# FSG3123 Perturbation Methods in Mechanics 7.5 credits

The course is designed for graduate students in the engineering sciences to provide to them 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.

Course offering missing for current semester as well as for previous and coming semesters
Headings with content from the Course syllabus FSG3123 (Spring 2019–) are denoted with an asterisk ( )

## Content and learning outcomes

### Course contents

The course is designed for graduate students in the engineering sciences to provide to them 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.

### Intended 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
• Apply perturbation methods to partial-differential problems

### Course disposition

The course consists of 14 two-hour lectures with 6 assignments. From the 3rd meeting, lectures will alternate with problem-solving classes, where students will discuss at the whiteboard their solution of pre-assigned problems from the Wilcox book, possibly showing comparisons with numerically integrated equations or analytical solutions. The discussion of at least one assigned problem is compulsory to pass the course.

The final exam shall consist of a home assignment to be delivered to the teachers.

## Literature and preparations

### Specific prerequisites

Basic knowledge of ordinary differential equations, Mechanics and Matlab.

### Recommended prerequisites

Basic knowledge of ordinary differential equations, Mechanics and Matlab.

### Equipment

No information inserted

### Literature

• 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

## Examination and completion

If the course is discontinued, students may request to be examined during the following two academic years.

P, F

### Examination

• DEL1 - Participation, 1.5 credits, grading scale: P, F
• INL1 - Assignment, 6.0 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

DEL1 Participating 1,5 hp (P, F)

INL1 Inlämningsuppgift 6,0 hp (P, F)

### Other requirements for final grade

• Active participation to the lectures and, in particular, to the problem-solving classes.
• Homework assignment.

### Opportunity to complete the requirements via supplementary examination

No information inserted

### Opportunity to raise an approved grade via renewed examination

No information inserted

### Ethical approach

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

## Further information

### Course web

Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

Course web FSG3123

SCI/Mechanics

### Main field of study

This course does not belong to any Main field of study.

### Education cycle

Third cycle

No information inserted

Antonio Segalini

### Supplementary information

Teachers:

·      Dr. Antonio Segalini (KTH Mechanics)

·      Docent Anders Dahlkild (KTH Mechanics)