Brian H Gilding - Academia.edu (original) (raw)

Papers by Brian H Gilding

$Research paper thumbnail of Pumping test analyses when the pumped well intercepts a vertical fracture$

Department of Water Affairs Technical Report 99, 1979

The classical analysis of pumping tests is based on the assumption that the pumped well acts as a... more The classical analysis of pumping tests is based on the assumption that the pumped well acts as an isolated point sink in an otherwise homogeneous aquifer. If the pumped well intercepts a facture, or other highly permeable zone, then this assumption is no longer realistic, for water is preferentially drawn from the highly permeable zone. Alternative methods of analysing the pumping test data are then needed. The report is devoted to a set of type-curves to describe the drawdown in an aquifer in which the pumped well intercepts a vertical fracture. As the practical interest in type-curves lies in their application as tools for estimating hydrogeological parameters, the report is divided in two parts. The first covers the application of the type-curves, and the second their theoretical development. The field practitioner interested solely in applications need refer only to the first part. Topics covered are the assumptions on which the type-curves are based, methods of interpretation and the optimal siting of observation wells. The second part of the report describes the origin of the type-curves, methods of approximation of the well-function, and a computer algorithm for purposes of calculation of the well-function.

Department of Water Affairs Technical Report 91, 1978

The report is of a project with two main objectives: (1) to become familiar with the finite eleme... more The report is of a project with two main objectives: (1) to become familiar with the finite element method as a tool for evaluating groundwater resources; and; (2) to appraise a modelling study of the Rawsonville-Goudini aquifer undertaken by an external consultant, in which a finite difference model was used. An introduction to the finite element method, descriptions of two computer programmes based on this method, and the execution of a verification study, are presented. Working through a documented case history, insight into conducting a groundwater modelling investigation is gained, a comparison is made between the finite difference and finite element techniques, and, the consultant's work is evaluated.

Department of Water Affairs Technical Report 90, 1979

Numerical methods for minimizing functions are being used in the Department of Water Affairs in a... more Numerical methods for minimizing functions are being used in the Department of Water Affairs in a number of situations. These include estimating parameters in a rainfall-runoff model, in a groundwater recharge model, and in a crop water-use model, and the calibration of finite-element aquifer models. This technical note surveys the numerical techniques for function minimization available at the time of writing. It is hoped that this report will serve as a reference manual for those interested in the further application of these techniques.

Comparison of Models of the Unsaturated Groundwater System and Evapotranspiration (in Dutch) Committee for Hydrological Research TNO Reports and Notes 13, 1984

SOMOF is a computer program for modelling the flow of water in an unsaturated soil. The name is a... more SOMOF is a computer program for modelling the flow of water in an unsaturated soil. The name is an acronym of Soil-Moisture Flow. The program is based on the solution of a nonlinear partial differential equation describing the water flow using a finite difference technique. A fluctuating watertable is treated by a transformation of the differential equation. The boundary condition applied at the phreatic surface is one of constant pressure. Water exchange at the land surface is interpreted as a potential flux condition. Water extraction by plants is incorporated through a sink function. Following a description of the characteristics of the program, its validation with data from an experimental catchment over a seven year period is presented.

Water Quality Modelling in the Inland Natural Environment, 1986

One of the options emerging from a recent master-planning study of the national water management ... more One of the options emerging from a recent master-planning study of the national water management of the Netherlands was that of, in times of need, diverting water from storage or from areas with a water surplus to areas with a water shortage via the existing surface water system. This mangement measure has however given cause for concern about the risk of contaminating natural groundwater reserves with alien water infiltrating out of the open channels. By alien water, water of a different chemical composition to that of the indigenous groundwater is understood. A preliminary study into the extent to which alien water could contaminate a groundwater body under the application of the management measure is described. The groundwater movement in a vertical cross-section perpendicular to an open channel has been simulated in three representarive situations. The tool used is a computer program for modelling the groundwater flow in a vertical profile with a phreatic surface, enhanced with software for plotting the position of a front of water introduced in the domain of interest. The results of the model simulations are presented as a sequence of pictures displaying the intrusion of the open-channel water into the subsurface as a function of time. The conclusion is, that under the conditions investigated, the threat of alien water contamination is minimal. This conclusion is bound up with the meteorological conditions prevailing in the Netherlands.

