© 2015

Methods of Mathematical Modelling

Continuous Systems and Differential Equations

  • Provides a self-contained and accessible introduction to mathematical modelling using ordinary and partial differential equations

  • Presents key approaches for formulating models and solution techniques via asymptotic analysis

  • Includes many challenging exercises and connections to classic models in applied mathematics including the Burgers equation, the Korteweg de Vries equation, Euler-Lagrange equations, pattern formation via Turing instabilities

  • Demonstrates a variety of solution techniques including boundary layer theory, self-similar solutions, fast/slow dynamical systems, and multiple scale analysis


Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Formulation of Models

    1. Front Matter
      Pages 1-1
    2. Thomas Witelski, Mark Bowen
      Pages 3-21
    3. Thomas Witelski, Mark Bowen
      Pages 23-45
    4. Thomas Witelski, Mark Bowen
      Pages 47-83
    5. Thomas Witelski, Mark Bowen
      Pages 85-110
  3. Solution Techniques

    1. Front Matter
      Pages 111-111
    2. Thomas Witelski, Mark Bowen
      Pages 113-125
    3. Thomas Witelski, Mark Bowen
      Pages 127-145
    4. Thomas Witelski, Mark Bowen
      Pages 147-165
    5. Thomas Witelski, Mark Bowen
      Pages 167-183
    6. Thomas Witelski, Mark Bowen
      Pages 185-200
    7. Thomas Witelski, Mark Bowen
      Pages 201-213
    8. Thomas Witelski, Mark Bowen
      Pages 215-232
  4. Case Studies

    1. Front Matter
      Pages 233-233
    2. Thomas Witelski, Mark Bowen
      Pages 235-249
  5. Back Matter
    Pages 251-305

About this book


This book presents mathematical modelling and the integrated process of formulating sets of equations to describe real-world problems. It describes methods for obtaining solutions of challenging differential equations stemming from problems in areas such as chemical reactions, population dynamics, mechanical systems, and fluid mechanics.

Chapters 1 to 4 cover essential topics in ordinary differential equations, transport equations and the calculus of variations that are important for formulating models. Chapters 5 to 11 then develop more advanced techniques including similarity solutions, matched asymptotic expansions, multiple scale analysis, long-wave models, and fast/slow dynamical systems.

Methods of Mathematical Modelling will be useful for advanced undergraduate or beginning graduate students in applied mathematics, engineering and other applied sciences.


Asymptotic Analysis Calculus of Variations Mathematical Modelling Multiple Scale Analysis Nonlinear Oscillators Ordinary Differential Equations Partial Differential Equations Perturbation Methods

Authors and affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.International Center for Science and Engineering ProgramsWaseda UniversityTokyoJapan

About the authors

Thomas Witelski is a Professor of Mathematics at Duke University specializing in nonlinear partial differential equations and fluid dynamics. He is a long-time participant in many study groups on mathematical modelling and industrial problems. He is the co-Editor-in-Chief of the Journal of Engineering Mathematics and also serves on the editorial board for the European Journal of Applied Mathematics. Witelski received his Ph.D. in Applied Mathematics from the California Institute of Technology in 1995 and was a postdoctoral fellow at the Massachusetts Institute of Technology.

Mark Bowen is an Associate Professor in the International Center for Science and Engineering Programs at Waseda University, where he teaches courses in differential equations and nonlinear dynamics. His expertise is in asymptotic analysis, nonlinear differential equations and fluid dynamics. He received his Ph.D. in Applied Mathematics in 1998 from the University of Nottingham.

Bibliographic information


“The text is well written. The authors have provided a clear and concise presentation of many important topics in a way that should be accessible to students following a first course in differential equations. … More advanced students could easily learn a significant amount of useful mathematics reading the text independently. … Methods of Mathematical Modelling is a welcome addition to the SUMS series and should prove to be useful for many instructors and students.” (Jason M. Graham, MAA Reviews,, February, 2016)

“The purpose of this text is to introduce the reader to the art of mathematical modeling … . The book provides an account of a number of useful for mathematical modelling techniques which are illustrated with examples and complemented with problems for self study.” (Yuriy V. Rogovchenko, zbMATH 1333.00025, 2016)