As I think of writing about the present work I am pleasantly reminded of a few names which occupy very special place in my academic and professional career. In 1966, while I was working at the Post Office Research Station in London (now most probably at Martlesham) I got introduced to computers. I cannot forget, with how much patience and perseverence, B.E. Surtees had not only helped but had almost taught me Algol programming. Later I used the computer extensively for solving scientific problems. Further, the Research Station generously granted me day-release to attend MSc (Comp.Sc.) course at the City University, London. Although I could not complete the course, I developed a strong liking for Numerical Analysis. It would be my privilege to mention the name of Professor V.E. Price who taught the subject with full devotion and dedication. I must admit that I learnt the basics of Numerical Analysis from there and much of it makes part of Chapters 1 to 7 and 10 of the book. I was greatly impressed by the book Modern Computing Methods by E.T. Goodwin, and still am. My intense desire for working in Numerical Analysis was fulfilled when I joined PhD in 1969 at Brunel University under the guidance of Professor J. Crank who was known internationally in the field. Luckily, a very challenging problem came my way to work upon from Hammersmith Hospital, London. The problem required knowledge for solving partial differential equations numerically. The first book on p.d.e., I read was G.D. Smith's who was coincidently teaching in the same department. Therefore no wonder, my treatment for solving p.d.e.'s in Chapter 11 may be biased towards his book. I worked with Professor Crank for five years – three years for my PhD and two years as a postdoctoral research fellow. It was only his constant inspiration that kept me going and galloping. Those five years, I may call the most precious years of my life. I came to know with deep sense of sorrow and grief that Professor Crank passed away in October 2006.

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.