Elements of Numerical Analysis

Introduction

In Chapter 11, we have dealt with parabolic equations defined over certain domain D with some kind of conditions prescribed on the boundary S. In these problems the boundary remains fixed for all times. However, there are very many problems of great practical importance where the boundary does not remain stationary; it moves. The boundary (whole or part) changes its position, shape and/or size with time. That is, S becomes a function of time in addition to the space variables. Considering Cartesian coordinates, the boundary can move along the x-axis only in case of one-dimensional problem, in the x-y plane in case of two-dimensional problem and in space in the case of three-dimensional problem. These problems are called moving boundary problems (mbp's). The most common example of a mbp is found in heat flow when a solid undergoes melting or a liquid changes its state under the process of solidification. In the melting or solidification problems, the solid/liquid interface (moving boundary) separating the solid and the liquid phases is a function of space and time. Another example from heat flow may be that of ablation, i.e., removal of the material from the surface of a solid body due to excessive heating – a practical example may be that of ablation from the space capsule at the time of its re-entry into the earth's atmosphere. Other familiar examples may be like consolidation of earth dam or diffusion of a gas in an absorbing medium. Besides, many more examples of moving boundary problems may be found in reference [1].

The free boundary problems (fbp's) are elliptic equations representing special type of steady state problems. They are defined over a fixed domain but location of the boundary is not known, a priori. A practical example of a fbp may be given from the context of ‘fluid flow through porous media’ is, the seepage of water into an earth dam.