Treatment of Singularities in a Harmonic Mixed Boundary Value Problem by Dual Series Methods

Whiteman, J. R.

doi:10.1093/qjmam/21.1.41

Cited by 24 publications

(5 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A range of mathematical techniques can be deployed to solve such problems [14,18]; contributions go back at least as far as the work of Weber [19], who presented what is effectively the vapour concentration field induced by a thin circular droplet. Subsequent work has employed methods including separation of variables [20], orthogonal polynomial expansions [21], Fourier or Hankel transforms [22,23], and Green's functions [24].…”

Section: Introductionmentioning

confidence: 99%

The shielding effect extends the lifetimes of two-dimensional sessile droplets

et al. 2020

View full text Add to dashboard Cite

We consider the diffusion-limited evaporation of thin two-dimensional sessile droplets either singly or in a pair. A conformal-mapping technique is used to calculate the vapour concentrations in the surrounding atmosphere, and thus to obtain closed-form solutions for the evolution and the lifetimes of the droplets in various modes of evaporation. These solutions demonstrate that, in contrast to in three dimensions, in large domains the lifetimes of the droplets depend logarithmically on the size of the domain, and more weakly on the mode of evaporation and the separation between the droplets. In particular, they allow us to quantify the shielding effect that the droplets have on each other, and how it extends the lifetimes of the droplets.

show abstract

Section: Introductionmentioning

confidence: 99%

The shielding effect extends the lifetimes of two-dimensional sessile droplets

et al. 2020

View full text Add to dashboard Cite

show abstract

“…This is desirable, since current analytic methods apply only to special equations, require the use of integral equation and special function theory, and do not lend themselves easily to numerical work; see, e.g. [1,2,6,8,9,10,11,12,13,14,15,16,17].…”

mentioning

confidence: 99%

Least squares approximations for dual trigonometric series

Kelman

Koper

1973

Glasgow Math. J.

View full text Add to dashboard Cite

1. Introduction. A systematic and easily automated least squares procedure, not using integral equations or special functions, is presented for approximating the solutions of general dual trigonometric equations. This is desirable, since current analytic methods apply only to special equations, require the use of integral equation and special function theory, and do not lend themselves easily to numerical work; see, e.g. [1,2,6,8,9,10,11,12,13,14,15,16,17].The series are described in § 2. The equation for least squares approximation is derived in § 3 and used to develop the computer program DUTSA (DUal Trigonometric Series Analyser). A few examples are presented in § 4 from amongst the several dozen dual trigonometric series to which DUTSA has been applied. These include examples from classes of dual equations for which solutions are not now available (save possibly for very special cases), e.g. arbitrary series not connected with applications, series associated with harmonic mixed boundary value problems in (bounded) rectangles and series with one of the mixed boundary conditions corresponding to a (linear) radiation condition.! The evidence from these computations indicates that most dual trigonometric series from applications can be solved with a relative least squares error (defined in § 5) smaller than 4% in 10 seconds or fewer on a computer with a 6 microsecond multiplication time.Our analysis, in common with earlier studies, is heuristic: the numerical evidence is suggestive, but it does not rigorously describe the limitations of the method. The least squares approach is so simple that we searched the literature carefully, but found no evidence of its prior use. We expect that it will be helpful in studying other dual and similar series.

show abstract

“…The problem(4.1) is derived using symmetry from a well known problem in a rectangle containing a slit which has been much studied; see for example Whiteman [ 6 ], [7 ] and Wait and Mitchell [ 5 ] . We define the two disjoint parts of the boundary The effect of the singularity is to reduce the accuracy of the Galerkin solution, particularly in the neighbourhood of the re-entrant corner.…”

Section: Galerkin Methods For Model Problemmentioning

confidence: 99%