Linear Partial Differential Equations for Scientists and Engineers

Abstract

Partial differential equations (PDEs) originated from the study of geometrical surfaces and a wide variety of problems in mechanics. Since till today they are essential in modeling of natural phenomena, and have a wide range of applications in different branches of applied mathematics, physics, engineering, biology and chemistry, the PDEs form an essential part of the core mathematical syllabus for scientists and engineers. The goal of the book here commented upon is to provide the readers with fundamental concepts, the underlying principles, wide range of applications and various methods of solution of linear PDEs, rather than an elegant exposition of general theory of PDEs in functional spaces. The book is divided into fifteen chapters. Historical background of the development of PDEs theory and applications, as well as basic concepts and ideas of the field are sketched in the introductory chapter. Next, basic types of the first and second order linear PDEs are provided. Laplace, Helmholtz, heat, wave, elasticity equations as well as linear Schrodinger and linearized Korteweg – de Vries equations are recalled and employed to formulate mathematical models of physical phenomena. Models of vibrating strings and membranes, heat conduction in solids, as well as waves in elastic medium are presented. Moreover, spherical and also cylindrical wave equations are discussed. The main part of the book is devoted to presentation of basic principles and analytical methods for solving linear PDEs. Fundamental principles, such as the superposition principle, conservation laws, the maximum and the minimum principles for boundary value problems are recalled. Results concerning the existence, uniqueness and well-posedness of solutions to linear boundary value problems are provided. The presentation of basic methods for solving linear PDEs include: the method of characteristics for solving first order linear and quasi-linear PDEs, the method of separation of variables, the Sturm – Liouville approach for selfadjoint equations consisting in solving the associated eigenvalue problems, the method of eigenvalue expansions, and the method of Green’s functions. Separate chapter is devoted to Fourier, Laplace, Hankel, Mellin integral