Partial differential equations : mathematical techniques for engineers

• Main Author: Epstein, M. (Author)
• Format: Electronic eBook
• Language: English
• Published: Cham, Switzerland : Springer, 2017.
• Series: Mathematical engineering.
• Subjects: Differential equations, Partial.; elektronické knihy; electronic books
• ISBN: 9783319552125; 9783319552118
• ISSN: 2192-4732
• Physical Description: 1 online resource (xiii, 255 pages) : illustrations (some color)

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....

Table of Contents:

• 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.