Perturbation methods in mechanics, 6 hp (FTN0415)

Picture of a diagram related to the topic of this course. Not of importance for the information of this page.

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 related to the research field of the graduate student. The total course workload corresponds to 6 hp.

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

The course starts the 3rd of October until late November.

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

Registration

To register to the course and for additional information please contact antonio.segalini@geo.uu.se.

Course schedule

Day Date Lecture Time Topic

Tue

3/10/2023

10-12 

Introduction, Heuristic analysis of a regular and singular problem

Wed

4/10/2023

10-12

Order relationship, Asymptotic series, Convergence, Asymptotic expansions

Tue

10/10/2023

10-12

Regular perturbation problems

Wed

11/10/2023

13-15

Problem Solving class                                                                                                                                      

Tue

17/10/2023

10-12

Regular perturbation problems (numerical methods for PDE) Singular perturbation examples

Wed

18/10/2023

13-15

Boundary-layer approach, Matching approaches

Tue

24/10/2023

13-15

General Boundary layer theory , Applications to PDE with BL

Wed

25/10/2023

10-12

Problem Solving class                                                                                                                                      

Tue

31/10/2023

13-15

Logaritms

Wed

01/11/2023

10-12

Method of strained coordinates, Method of Multiple scales

Tue

07/11/2023

13-15

Nonlinear oscillators, boundary-layer method with multiple scales

Thur 09/11/2023 10-12 Resonances, PDE analysis with multiple scales
Tue 21/11/2023 10-12 WKB approximation
Tue 28/11/2023 13-15 Problem Solving class    

Last modified: 2023-08-17