Proceedings of the Royal Society of Edinburgh Section A Mathematics, 1996

A degenerate parabolic partial differential equation with a time derivative and first- and second... more A degenerate parabolic partial differential equation with a time derivative and first- and second-order derivatives with respect to one spatial variable is studied. The coefficients in the equation depend nonlinearly on both the unknown and the first spatial derivative of a function of the unknown. The equation is said to display finite speed of propagation if a non-negative weak solution which has bounded support with respect to the spatial variable at some initial time, also possesses this property at later times. A criterion on the coefficients in the equation which is both necessary and sufficient for the occurrence of this phenomenon is established. According to whether or not the criterion holds, weak travelling-wave solutions or weak travelling-wave strict subsolutions of the equation are constructed and used to prove the main theorem via a comparison principle. Applications to special cases are provided.

Journal, of Mathematical Analysis and Applications, 1980

The paper continues work of the author and L.A.Peletier on self-similar solutions of the nonlinea... more The paper continues work of the author and L.A.Peletier on self-similar solutions of the nonlinear diffusion equation u_t = (u^m)_xx in which u is the unkown, and m is a constant greater than 1. Self-similar solutions of three types are considered. Substitution of the self-similar forms into the partial differential equation leads to a class of ordinary differential equations with two parameters. In two earlier papers it was shown that this ordinary differential equation admits weak solutions of two kinds. The first is a non-classical solution with bounded support. The second is a bounded, positive, classical solution. For both types, necessary and sufficient conditions for existence were determined. Uniqueness for all relevant parameter values has been established only for solutions of the first type. In the present paper, the open question of the uniqueness of solutions of the second type is resolved. Furthermore, results on the asymptotic behaviour of solutions of this type are obtained.

SIAM Journal on Mathematical Analysis, 1979

The solution of the Cauchy-Dirichlet problem for the porous media equation in one space dimension... more The solution of the Cauchy-Dirichlet problem for the porous media equation in one space dimension in the quarter-plane is studied. An estimate is obtained for its behaviour for large time, in terms of the prescribed lateral boundary data.

Journal of Mathematical Analysis and Applications, 1977

The authors consider the nonlinear diffusion equation u_t = (u^m)_xx where u is the density of a ... more The authors consider the nonlinear diffusion equation u_t = (u^m)_xx where u is the density of a polytropic gas in one-dimensional motion through a homogeneous porous medium, t is the time, and m a constant greater than 1. Self-similar solutions of three types are looked for. Substitution of the self-similar forms into the partial differential equation leads to a class of ordinary differential equations containing two parameters. Weak solutions with compact support of this class of equation were studied in an earlier paper. In the present paper the authors investigate weak solutions that do not have compact support. First it is shown that any such solution is a bounded, positive, classical solution. Then some nonexistence results are obtained, and existence and uniqueness theorems for different values of the parameters are proven.

Annali della Scuola Normale Superiore di Pisa Classe di Scienza, 1977

The author deals with the equation u_t = (a(u) u_x)_x + b(u) u_x in which subscripts denote parti... more The author deals with the equation u_t = (a(u) u_x)_x + b(u) u_x in which subscripts denote partial differentiation. The Cauchy problem, the Cauchy-Dirichlet problem in a semi-infinite strip, and the Cauchy-Dirichlel problem in a rectangle are studied. Weak solutions are formulated. The main results are existence, uniqueness, and local regularity theorems. Further it is shown how it is possible to extract maximum principles for weak solutions of the problems from existence proofs. The author also presents necessary and sufficient conditions for weak solutions of the problems to vanish in an open subset of their domain of definition.

Journal of Mathematical Analysis and Applications, 1976

Ordinary and Partial Differential Equations, 1974

The porous medium equation takes its name from its description of the laminar flow of a homogeneo... more The porous medium equation takes its name from its description of the laminar flow of a homogeneous gas in a homogeneous isotropic porous medium. Solutions to the Cauchy problem exist only in a generalized sense. An estimate of the Holder continuity of such a solution with respect to the spatial variable has been obtained by D.G.Aronson. The exponent is the best possible globally. Building on the aforementioned result, S.N.Kruzhkov has shown that a generalized solution is also Holder continuous with respect to the time variable, using a technique based on the maximum principle. An extension of this technique leads to the establishment of the Holder continuity of a generalized solution with respect to the time variable with the optimum exponent.

