## Asymptotic Expansions & Perturbation Methods

 Unit code: MATH34011 Credit Rating: 10 Unit level: Level 3 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

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#### Aims

To demonstrate the power and the beauty of modern asymptotic methods via discussing the main ideas and approaches used in the theory of asymptotic expansions to simplify and to solve different mathematical problems which involve large or small parameters.

#### Overview

The development of the theory of asymptotic expansions, which serves as a foundation for perturbation methods, is one of the most important achievements in applied mathematics in the twentieth century. Perturbation methods represent a very powerful tool in modern mathematical physics and in particular, in fluid dynamics. This course unit will introduce students to a range of modern asymptotic techniques and illustrate their use in model problems involving ordinary and partial differential equations.

#### Learning outcomes

On successful completion of the course unit the students will be able to:

• Identify the properties of asymptotic expansions and distinguish them from other series expansions.
• Recall the properties of Geometric progressions, Taylor series, and use them to derive asymptotic expansions of integrals.
• Describe Watson's Lemma and Laplace's method, and use them to analyse the asymptotic behaviour of Laplace type integrals.
• Apply the method of stationary phase to derive asymptotic expansions of Fourier type integrals.
• Use the method of steepest descent to analyse the behaviour of integrals in the complex plane.
• Evaluate the convergence of asymptotic expansions and obtain simple bounds on the remainder term.
• Classify saddle-points, and identify and sketch paths of steepest descent/ascent.
• Recall the WKB method and use it to obtain the WKB approximation to solutions of ordinary differential equations.
• Discuss boundary-layer theory and calculate uniform approximation to solutions of ordinary diferential equations.
• Apply the method of multiple scales to derive long time uniform approximations to solutions of differential equation.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Mid-semester coursework: weighting 20%
• End of semester examination: two hours weighting 80%

#### Syllabus

• Asymptotic expansions. Taylor expansion as a conventional converging power series and as an example of an asymptotic expansion . Asymptotic expansions for definite integrals with the upper or lower limits of integration depending on small or large parameters. Functions defined by real integrals. Laplace's method for definite integrals the integrand being of the form f(t)exp(Î»t), where the parameter Î» is large; Watson's Lemma. Generalisation for functions defined by contour integrals. Steepest descent. Applications.
• Regular asymptotic expansions for functions depending on the coordinate x, scalar or vector, and on a small parameter Îµ. Solution of ordinary and partial differential equations with small parameters.
• Singular perturbations (algebraic example). Notion of the boundary layer. Inner and outer solutions. Overlap region. Matching of the asymptotic expansions. Ordinary differential equations with singular perturbations. Methods to determine location of the boundary layer.
• Method of multiple scales. Quasi-periodic solutions of second order ordinary differential equations developing non-uniformity at large time. Uniformly valid solutions. Amplitude equations. WKB Method.

• R. Wong, Asymptotic Approximation of Integrals, Academic Press 1989.
• E.J. Hinch, Perturbation Methods, Cambridge 1991.
• O.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers.
• M. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press 1964.
• J. Kevorkian and J.D. Cole, Perturbation Methods in Applied Mathematics, Springer 1985.
• A.H. Nayfeh, Perturbation Methods, Wiley 1973.

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 24 hours
• Tutorials - 12 hours
• Independent study hours - 64 hours

#### Teaching staff

Jitesh Gajjar - Unit coordinator