A moving boundary problem arising from the diffusion of oxygen in absorbing tissue

Approximate analytical and numerical solutions of a partial differential equation are obtained which describe the diffusion of oxygen in an absorbing medium. Essential mathematical difficulties are associated with the presence of a moving boundary which marks the furthest penetration of oxygen into the medium and also with the need to allow for an initial distribution of oxygen through the medium

The numerical solution of elliptic and parabolic partial differential equations with boundary singularities

A general numerical method is described for the solution of linear elliptic and parabolic partial differential equations in the presence of boundary singularities. The method is suitable for use with either a finite-difference or finite element scheme. Modified approximations for the derivatives are developed using the local analytical form of the singularity. General guidelines are given showing how the local analytical form can be found and how the modified approximations can be developed for many problems of mathematical physics. These guidelines are based on the reduction of the differential equation to the form Δu = gu + f. The potential problem treated by Motz and Woods is taken as a numerical example. The numerical results compare favourably with those obtained by other techniques

Solution of three-dimensional free boundary problems by variable interchange

The three-dimensional problem of the seepage of fluid through a circular earth dam is solved to illustrate a new approach to more general free boundary problems. The method is based on Boadway's transformation in which the dependent variable, representing velocity potential, is interchanged with one of the independent space variables, which then becomes the new dependent variable to be computed. The need to determine the position of the whole of the free surface in the three-dimensional physical space is reduced to locating the position of the separation line on a fixed plane boundary in the transformed domain. An iterative algorithm approximates within each single loop both a finite-difference solution of the partial differential equation and the position of the separation line

A method for solving moving boundary problems in heat flow Part I: Using cubic splines

A new approach to a heat-flow problem involving a moving boundary makes use of a grid system which moves with the boundary. The necessary interpolations are performed by using cubic splines. The method smooths out irregularities in the motion of the boundary which were evident in previous calculations based on a fixed grid system

A method for solving moving boundary problems in heat flow part ii: Using cubic polynomials

A moving grid system has been used to get the solution of the moving boundary problem discussed earlier in Part I, but basing the necessary interpolations on ordinary cubic polynomials rather than splines. The computations are much more economical and the results obtained are also found to he more satiafactory

Numerical solution of a free boundary problem by interchanging dependent and independent variables

The classical problem of seepage of fluid through a porous dam is solved to illustrate a new approach to more general free boundary problems. The numerical method is based on the interchange of the dependent variable, representing velocity potential, with one of the independent space variables, which becomes the new variable to be computed. The need to determine the position of the whole of the free boundary in the physical plane is reduced to locating the position of the separation point on a fixed straight—line boundary in the transformed plane. An iterative algorithm approximates within each single loop both a finite-difference solution of the partial differential equation and the position of the free boundary. The separation point is located by fitting a 'parabolic tail' to the finite-difference solution

Isotherm migration along orthogonal flow lines in two dimensions

A novel approach to the solution of transient heat flow problems in two dimensions is described. The movements of isotherms along orthogonal flow lines are tracked in successive small intervals of time by solving a locally one-dimensional IMM form of radial heat equation. The determination of the new orientation of the orthogonal system at the end of each time interval is based on geometrical considerations. The method is applied to the moving boundary problem presented by the solidification of a square prism of fluid and the results compared with those obtained by the enthalpy method

A simple finite-difference modification for improving accuracy near a corner in heat flow problems

Traditional numerical methods of solution of heat flow problems are subject to inaccuracies near sharp corners, where the derivatives of the exact solution may "become unbounded (Jeffreys1). 2Previous methods of overcoming this difficulty (Motz, Woods3

The treatment of boundary singularities in axially symmetric problems containing discs

Axially symmetric problems (e.g. Laplace's equation in cylindrical co-ordinates) containing discs possess boundary singularities arising from the mixed boundary conditions that occur across the disc edge. A modified finite-difference method is presented which effectively eliminates the inaccuracies that occur in the standard numerical solution near such singularities. Techniques for developing the analytical forms of such singularities are given and modified finite-difference approximations are obtained. The steady-state diffusion of oxygen around a circular electrode is taken as the model problem and a modified quadrature method is presented for the calculation of the oxygen flux through the electrode