earlier [Ü. Aksoy, H. Begehr, A.O. Çelebi, AV Bitsadze’s observation on bianalytic functions and the Schwarz problem. Complex Var Elliptic Equ 64(8): 1257–1274 (2019)], just a modification is mentioned. While the Dirichlet problem is completely discussed for arbitrary order, the Neumann problem is just handled for order up to three. But a generalization to arbitrary order is likely.

Evaluation of the Wave-Resistance Green Function: Part 1—The Double Integral

1987 ◽  
Vol 31 (02) ◽  
pp. 79-90
Author(s):  
J. N. Newman
Keyword(s):  
Asymptotic Behavior ◽  
Three Dimensional ◽  
Steady Motion ◽  
Wave Resistance ◽  
Integral Representations ◽  
Far Field ◽  
Double Integral ◽  
Numerical Studies ◽  
Single Integral ◽  
Dimensional Domain

Analytical and numerical studies are made of the source potential for steady motion beneath a free surface. Various alternative integral representations are reviewed, and attention is focused on the component which usually is expressed as a double integral. A particular form is selected for numerical applications, where the double integral represents a symmetrical nonradiating disturbance, and the far-field waves are accounted for separately in the complementary single integral. Systematic expansions are derived for the singularity of the double integral at the origin, and for its asymptotic behavior far from the origin. Guided by these expansions, numerical approximations of the double integral are derived in terms of three-dimensional polynomials, which greatly facilitate the computation of the double integral. Tables of the coefficients in these approximations are presented, permitting the double integral to be evaluated throughout the three-dimensional domain with an accuracy of five to six decimal places. Greater accuracy can be achieved by using extended tables of the same coefficients. Algorithms for evaluating the Chebyshev polynomial approximations and a description of the computational methods used to derive the coefficients are included in the Appendices.

Evaluation of the Wavelike Disturbance in the Kelvin Wave Source Potential

1988 ◽  
Vol 32 (01) ◽  
pp. 44-53 ◽  
Author(s):  
J. J. M. Baar ◽  
W. G. Price
Keyword(s):  
Numerical Evaluation ◽  
Field Component ◽  
Near Field ◽  
Neumann Series ◽  
Series Expansions ◽  
Effective Solution ◽  
Kelvin Wave ◽  
Wave Source ◽  
Series Representations ◽  
Single Integral

This paper discusses the numerical evaluation of the characteristic Kelvin wavelike disturbance trailing downstream from a translating submerged source. Mathematically the function describing the wavelike disturbance is expressed as a single integral with infinite integration limits and a rapidly oscillatory integrand. Numerical integration of such integrals is both cumbersome and time-consuming. Attention is therefore focused on two complementary Neumann-series expansions which were originally derived by Bessho [1].2 Numerically stable algorithms are presented for the accurate and efficient evaluation of the two series representations. When used in combination with the Chebyshev expansions for the nonoscillatory near-field component which were recently obtained by Newman [2], the present algorithms provide an effective solution to the numerical difficulties associated with the evaluation of the Kelvin wave source potential.

scholarly journals CLOSED NEWTON-COTES CUBATURE SCHEMES FOR TRIPLE INTEGRALS WITH ERROR ANALYSIS

2021 ◽  
Vol 16 (1) ◽  
Author(s):  
Kamran Malik
Keyword(s):  
Applied Sciences ◽  
Error Analysis ◽  
Higher Dimensions ◽  
The Other ◽  
One Dimension ◽  
Local Error ◽  
Basic Form ◽  
Double Integral ◽  
Single Integral ◽  
Error Terms

Most of the problems in applied sciences in engineering contain integrals, not only in one dimension but also in higher dimensions. The complexity of integrands of functions in one variable or higher variables motivates the quadrature and cubature approximations. Much of the work is focused on the literature on single integral quadrature approximations and double integral cubature schemes. On the other hand, the work on triple integral schemes has been quite rarely focused. In this work, we propose the closed Newton-Cotes-type cubature schemes for triple integrals and discuss consequent error analysis of these schemes in terms of the degree of precision and local error terms for the basic form approximations. The results obtained for the proposed triple integral schemes are in line with the patterns observed in single and double integral schemes. The theorems proved in this work on the local error analysis will be a great aid in extending the work towards global error analysis of the schemes in the future.

