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Preface; Contents; Part I Background; 1 Vector Fields and Ordinary Differential Equations; 1.1 Introduction; 1.2 Curves and Surfaces in mathbbRn; 1.2.1 Cartesian Products, Affine Spaces; 1.2.2 Curves in mathbbRn; 1.2.3 Surfaces in mathbbR3; 1.3 The Divergence Theorem; 1.3.1 The Divergence of a Vector Field; 1.3.2 The Flux of a Vector Field over an Orientable Surface; 1.3.3 Statement of the Theorem; 1.3.4 A Particular Case; 1.4 Ordinary Differential Equations; 1.4.1 Vector Fields as Differential Equations; 1.4.2 Geometry Versus Analysis; 1.4.3 An Example.

1.4.4 Autonomous and Non-autonomous Systems1.4.5 Higher-Order Equations; 1.4.6 First Integrals and Conserved Quantities; 1.4.7 Existence and Uniqueness; 1.4.8 Food for Thought; References; 2 Partial Differential Equations in Engineering; 2.1 Introduction; 2.2 What is a Partial Differential Equation?; 2.3 Balance Laws; 2.3.1 The Generic Balance Equation; 2.3.2 The Case of Only One Spatial Dimension; 2.3.3 The Need for Constitutive Laws; 2.4 Examples of PDEs in Engineering; 2.4.1 Traffic Flow; 2.4.2 Diffusion; 2.4.3 Longitudinal Waves in an Elastic Bar; 2.4.4 Solitons.

2.4.5 Time-Independent Phenomena2.4.6 Continuum Mechanics; References; Part II The First-Order Equation; 3 The Single First-Order Quasi-linear PDE; 3.1 Introduction; 3.2 Quasi-linear Equation in Two Independent Variables; 3.3 Building Solutions from Characteristics; 3.3.1 A Fundamental Lemma; 3.3.2 Corollaries of the Fundamental Lemma; 3.3.3 The Cauchy Problem; 3.3.4 What Else Can Go Wrong?; 3.4 Particular Cases and Examples; 3.4.1 Homogeneous Linear Equation; 3.4.2 Non-homogeneous Linear Equation; 3.4.3 Quasi-linear Equation; 3.5 A Computer Program; References; 4 Shock Waves; 4.1 The Way Out.

4.2 Generalized Solutions4.3 A Detailed Example; 4.4 Discontinuous Initial Conditions; 4.4.1 Shock Waves; 4.4.2 Rarefaction Waves; References; 5 The Genuinely Nonlinear First-Order Equation; 5.1 Introduction; 5.2 The Monge Cone Field; 5.3 The Characteristic Directions; 5.4 Recapitulation; 5.5 The Cauchy Problem; 5.6 An Example; 5.7 More Than Two Independent Variables; 5.7.1 Quasi-linear Equations; 5.7.2 Non-linear Equations; 5.8 Application to Hamiltonian Systems; 5.8.1 Hamiltonian Systems; 5.8.2 Reduced Form of a First-Order PDE; 5.8.3 The Hamilton -- Jacobi Equation; 5.8.4 An Example.

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.