Partial Differential Equations

1. Front Matter

   Pages i-xiii

   PDF

2. Background

   1. Front Matter

      Pages 1-1

      PDF

   2. Vector Fields and Ordinary Differential Equations

      • Marcelo Epstein

      Pages 3-24

   3. Partial Differential Equations in Engineering

      • Marcelo Epstein

      Pages 25-47

3. The First-Order Equation

   1. Front Matter

      Pages 49-49

      PDF

   2. The Single First-Order Quasi-linear PDE

      • Marcelo Epstein

      Pages 51-74

   3. Shock Waves

      • Marcelo Epstein

      Pages 75-88

   4. The Genuinely Nonlinear First-Order Equation

      • Marcelo Epstein

      Pages 89-112

4. Classification of Equations and Systems

   1. Front Matter

      Pages 113-113

      PDF

   2. The Second-Order Quasi-linear Equation

      • Marcelo Epstein

      Pages 115-130

   3. Systems of Equations

      • Marcelo Epstein

      Pages 131-153

5. Paradigmatic Equations

   1. Front Matter

      Pages 155-155

      PDF

   2. The One-Dimensional Wave Equation

      • Marcelo Epstein

      Pages 157-182

   3. Standing Waves and Separation of Variables

      • Marcelo Epstein

      Pages 183-208

   4. The Diffusion Equation

      • Marcelo Epstein

      Pages 209-238

   5. The Laplace Equation

      • Marcelo Epstein

      Pages 239-252

6. Back Matter

   Pages 253-255

   PDF

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.

• Continuum physics
• Diffusion equations
• Non-linear models
• Discrete vibrating systems
• Hyperbolic equations
• Fourier integral
• Duhamel's principle
• Green's functions
• Dirichlet problems
• partial differential equations

“The book would be accessible to strong undergraduates with some multivariable calculus, basic linear algebra and ordinary differential equations. The author provides references at each stage to several of the standard texts, including those of Garabedian and John. This would be a good textbook for an introduction to PDEs or as a supplement to a more standard mathematical treatment.” (William J. Satzer, MAA Reviews, maa.org, July, 2017)

• Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Canada

  Marcelo Epstein

Policies and ethics