Evaluation of the Wave-Resistance Green Function: Part 3— The Single Integral Near the Singular Axis

1994 ◽  
Vol 38 (01) ◽  
pp. 1-8
Author(s):  
J.-M. Clarisse ◽  
J.N. Newman
Keyword(s):  
Asymptotic Expansion ◽  
Free Surface ◽  
Green Function ◽  
Convergent Series ◽  
Wave Resistance ◽  
Complementary Region ◽  
Single Integral

The single integral part of the wave-resistance Green function is considered near its singular axis. Completing the work of Ursell (1988) an asymptotic expansion is proposed that is valid in the vicinity of the source when this lies on or near the free surface. For the complementary region away from the singular axis an improvement of Bessho's (1964) convergent series is also obtained. Numerically, both expres¬sions along with previous results guarantee fast and accurate evaluations of the single integral within two wavelengths downstream of the source. For larger distances downstream, the result of Ursell (1988) is shown to be still valid within bounded distances of the singular axis. Hence the single integral can be evaluated analytically or numerically in a whole neighborhood of its singular axis.

4. On some Quaternion Integrals

1872 ◽  
Vol 7 ◽  
pp. 318-320
Author(s):  
Tait
Keyword(s):  
Scalar Function ◽  
Closed Curve ◽  
Normal Vector ◽  
Double Integral ◽  
Single Integral ◽  
Image Position

AbstractIn my paper on “Green's and other allied theorems” (Trans. R. S. E. 1869–70), I showed thatwhere P is any scalar function of ρ, and the single integral is extended round any closed curve, while the double integral extends over any surface bounded by the curve, ν being its normal vector.Writingthis gives at onceof which the scalar and vector parts respectively were, in the paper referred to, shown to be equal.

An Iterative Technique for Minimising a Double Integral with Applications in Elasticity

1967 ◽  
Vol 71 (680) ◽  
pp. 573-575 ◽  
Author(s):  
J. P. H. Webber
Keyword(s):  
A Priori ◽  
Minimum Energy ◽  
Plane Elasticity ◽  
Iterative Technique ◽  
Double Integral ◽  
Energy Principle ◽  
Minimum Energy Principle ◽  
Undetermined Function ◽  
Single Integral ◽  
Initial Choice

The solution of a problem in plane elasticity can be associated with a minimum energy principle, involving the minimisation of a double integral subject to certain boundary conditions. In 1933, Kantorovich proposed a method which reduced this problem to that of finding the minimum of a single integral. He chose part of the solution a priori (in accordance with the character of the problem) which left only one undetermined function in one of the variables. This unknown function could then be found through the solution to an ordinary differential equation. He then constructed a second approximation by introducing two unknown functions which led to the solution of two simultaneous differential equations. Again the accuracy of the solution was dependent on the initial choice for part of the function; so also was the rate of convergence.

scholarly journals Consensus of Multi-Integral Fractional-Order Multiagent Systems with Nonuniform Time-Delays

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-24 ◽  
Author(s):  
Jun Liu ◽  
Wei Chen ◽  
Kaiyu Qin ◽  
Ping Li
Keyword(s):  
Numerical Simulations ◽  
Frequency Domain ◽  
Multiagent Systems ◽  
Fractional Order ◽  
Time Delays ◽  
Multiple Integral ◽  
Frequency Domain Method ◽  
Double Integral ◽  
Single Integral ◽  
Special Case

Consensus of fractional-order multiagent systems (FOMASs) with single integral has been wildly studied. However, the dynamics with multiple integral (especially double integral to sextuple integral) also exist in FOMASs, and they are rarely studied at present. In this paper, consensus problems for multi-integral fractional-order multiagent systems (MIFOMASs) with nonuniform time-delays are addressed. The consensus conditions for MIFOMASs are obtained by a novel frequency-domain method which properly eliminates consensus problems of the systems associated with nonuniform time-delays. Besides, the method revealed in this paper is applicable to classical high-order multiagent systems which is a special case of MIFOMASs. Finally, several numerical simulations with different parameters are performed to validate the correctness of the results.

scholarly journals On singular and highly oscillatory properties of the Green function for ship motions

2001 ◽  
Vol 445 ◽  
pp. 77-91 ◽  
Author(s):  
XIAO-BO CHEN ◽  
GUO XIONG WU
Keyword(s):  
Free Surface ◽  
Green Function ◽  
Velocity Potential ◽  
Error Function ◽  
Dispersion Curves ◽  
Ship Motions ◽  
Oscillatory Behaviour ◽  
Highly Oscillatory ◽  
Single Integral ◽  
The Green Function

The Green function used for analysing ship motions in waves is the velocity potential due to a point source pulsating and advancing at a uniform forward speed. The behaviour of this function is investigated, in particular for the case when the source is located at or close to the free surface. In the far field, the Green function is represented by a single integral along one closed dispersion curve and two open dispersion curves. The single integral along the open dispersion curves is analysed based on the asymptotic expansion of a complex error function. The singular and highly oscillatory behaviour of the Green function is captured, which shows that the Green function oscillates with indefinitely increasing amplitude and indefinitely decreasing wavelength, when a field point approaches the track of the source point at the free surface. This sheds some light on the nature of the difficulties in the numerical methods used for predicting the motion of a ship advancing in waves.

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