Analytic formulation of Cauchy integrals for boundaries with curvilinear geometry

Abstract

A general framework for analytic evaluation of singular integral equations with a Cauchy kernel is developed for higher order line elements of curvilinear geometry. This extends existing theory which relies on numerical integration of Cauchy integrals since analytic evaluation is currently published only for straight lines, and circular and hyperbolic arcs. Analytic evaluation of Cauchy integrals along straight elements is presented to establish a context coalescing new developments within the existing body of knowledge. Curvilinear boundaries are partitioned into sectionally holomorphic elements that are conformally mapped from a local curvilinear Z-plane to a straight line in the Z-plane. Cauchy integrals are evaluated in these planes to achieve a simple representation of the complex potential using Chebyshev polynomials and a Taylor series expansion of the conformal mapping. Bell polynomials and the Faà di Bruno formula provide this Taylor series for mappings expressed as inverse mappings and/or compositions. Examples illustrate application of the general framework to boundary-value problems with boundaries of natural coordinates, Bezier curves and B-splines. Strings formed by the union of adjacent curvilinear elements form a large class of geometries along which Dirichlet and/or Neumann conditions may be applied. This provides a framework applicable to a wide range of fields of study including groundwater flow, electricity and magnetism, acoustic radiation, elasticity, fluid flow, air flow and heat flow.

An analytic solution for groundwater uptake by phreatophytes spanning spatial scales from plant to field to regional

“…24]]. This convention is adopted here so the kernel functions of the line elements presented later reproduce those in [25].…”

“…The discharge per width, with components v x and v y , may be obtained from the complex conjugate of the derivative of with respect to z [25] w…”

“…The values of the coefficients B k j (a i ) may be computed recursively for specified values of a i following [25] using…”

“…38] and are fully derived in [25] and take on the following forms in the near-field and far-field of a line:…”

“…The complex potential for the phreatophytes within a field is obtained by summing the complex potentials for all plants within the field (18) for the line elements along the fields (25) and for a point-sink (36). This is illustrated in Fig.…”