**Course Code: MATH 6191**

**Course Title: ASYMPTOTIC AND PERTURBATION ANALYSIS**

Level: Graduate

Level: Graduate

**Course**

Type: Elective

Type: Elective

No. of Credits: 4

No. of Credits: 4

Pre-requisites: None

Pre-requisites: None

**Course Rationale**

Perturbation methods are often used by applied mathematicians, physicists and engineers to systematically construct an approximation of the solution of a problem that is otherwise unsolvable. It is a fundamental tool of applied mathematics. The methods rely on there being a parameter in the problem that is relatively small. Small parameters are inherent in several applications, such as population dynamics, solid and fluid mechanics, aerodynamics, electrodynamics, wave propagation, detonation theory, heat transfer, chemically reactive systems, combustion, celestial mechanics, biological systems and nonlinear oscillators.

Although computational techniques may be utilized to solve highly nonlinear, inhomogeneous and multidimensional problems, purely numerical solutions often do not provide physical insight into the physics of the problem. Analytical perturbation techniques are often used in conjunction with computer-generated solutions to provide a more complete analysis of the solution, as well as the influence of individual control parameters. This course will be an asset for students doing their final MSc project in the broad field of applied mathematics.

**Course Description**

This course will serve as a survey of the more important asymptotic and multiple scale analysis methods utilized in mathematics and the applied sciences. These methods are used to systematically construct approximations to problems that are otherwise intractable.

Students taking this course must have a thorough understanding of ordinary differential equations. Prior knowledge of numerical techniques for ordinary differential equations, and the fundamentals of matrix theory (eigenvalues, eigenvectors and determinants) will be assumed.

The emphasis throughout will be on deriving explicit analytical results rather than on the abstract properties of the solutions. Both regular and singular perturbation techniques will be addressed during lectures, and students will utilize these methods to analyse model problems provided in coursework assignments. Simple illustrative examples and more challenging real-world applications will be addressed during lectures.

Throughout the course, asymptotic methods will also be contrasted and compared with numerical methods for solving similar systems. This course will serve to illustrate the complementary nature of perturbative and numerical techniques.

**Learning Outcomes**

Upon successful completion of the course, students will be able to:

- solve problems using the fundamental ideas of asymptotic and multiple scale analysis
- develop practical skills in applying asymptotic methods for analysing mathematical and physical problems
- identify small model parameters that influence the overall behaviour of systems
- practice techniques for incorporating multiple scales into complicated nonlinear equations, and for solving the resulting simplified systems methodically to obtain increasingly accurate approximations of the true solution
- illustrate how perturbative and numerical methods often complement each other
- start their own research projects in modern applications of asymptotic theory

** **

**Content (48 hours)**

- Introduction to asymptotic approximations (1.5 hours).
- Regular and singular perturbation methods for ordinary and partial differential equations (9 hours).
- Matched asymptotic expansions: Boundary layer theory, outer and inner solutions with matching principles, interior layers, corner layers (9 hours).
- Introduction to Multiple Scales: Slowly varying coefficients, forced motion near resonance, Floquet theory, Wittaker’s method (6 hours).
- Boundary layers by multiple scales (3 hours).
- Nonlinear oscillators (1.5 hours).
- Bifurcation Theory: Hopf bifurcations, weakly non-linear analysis (9 hours).
- Two-time and uniform expansions (9 hours).

**Teaching Methodology**

__Lectures:__ Three (3) lectures each week (50 minutes each).

__Tutorials:__ One (1) weekly tutorial session (50 minutes of problem solving, based on theory covered during lectures).

** **

**Assessment**

Coursework (3 assignments each worth 10%: solving problems directly related to the theory covered during lecture hours) – 30%

Midterm Examination (one week take-home examination) – 35%

Final Examination (one 3-hour written paper) – 35%

**Course Calendar**

Week |
Lecture subject |
Assignments |
---|---|---|

1 |
Preliminaries: Introduction to asymptotic approximations |
- |

2 |
Regular perturbation methods for ordinary and partial differential equations |
Assignment 1 |

3 |
Singular perturbation methods for ordinary and partial differential equations |
- |

4 |
Matched asymptotic expansions: Boundary layer theory, outer and inner solutions with matching principles, interior layers, corner layers |
- |

5 |
Matched asymptotic expansions: Boundary layer theory, outer and inner solutions with matching principles, interior layers, corner layers |
Assignment 2 |

6 |
Introduction to Multiple Scales: Slowly varying coefficients, forced motion near resonance, Floquet theory, Wittaker’s method |
- |

7 |
Introduction to Multiple Scales: Slowly varying coefficients, forced motion near resonance, Floquet theory, Wittaker’s method |
- |

8 |
Boundary layers by multiple scales |
Mid-term Examination (35%) |

9 |
Bifurcation Theory: Hopf bifurcations, weakly non-linear analysis |
- |

10 |
Bifurcation Theory: Hopf bifurcations, weakly non-linear analysis |
- |

11 |
Two-time and uniform expansions |
Assignment 3 |

12 |
Two-time and uniform expansions |
- |

13 |
Revision |
- |

**Required Reading**

__ __

**Essential Text**

- Introduction to Perturbation Methods, M. H. Holmes. Springer-Verlag, ISBN 0-387-94203-3.

**Other Suggested Texts / References**

- Advanced Mathematical Methods for Scientists and Engineers, C. M. Bender and S. A. Orzag, Springer-Verlag, ISBN 0-387-98931-5.
- Multiple Scale and Singular Perturbation Methods, J. Kevorkian and J. D. Cole, Springer-Verlag, ISBN 0-387-94202-5.
- Perturbation Methods in Applied Mathematics, J. Kevorkian and J. D. Cole, Springer-Verlag, ISBN 0-387-90507-3.