## MATH4AA Asymptotic Analysis

Second Semester |

Asymptotic analysis is a branch of mathematical analysis concerned with the limiting behavior of functions, in particular the behavior of functions when a variable or parameter is "large" or "small". The tools of asymptotic analysis are useful in describing the unknown solution to a number of algebraic, differential, or integral equations, particularly for those cases where solving such an equation exactly is not possible. One can, in such contexts, view the approach as yielding results which lie somewhere between exact solutions (which are nice, yet uncommon for many equations) and numerical simulations (which can be quite useful for understanding the behavior of a solution for a specific collection of parameters, but often less useful for deducing general behavior or structure). In a number of cases, an asymptotic approximation for a given function may even provide more insight than does the exact solution.

We provide a survey of techniques in asymptotic analysis, and you will be exposed to a wide assortment of tools related to the asymptotic approximation of functions and perturbation methods for the solution of differential equations. In order to make the material self-contained, we first review useful preliminaries, before moving on to the core material involving the asymptotic approximation of functions (including infinite series, products, and integrals) as well as the approximation of solutions to algebraic equations. We then survey various tools which are useful in the study of solutions to ODE and PDE.

### Prerequisites

At some point in your study of calculus, you were likely asked to find a Taylor series representation for a given function. You may have even used Taylor series in order to solve an ODE for an unknown function, when taking an ODE paper. Later, if you took a PDE paper, you were likely asked to find a Fourier series for some unknown function which satisfied a PDE, and you might have wondered why such an infinite series actually converged, and more importantly, converged to a solution of whatever PDE you were solving. In all of these cases, one is concerned with approximating local behavior of a function through a succession of terms, with the vague notion that adding some number of terms will make the approximation "better". Through these examples, you were already exposed to some of the key ideas of this paper.

My goal is to make this paper useful for a wide variety of students, and as such there are no formal prerequisites. MATH 203, COMO 204 would be desirable, but can be considered optional, as the lectures and problem sets will be self-contained.

### Course Outline

This paper will cover a selection of topics related to asymptotic analysis and perturbation methods. Topics included in lectures will be selected from the following four lists:

I. Preliminaries

Review of topics from calculus (infinite series, infinite products, Taylor's theorem, convergence and divergence of sequences and series), linear algebra (eigenvalues, Hurwitz matrix), ODE (Laplace transform, series solution of ODE, Sturm-Liouville theory), PDE (Fourier transform, Fourier series, heat kernels).

II. Basic definitions and results in asymptotic analysis

A. Asymptotic expansion of functions and roots of algebraic equations.

B. Expansion and approximation of functions taking the form of infinite series, products, and integrals.

C. Approximation of integrals with a large or small parameter (integration by parts, Watson's lemma, Laplace's method, method of stationary phase, method of steepest descent).

D. Divergence and resummation of asymptotic series.

III. Methods for ODE

A. Expansion of ODE solutions in dependent variables.

B. Large-time behavior of ODE solutions and linear stability analysis.

C. Expansion of ODE solutions in small or large parameters (regular and singular perturbation, method of matched asymptotic expansions, WKB method, Poincaré-Lindstedt method).

D. Approaches for ODE when there are no small parameters (delta-expansion method, homotopy analysis method, iterative methods).

IV. Methods for PDE

A. Generalized Fourier series. Approximation of Sturm-Liouville eigenvalues.

B. Non-dimensionalization. Symmetries and self-similarity.

C. Approximation of exact PDE solutions given in integral or series form.

D. Long-time behavior and instabilities in diffusion processes.

E. Multiscale analysis, averaging, and homogenization.

### Lecturer

R. A. Van Gorder

### Literature

Lecture notes will be the primary reference for this paper. Other literature may be provided, but only the material in the lecture notes and take-home problem sets will be examinable. For those wishing for additional sources or additional practice problems, several useful books include:

O.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers.

E.J. Hinch, Perturbation Methods, Cambridge 1991.

J. Kevorkian and J.D. Cole, Perturbation Methods in Applied Mathematics, Springer 1985.

A.H. Nayfeh, Perturbation Methods, Wiley 1973.

M. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press 1964.

### Assessment

There will be three take-home problem sets, with problem set grouped according to topics II, III, and IV, above. Mastery of mathematics follows from doing rather than by watching others do mathematics, and these problems sets should be viewed as a learning activity no less valuable than lectures. In addition to their contribution to your final mark, these problem sets will serve as useful preparation for the exam.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = max(E, (E + 3A)/4)**

where:

- E is the Exam mark
- A is the Assignments mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.