The re-iterated Galerkin method

The iterated Galerkin method introduced by Sloan is modifiedto allow the improvement in the Galerkin approximation whichiteration produces to be repeated. It is shown in principle,and verified in illustration, that the repeated improvementsmay be achieved under weaker conditions than those requiredby Sloan. The sequence of approximations obtained is appliedto variational principles and in the case when a maximum principleis available it is shown that each step in the process improvesthe estimate of the maximum value of the associated functional.Applications to integral equations are given.     

The Edge Effect of a Surface Film on Gravity Waves

A mathematical model is developed for surface waves which propagatefrom a region with a clean free surface into a region beneathan inextensible surface film. Matched asymptotic expansionsare used and the Wiener-Hopf technique is invoked in order todetermine the velocity field in a small boundary-layer regionadjacent to the edge of the film. Within this transition region,where the boundary-layer length and the capillary length arerequired to be comparable, the local form of the free surfaceis determined.     

The Numerical Determination of Fundamental Solutions of Elliptic Equations

A method is developed for the numerical determination of thefundamental solution of a second-order, elliptic partial differentialequation. It is shown that existing formulations based on complex-variabletechniques can be modified to produce a Volterra-type integralequation for a complex-valued function related to the fundamentalsolution. The integral equation is easily solved by using arecurrence algorithm and two examples are given as illustrations.The principal application of the method, to subsonic compressibleflow problems, will be reported in due course.     

On Some Integral Equations with a Hankel Function Kernel

A method is presented for converting integral and integro-differentialequations whose kernel is the Hankel function into Cauchy singular equations. The latter may be solved bystandard function-theoretic methods or by implementing a secondtransformation to convert them into Abel-type equations, whichare readily inverted. Examples are drawn from wave diffractiontheory in two dimensions and include a mixed boundary-valueproblem which may be reduced to a pair of coupled integro-differentialequations. It is shown that the conversion to a singular equationpair and thence to Abel form permits uncoupling and a directsolution. Other applications of the method are briefly indicated.     

THE ECONOMIC VALUE OF ENVIRONMENTAL RESEARCH IN UNDERSTANDING THE RELATIVE CONTRIBUTIONS OF SOURCES OF NUTRIENTS TO COASTAL WATERS

ABSTRACT. We model the value of environmental research in the presence of uncertainty about thesources of environmental pollutants and natural processes affecting the level of pollution. The model may be used to estimate the value of environmental research directed at resolving the uncertainty. We illustrate the model using a numerical simulation of a hypothetical case involving nutrient pollution of coastal waters. We show that the ex ante value of research is positively related to the level of uncertainty. There is a diminishing return with respect to the level of research investment. We find that research is more valuable ex post if it leads to unexpected findings.     

OPTIMAL RESPONSES TO SHORELINE CHANGES: AN INTEGRATED ECONOMIC AND GEOLOGICAL MODEL WITH APPLICATION TO CURVED COASTS

The placement of sandy fill on an eroding beach, also known as “beach nourishment,” is a soft‐structure response to property loss often undertaken by coastal property owners. Here, we examine optimal beach nourishment along a curved shoreline with a model that couples geomorphic shoreline evolution with human responses to that evolution. The model includes the exogenous effects of shoreline recession due to sea‐level rise and the endogenous shoreline changes due to gradients in alongshore sediment transport. Human responses (nourishments) to shoreline changes that are not constant along the coast will change the shoreline curvature, thereby influencing local rates of shoreline change. We examine the net benefits of two alternative strategies to undertake soft structural responses to erosion: (i) the decisions of property owners to respond independently and (ii) the decisions of property owners to coordinate their individual responses. Under plausible parameterization, we find that a strategy to coordinate can dominate a strategy to act independently, and the benefits of coordination increase with greater shoreline curvature. This integration of economic and geological models demonstrates how feedbacks between human activities and natural processes can affect the evolution of the shoreline.