High-order asymptotic methods provide accurate, analytic solutions to intractable potential problems

The classical problem of determining the density and capacity of arrays of potential sources is studied. This corresponds to a wide variety of physical problems such as electrostatic capacitance, stress in elastostatics and the evaporation of fluid droplets. An asymptotic solution is derived that is shown to give excellent accuracy for arbitrary arrays of sources with non-circular footprints, including polygonal footprints. The solution is extensively validated against both experimental and numerical results. We illustrate the power of the solution by showcasing a variety of newly accessible classical problems that may be solved in a rapid, accurate manner.

for r < a(θ), where a(θ) = 1 + ϵf (θ), and f can be decomposed into a suitable finite Fourier series as where M ≥ 2. We note that the series may be chosen to start at i = 2 without loss of generality by selecting a suitable centre for the coordinate system for the source, as seen in Wray and Moore [1].
Then using a decomposition due to Copson [2] and Fabrikant [3], this can be written as where the L operator, as described by Fabrikant [3], is given by where for our purposes it suffices to truncate at K = 2M .Note that, in Eq. ( 3), we have introduced the variable θ 1 for convenience in the inversions that follow: it is simply the standard planar polar angle.
The key property to be used repeatedly is that Throughout, L-operators will be inverted using Eq. ( 5), while Abel-type operators will be inverted in the standard manner of Copson [2].The expression given in Eq. ( 3) is the integral equation to be inverted for σ.We now demonstrate how to do this for the case c S ≡ 1, although other choices proceed in a similar manner.Inverting the outermost L and Abel operators yields and making use of ( 5) yields This is now solved order by order by expanding Note that the solutions for each of these are given below explicitly in Eq. ( 11), Eq. ( 20) and Eq. ( 30) respectively.

Leading order
At leading order, the solution trivially coincides with that of a circular disk at uniform potential, as first demonstrated by [4].In order to correctly locate the square root singularity at the contact line while retaining smoothness at the origin, we use the asymptotically equivalent solution First order At first order, Eq. ( 8) yields The terms on the first line can be thought of as forcing terms for the standard Abel-type problem on the second line.
We therefore compute Making use of the helpful result yields where c(x) = 2 F 1 1 2 , 1 − j 2 ; 1; 1 − x 2 is independent of θ ′ and so does not contribute to the integral.Then where we have made use the Fourier series expansion for f (θ).Thus Eq. ( 13) yields which can be inverted to find Again, we may re-write this in a suitable asymptotically equivalent form,

Second order
The forcing terms up to second order are where Then, we may expand as ϵ → 0, where we find that At leading and first order all terms cancel, verifying Eq. ( 11) and Eq. ( 20), and hence the remaining system to solve is Inverting the Abel operator yields Then, and hence where the c j (r) and d j (r) are the Fourier coefficients corresponding to the terms in square brackets as given by Eq. (29).Note that this recovers the monochromatic solution given by Wray and Moore [1] in the appropriate case, but accurately treats the second order terms which were dealt with approximately therein.In line with Eq. (11) and Eq.(20), we map r to r/a(θ) in practical applications to ensure that singularities remain at the contact line while preserving appropriate smoothness properties at r = 0.
For the large values of j and ϵ encountered in extending outside the asymptotic limit to accommodate shapes such as polygons, Eq. (30) can occasionally yield unphysical divergence to negative infinity [1].Experimentation suggests that when determining the density for shapes with Fourier expansions that correspond to large values of ϵ (i.e.ones that fall far outside the asymptotic limit, such as those found for polygons), it is in fact best to truncate the sum Eq. (30) at twice the rotational symmetry of the source boundary.