Heat Kernel Method and its Applications

This book deals with the partial differential equations and some of their applications. The book consists of four parts. The first part presents some background material and an introduction to partial differential equations. The second part is a short introduction to various aspects of differential geometry.The third part deals with a systematic treatment of singularly perturbed parabolic partial differential equations. The main theme in this part is the development of a short-time asymptotic expansion for the heat kernel. Some advanced methods and extensions are discussed as well. The forth part is devoted to applications. It starts with a short introduction to financial mathematics including stochastic differential equations and the description of some basic models. The advanced perturbational techniques developed earlier are applied to some models of mathematical finance. The book is addressed primarily t o non-mathematicians interested in partial differential equations. The presentation is rather informal and is as self-contained as possible from some elementary introductory concepts to rather advanced technical methods. The style of the presentation makes it accessible for a wider audience. Although this book is intended for advanced undergraduate or beginning graduate students in, it should also provide a useful reference for professional physicists, applied mathematicians as well as quantitative analysts with an interest in partial dif ferential equations, mathematical physics, differential geometry, singular perturbations and mathematical finance.