<i>Mixed Boundary Value Problems in Potential Theory</i>

Sneddon, I. N.; Gillis, J.

doi:10.1063/1.3035123

Cited by 366 publications

(243 citation statements)

References 0 publications

Supporting

Mentioning

237

Contrasting

Unclassified

Order By: Relevance

“…For example the classic book of Sneddon (1966) has references only on dual series, the book of Fabrikant (1991) doesn't mention dual or triple trigonometrical series, while the last book on the subject by Duffy (2008), has only one specific example on sine series. Examples on triple trigonometrical series are the studies of Tranter (1969) and Kerr et al (1994).…”

Section: ! Mixed Boundary Value Problems Involving Triple Trigonomementioning

confidence: 99%

“…Systems of that kind are well treated in the literature and there have been several authors who provided proper solutions. For a summary the reader is referred to the classical book of Sneddon (1966). However, it should be mentioned that the suggested solutions were derived without being complemented by numerical computations.…”

Section: Computation Of the Expansion Coefficients Of The Dual Trigonmentioning

confidence: 99%

“…However, it should be mentioned that the suggested solutions were derived without being complemented by numerical computations. The solution provided in Sneddon (1966) for instance, although accurate, does not allow efficient numerical performance as it requires the computation of the derivative of an integral for which no analytical solution exists and accordingly the only feasible way is to elaborate numerically. This type of complications however are prone to numerical inaccuracies and accordingly the system of (6.8) and (6.9) is processed further using the methodology suggested by Tranter (1969).…”

Section: Computation Of the Expansion Coefficients Of The Dual Trigonmentioning

confidence: 99%

See 2 more Smart Citations

Three-dimensional steep wave impact on a vertical cylinder

2016

View full text Add to dashboard Cite

The present study treats the three-dimensional hydrodynamic slamming problem on a vertical plate subjected to the impact of a steep wave moving towards the plate with a constant velocity. The problem is complicated significantly by assuming that there is a rectangular opening on the plate which allows the discharge of the liquid. The analysis is conducted analytically assuming linear potential theory. The examined configuration determines two boundary value problems with mixed conditions which should be taken into account properly. The mathematical process assimilates the plate with a degenerate elliptical cylinder allowing thus the employment of elliptical harmonics that ensure the satisfaction of the free-surface boundary condition of the front face of the steep wave, beyond the plate. This assumption leads to an additional boundary value problem with mixed conditions in the vertical direction. The associated problem involves triple trigonometrical series and it is solved through a transformation into integral equations. To tackle the boundary value problem in the vertical direction a perturbation technique is employed. Extensive numerical calculations are presented as regards the variation of the velocity potential on the plate at the instant of the impact which reveals the influence of the opening. The theory is extended to the computation of the total impulse exerted on the plate using pressure-impulse theory.

show abstract

Section: ! Mixed Boundary Value Problems Involving Triple Trigonomementioning

confidence: 99%

Section: Computation Of the Expansion Coefficients Of The Dual Trigonmentioning

confidence: 99%

Section: Computation Of the Expansion Coefficients Of The Dual Trigonmentioning

confidence: 99%

See 1 more Smart Citation

Three-dimensional steep wave impact on a vertical cylinder

2016

View full text Add to dashboard Cite

show abstract

“…In this respect, particularly in dealing with somewhat unusual mixed boundary value problems, the methods of complex potentials and singular integral equations appear to be far superior to the standard operational techniques. An extensive treatment of the opera--ti.onal techniques for the solution of dua'L series and dual integral equations may be found in a recent book by Sneddon [1]. [2][3][4][5][6][7][8][9][10][11][12][13] are some of the outstanding references on the theory and applications of the complex potentials and the singular integral equations.…”

Section: Multiple Integral Equationsmentioning

confidence: 99%

“…commonly the case, one may directly apply the j equilibrium principles and reduce the problem to a boundary value problem which consists of a (system of) differential equation(s) subject to cer-: tain boundary conditions. Even though in most practical applications the -minimization problem is further reduced to a boundary value problem, it may also be solved directly by using an approximate technique; such as the ing u and its normal derivatives) of order (at most) 2m-1, f and g are known functions, and s is any convenient coordinate defining the point on the boundary (say, the arc length)._ The domain D may contain the point at 1 'infinity and may be multiplyconnected. Contours forming the boundary are assumed to consist of piecewise smooth arcs.…”

mentioning

confidence: 99%

Mixed Boundary-value Problems in Mechanics

1978

View full text Add to dashboard Cite

Correspondence between Electrostatics and Contact Mechanics with Further Results in Equilibrium Charge Distributions

Batle,

Vlasiuk,

Ciftja

2023

Annalen der Physik

View full text Add to dashboard Cite

It is common in mesoscopic systems to find instances where several charges interact among themselves. These particles are usually confined by an external potential that shapes the symmetry of the equilibrium charge configuration. In the case of classical charges moving on a plane and repelling each other via the Coulomb potential, they possess a ground state à la Thomson or Wigner crystal. As the number of particles N increases, the number of local minima grows exponentially and direct or heuristic optimization methods become prohibitively costly. Therefore the only feasible approximation to the problem is to treat the system in the continuum limit. Since the underlying framework is provided by potential theory, we shall by‐pass the corresponding mathematical formalism and list the most common cases found in the literature. Then we prove a (albeit known) mathematical correspondence that will enable us to re‐discover analytical results in electrostatics. In doing so, we shall provide different methods for finding the equilibrium surface density of charges, analytical and numerical. Additionally, new systems of confined charges in three‐dimensional surfaces will be under scrutiny. Finally, we shall highlight exact results regarding a modified power‐law Coulomb potential in the d‐dimensional ball, thus generalizing the existing literature.

show abstract

Mixed Boundary Value Problems in Potential Theory

Cited by 366 publications

References 0 publications

Three-dimensional steep wave impact on a vertical cylinder

Three-dimensional steep wave impact on a vertical cylinder

Mixed Boundary-value Problems in Mechanics

Correspondence between Electrostatics and Contact Mechanics with Further Results in Equilibrium Charge Distributions

Contact Info

Product

Resources

About