Partial Differential Equations-An Introduction

This book presents an introduction to the theory of partial differential equations(PDEs). The book is suitable for all types of basic courses on PDEs. Chapter 2 is devoted on first order PDEs. They are considered classiffcation of first order PDEs, solvability of quasilinear first order PDEs, the Cauchy problem for quasilinear first order PDEs, the Pfaffan equation and some special systems. In Chapters 3 and 4 are considered the classiffcation and canonical forms of second order PDEs. Chapter 5 is concerned with the wave equation. They are investigated even and odd dimensional wave equations, method of separation of variables, energy method. It is introduced the Riemann functions. Chapter 6 deals with the heat equation. They are considered the weak and strong maximum principles, the Cauchy problem, the mean value formula, the method of separation of variables, the energy method. The Laplace equation is introduced in Chapter 7. They are given the basic properties of elliptic problems, the fundamental solutions, integral representation of harmonic functions, mean-value formulas, strong principle of maximum. Chapter 8 is devoted on Cauchy-Kovalevskay theorem.