Annali della Scuola Normale Superiore di Pisa Classe di Scienze, 1990

This note corrects the typesetting errors in "Improved theory for a nonlinear degenerate paraboli... more This note corrects the typesetting errors in "Improved theory for a nonlinear degenerate parabolic equation" which appeared in the same journal the year before.

Uspekhi Matematicheskikh Nauk, 1998

The paper is concerned with the behaviour of a solution of the initial-value problem for a nonlin... more The paper is concerned with the behaviour of a solution of the initial-value problem for a nonlinear degenerate parabolic equation in several space variables. The equation contains a diffusive term and a convective term, involving nonlinear dependence on the unknown and the gradient of the unknown. Supposing that the support of the initial data is bounded on one side in a certain spatial direction, estimates on the growth of the of the support of the solution in that particular direction for large values of time are reported.

Journal of Mathematical Analysis and Applications, 2003

With q a positive real number, the nonlinear partial differential equation in the title of the pa... more With q a positive real number, the nonlinear partial differential equation in the title of the paper arises in the study of the growth of surfaces. In that context it is known as the generalized deterministic KPZ equation. The paper is concerned with the initial-value problem for the equation under the assumption that the initial-data function is bounded and continuous. Results on the existence, uniqueness, and regularity of solutions are obtained.

Dental materials : official publication of the Academy of Dental Materials, 2018

Miniature magnets are used in dentistry, principally for the retention of prosthetic devices. The... more Miniature magnets are used in dentistry, principally for the retention of prosthetic devices. The relationship between force and separation of a magnet and its keeper, or, equivalently, two such magnets, has been neither defined theoretically nor described practically in any detail suitable for these applications. The present paper addresses this lacuna. A magnet is considered as a conglomeration of magnetic poles distributed over a surface or a solid in three-dimensional space, with the interaction of poles governed by the Coulomb law. This leads to a suite of mathematical models. These models are analysed for their description of the relationship between the force and the separation of two magnets. It is shown that at a large distance of separation, an inverse power law must apply. The power is necessarily integer and at least two. All possibilities are exhausted. Complementarily, under reasonable assumptions, it is shown that at a small distance of separation, the force remains f...

Comptes Rendus De L Academie Des Sciences Serie 1 Mathematique, 1991

The property of finite speed of propagation for the general nonlinear diffusion-convection-reacti... more The property of finite speed of propagation for the general nonlinear diffusion-convection-reaction equation of the form u t =(a(u)) xx +(b(u)) x +c(u) is characterized. This is achieved utilizing travelling-wave solutions of the equation. The study of the travelling waves is reduced to the analysis of a singular nonlinear Volterra integral equation.

Nonlinear Analysis: Theory, Methods & Applications, 2015

Nonlinear second-order hyperbolic equations are gaining ground as models in many areas of applica... more Nonlinear second-order hyperbolic equations are gaining ground as models in many areas of application, as extensions of parabolic reaction–diffusion equations that might otherwise be used. The theory of travelling-wave solutions of such reaction–diffusion equations is well established. The present paper is concerned with its counterpart for the wider class of equations in the particular case that the reaction term is bistable. Conditions that are necessary and sufficient for the existence and uniqueness of these solutions are determined. A combination of traditional ordinary differential equation techniques and an innovatory integral equation approach is employed.

Proceedings of the American Mathematical Society, 1990

The Cauchy problem for a nonlinear diffusion-convection equation is studied. The equation may be ... more The Cauchy problem for a nonlinear diffusion-convection equation is studied. The equation may be classified as being of degenerate parabolic type with one spatial derivative and a time derivative. It is shown that under certain conditions solutions of the initial-value problem exhibit instantaneous shrinking. This is to say, at any positive time the spatial support of the solution is bounded above, although the support of the initial data function is not. This is a phenomenon which is normally only associated with nonlinear diffusion with strong absorption. In conjunction, a previously unreported phenomenon is revealed. It is shown that for a certain class of initial data functions there is a critical positive time such that the support of the solution is unbounded above at any earlier time, whilst the opposite is the case at any later time.

Nonlinearity, 2003

This note concerns reaction-diffusion processes which display remarkable behaviour. Everywhere th... more This note concerns reaction-diffusion processes which display remarkable behaviour. Everywhere the concentration, density or temperature exceeds some critical level until at some moment in time it decreases to the critical level at one point in space. At this instant, the complete profile immediately drops to the critical level at every point in space, and then remains there.