MMath&Phys Mathematics and Physics

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This unit provides an overview of analytical and asymptotic mathematical methods of use in problems that arising across applied mathematics. Starting from assumed knowledge of analytical methods for ODEs, asymptotic and perturbation methods will be developed for ODE problems. Exact solution techniques will be reviewed for linear PDEs on their own and as a motivation for integral transforms. Finally the course will culminate with a selection of classic examples of the use of approximate and asymptotic methods in PDE problems. 

For MSc Applied Mathematics: none.

For MMath students,  

MATH20401/20411/24420 (PDEs and vector calculus)

and MATH34011 (complex analysis and applications).  

MATH35041 recommended. 

Maths & Physics or Physics students can replace the prerequisite MATH24420 by PHYS20171.

The unit aims to:

To provide training in a variety of exact and asymptotic methods and techniques in order that they may be applied to a wide range of partial differential equation problems in applied mathematics and numerical analysis.

• Apply and interpret nondimensional analysis and relate to similarity solutions of PDE problems
• Construct series and asymptotic solutions for ODEs including matching regions where appropriate
• Select and construct series solutions for homogeneous and inhomogeneous PDE problems
• Select and apply Fourier transforms to solve linear PDEs in infinite domains
• Use complex variable theory with contour deformation to evaluate integrals, calculating residues as necessary
• Relate asymptotic and transform methods to applications in PDE problems, and derive results in similar problems with appropriate guidance 

Syllabus:

1. Dimensional analysis, dimensionless parameters and similarity solutions for PDEs.
2. Series expansions of ODEs, dominant balance and singular points.
3. Regular and singular asymptotic expansions. Matching regions for ODE problems.
4. Series solutions for PDEs, motivated by separation of variables. Applications to homogeneous and inhomogeneous problems.
5. Definition and properties of Fourier transforms. Application to solution of PDEs in infinite domains.
6. Complex variable theory. Role of contour deformation and Cauchy's residue theorem in evaluating integrals. 
7. Application of asymptotic methods for PDE problems. Examples to include elasticity and fluid mechanics, in range of real and transform methods.
 

The course will be timetabled with three-in person hours per week. Two are dedicated lectures, with the final hour alternating between tutorial and lecture. 

Written exam during final exam period (closed book) - Marked exam script

One coursework assignment/take home test - Overall comment on work and annotations as necessary. Also group feedback. 

• 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 Engi- 
neers. 
• 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 
• Ockendon, Howison, Lacey and Movchan. Applied Partial Differential Equations, Oxford 
University Press, 2003. 
• J. Kevorkian, Partial Differential Equations, second edition. Springer, 1999.