N.L. Gol'dman
Inverse Stefan Problems
Dordrecht: Kluwer Academic Publishers, 1997

This monograph presents a new theory and methods of solving inverse Stefan problems for quasilinear parabolic equations in domains with free boundaries. Such problems arise in the modelling and control of nonlinear processes with phase transitions in thermophysics and mechanics of continuous media. The research in inverse Stefan problems is important for the perfection of technologies both in high temperature processes (e.g., metallurgy, aircraft, astronautics and power engineering) and in hydrogeology, exploitation of oil-gas fields, etc.

The statements of such inverse problems on the determination of boundary functions and coefficients of the equation are considered for different types of additional information about their solution.

The regularization variational method is proposed to obtain stable approximate solutions for this class of ill-posed problems. The numerical descriptive regularization algorithms implementing this method are developed. They utilize a priori knowledge of the qualitive structure of the sought solution and ensure substantial savings in computational costs. Results of calculations for important applications in a continuous casting and for the treatment of materials using laser technology are also given.

This book will be of interest to post-graduate students and researchers whose work involves partial differetial equations, numerical analysis, phase transformation and the mathematics of physics.