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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.

Electrons --- Emission. --- Electron emission --- Electronic work function --- Emission of electrons --- Work function, Electronic --- Electric discharges through gases --- Electron work function --- Free electron theory of metals --- Internal conversion (Nuclear physics) --- Physics. --- Thermodynamics. --- Heat engineering. --- Heat transfer. --- Mass transfer. --- Optical materials. --- Electronic materials. --- Semiconductors. --- Nanotechnology. --- Numerical and Computational Physics, Simulation. --- Engineering Thermodynamics, Heat and Mass Transfer. --- Optical and Electronic Materials. --- Nanotechnology and Microengineering. --- Molecular technology --- Nanoscale technology --- High technology --- Crystalline semiconductors --- Semi-conductors --- Semiconducting materials --- Semiconductor devices --- Crystals --- Electrical engineering --- Electronics --- Solid state electronics --- Electronic materials --- Optics --- Materials --- Mass transport (Physics) --- Thermodynamics --- Transport theory --- Heat transfer --- Thermal transfer --- Transmission of heat --- Energy transfer --- Heat --- Mechanical engineering --- Chemistry, Physical and theoretical --- Dynamics --- Mechanics --- Physics --- Heat-engines --- Quantum theory --- Natural philosophy --- Philosophy, Natural --- Physical sciences