main-content

This book deals with mathematical modeling, namely, it describes the mathematical model of heat transfer in a silicon cathode of small (nano) dimensions with the possibility of partial melting taken into account. This mathematical model is based on the phase field system, i.e., on a contemporary generalization of Stefan-type free boundary problems. The approach used is not purely mathematical but is based on the understanding of the solution structure (construction and study of asymptotic solutions) and computer calculations. The book presents an algorithm for numerical solution of the equations of the mathematical model including its parallel implementation. The results of numerical simulation concludes the book. The book is intended for specialists in the field of heat transfer and field emission processes and can be useful for senior students and postgraduates.​

### Chapter 1. Introduction

Abstract
In this chapter, a general description of the mathematical model of heat transfer and field emission is presented.

### Chapter 2. Physical Basis for Field Emission

Abstract
In this chapter, without claiming to be original, we recall some basic notions of the solid-state physics which will be used to construct the mathematical model of the field emission cathode. More detailed descriptions of these facts can be found in any literature on the solid-state physics. Our description is based on [2, 3, 23, 27, 28, 30, 41, 42]. In this chapter, we also present some known notions from the theory of field emission based on the papers and monographs [4, 1012, 17, 20, 34, 36, 37, 39, 43] and several others which will be mentioned in the course of presentation.

### Chapter 3. Mathematical Model

Abstract
This chapter is a “mathematical” one. Here we collect the mathematical background related to the mathematical model of phase transition based on the phase field system introduced by G. Caginalp. Sections 3.1 and 3.2 of the chapter contain some preliminaries and considerations about mathematical models from the physical viewpoint. In Sect. 3.3, we give the results of asymptotic analysis applied to the phase field system. In Sect. 3.4, we discuss a new definition of the generalized solution to the phase field system which is stable under passing to the limiting Stefan–Gibbs–Thomson problem. Finally, in Sect. 3.5, we discuss an approach which is a combination of mathematical (asymptotic) investigation and numerical analysis.