Epstein M. Partial Differential Equations: Mathematical Techniques for Engineers

Springer International Publishing AG, 2017. — 261 p. — (Mathematical Engineering) — ISBN: 3319552112.This monograph presents a graduate-level treatment of partial differential equations (PDEs) for engineers. The book begins with a review of the geometrical interpretation of systems of ODEs, the appearance of PDEs in engineering is motivated by the general form of balance laws in continuum physics. Four chapters are devoted to a detailed treatment of the single first-order PDE, including shock waves and genuinely non-linear models, with applications to traffic design and gas dynamics. The rest of the book deals with second-order equations. In the treatment of hyperbolic equations, geometric arguments are used whenever possible and the analogy with discrete vibrating systems is emphasized. The diffusion and potential equations afford the opportunity of dealing with questions of uniqueness and continuous dependence on the data, the Fourier integral, generalized functions (distributions), Duhamel's principle, Green's functions and Dirichlet and Neumann problems. The target audience primarily comprises graduate students in engineering, but the book may also be beneficial for lecturers, and research experts both in academia in industry.Background
Vector Fields and Ordinary Differential Equations
Partial Differential Equations in Engineering
The First-Order Equation
The Single First-Order Quasi-linear PDE
Shock Waves
The Genuinely Nonlinear First-Order Equation
Classification of Equations and Systems
The Second-Order Quasi-linear Equation
Systems of Equations
Paradigmatic Equations
The One-Dimensional Wave Equation
Standing Waves and Separation of Variables
The Diffusion Equation
The Laplace Equation