Perturbation methods in mechanics, 6 hp (FTN0415)

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Course description

The course is designed to provide to the students familiarity with perturbation methods, with special focus on how these methods provide useful insight in mathematical problems encountered in physics and engineering. The solution of ordinary differential equations with one small/large parameter will be analyzed, both within the framework of regular- or singular- perturbation theory, with special attention on boundary-layer theory, WKB approaches and multiple-scale analyses. The extension of the methods to partial-differential equations will also be discussed. Special focus will be devoted to nonlinear oscillators in different operating conditions.

The course consists of 14 two-hour lectures with 6 assignments that the students will present in problem-solving classes. A final home assignment is also expected. The total course workload corresponds to 6 hp.

Basic knowledge of ordinary differential equations, mechanics and Matlab/Python is required to attend the course.

The course starts the 20th of March until late May.

Learning outcomes

Once the course will be completed, the student should be able to:

  • Explain basic concepts of perturbation techniques, such as order relationships, asymptotic sequences, asymptotic expansions and convergence issues.
  • Propose a solution method for regular perturbation problems.
  • Explain the difference between a regular and a singular perturbation problem.
  • Analyze a singular problem by means of a balancing method, methods of strained coordinates and boundary-layer theory.
  • Determine inner and outer solutions for singular perturbation problems by means of boundary-layer theory and the composite form.
  • Use WKB methods to solve linear ordinary differential equations subjected to different length or time scales.
  • Perform a multiple-scale analysis on linear and non-linear problems.
  • Nonlinear oscillators.
  • Apply perturbation methods to partial-differential problems

Course literature

  • M. Holmes (2013) Introduction to Perturbation Methods, Second Edition
  • D. Wilcox (1995) Perturbation methods in the computer age. DWC Industries Inc.
  • E. J. Hinch (1991) Perturbation methods. Cambridge University Press.
  • C. Bender & S. Orszag (2010) Advanced mathematical methods for scientists and engineers. Springer


To register to the course and for additional information please contact

Last modified: 2023-01-19