Applied Mathematics

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Applied Mathematics (1 Year) [MSc] course units Course description The Applied Mathematics group in the School of Mathematics at the University of Manchester has a long-standing international reputation for its research. Expertise in the group encompasses a broad range of topics, including Continuum Mechanics, Analysis & Dynamical Systems, Industrial & Applied Mathematics, Inverse Problems, Mathematical Finance, and Numerical Analysis & Scientific Computing. The group has a strongly interdisciplinary research ethos, which it pursues in areas such as Mathematics in the Life Sciences, Uncertainty Quantification & Data Science, and within the Manchester Centre for Nonlinear Dynamics. The Applied Mathematics group offers the MSc in Applied Mathematics as an entry point to graduate study. The MSc has two pathways, reflecting the existing strengths within the group in numerical analysis and in industrial mathematics. The MSc consists of five core modules (total 75 credits) covering the main areas of mathematical techniques, modelling and computing skills necessary to become a modern applied mathematician. Students then choose three options, chosen from specific pathways in numerical analysis and industrial modelling (total 45 credits). Finally, a dissertation (60 credits) is undertaken with supervision from a member of staff in the applied mathematics group with the possibility of co-supervision with an industrial sponsor. Aims The course aims to develop core skills in applied mathematics and allows students to specialise in industrial modelling or numerical analysis, in preparation for study towards a PhD or a career using mathematics within industry. An important element is the course regarding transferable skills which will link with academics and employers to deliver important skills for a successful transition to a research career or the industrial workplace. Special features The course features a transferable skills module, with guest lectures from industrial partners. Some dissertation projects and short internships will also be available with industry. Teaching and learning Students take eight taught modules and write a dissertation. The taught modules feature a variety of teaching methods, including lectures, coursework, and computing and modelling projects (both individually and in groups). The modules on Scientific Computing and Transferable Skills particularly involve significant project work. Modules are examined through both coursework and examinations. Coursework and assessment Assessment comprises course work, exams in January and May, followed by a dissertation carried out and written up between June and September. The

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dissertation counts for 60 credits of the 180 credits and is chosen from a range of available projects, including projects suggested by industrial partners. Course unit details CORE (75 credits) 1. Mathematical methods  2. Partial Differential Equations  3. Scientific Computing  4. Dynamical Systems  5. Transferrable skills for mathematicians Industrial modelling pathway  1 Continuum mechanics  2. Stability theory  3. Conservation and transport laws Numerical analysis pathway  1. Numerical linear algebra  2. Finite Elements  3. Optimization and variational calculus Course unit list (Printed: 29Mar 2nd Semester 2017) The course unit details given below are subject to change, and are the latest example of the curriculum available on this course of study. 

Credit rating

Title

Code

Mandatory/optional

Applied Dynamical Systems

MATH64041 15

Mandatory

Mathematical Methods (as MAGIC022)

MATH64051 15

Mandatory

PDEs: Theory and Practice (MAGIC058)

MATH64062 15

Mandatory

Transferable Skills for Applied Mathematicians

MATH65740 15

Mandatory

Scientific Computing

MATH69111 15

Mandatory

Continuum Mechanics

MATH65061 15

Optional

Transport Phenomena and Conservation Laws

MATH65122 15

Optional

3

Credit rating

Title

Code

Mandatory/optional

Stability Theory

MATH65132 15

Optional

Approximation Theory and Finite Element Analysis

MATH66052 15

Optional

Numerical Linear Algebra

MATH66101 15

Optional

Numerical Optimization and Inverse Problems MATH66132 15

Optional

Facilities Modern computing facilities are available to support the course. Disability support Practical support and advice for current students and applicants is available from the Disability Advisory and Support Service. Email: [email protected] Structure of the programme Students can take the MSc in Applied Mathematics or alternatively can choose to take one of the structured pathways, leading to an MSc in Applied Mathematics with Numerical Analysis or an MSc in Applied Mathematics with Industrial Modelling. Students will take 8 taught course units (120 credits) throughout semesters 1 and 2. This will give a broad training in advanced Applied Mathematics. For the MSc in Applied Mathematics there are 5 compulsory units and 3 optional units (chosen from 6 optional courses). For students taking the pathway courses, all 8 taught courses are compulsory. Dissertations undertaken over the summer can be taken in collaboration with industry and various sponsored projects are available. Choices of dissertations will be made after the January exams. Some of the work undertaken in the transferable skills will then be focused on the areas of importance for the dissertation topic chosen. Aims of the programme The aims of the programme are to train students in a broad range of Applied Mathematical Methods and techniques both analytical and computational with a focus on application areas. The aim is that students will pick up a variety of skills of great use for entrance onto a PhD programme or entrance into employment within an industrial sector where knowledge of applied mathematics is of great use.

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Emphasis is on engagement with industry where possible and to train students not only to do mathematics but also to gain additional transferable skills of importance in academia and industry. Notably, in the transferable skills unit, students will focus on group work, mathematical modelling problems, communication of work undertaken via written projects and oral presentations and develop their research skills. Level 6 course units (printed: Beginning of 2017-1st Semester)

Semester

Requirement

Credit Rating

Level

MATH64041 - Applied Dynamical Systems

1

Mandatory

15

6

MATH64051 - Mathematical Methods (as MAGIC022)

1

Mandatory

15

6

MATH65061 - Continuum Mechanics

1

Optional

15

6

MATH66101 - Numerical Linear Algebra

1

Optional

15

6

MATH69111 - Scientific Computing

1

Mandatory

15

6

MATH64062 - PDEs: Theory and Practice (MAGIC058)

2

Mandatory

15

6

MATH65122 - Transport Phenomena and Conservation Laws

2

Optional

15

6

MATH65132 - Stability Theory

2

Optional

15

6

MATH66052 - Approximation Theory and Finite Element Analysis

2

Optional

15

6

MATH66132 - Numerical Optimization and Inverse Problems

2

Optional

15

6

1 and 2

Mandatory

15

6

Description

MATH65740 - Transferable Skills

5

Description

Semester

Requirement

Credit Rating

Level

for Applied Mathematicians

Courses’ Descriptions Applied Dynamical Systems Unit code:

MATH64041

Credit Rating:

15

Unit level:

Level 6

Teaching period(s):

Semester 1

Offered by

School of Mathematics

Available as a free choice unit?:

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Requisites None Aims To develop a basic understanding dynamical systems theory, particularly those aspects important in applications. To describe and illustrate how the basic behaviours found in dynamical systems may be recognized and analyzed. Overview Dynamical systems theory is the mathematical theory of time-varying systems; it is used in the modelling of a wide range of physical, biological, engineering, economic and other phenomena. This module presents a broad introduction to the area, with emphasis on those aspects important in the modelling and simulation of systems. General dynamical systems are described, along with the most basic sorts of behaviour that they can show. The dynamical systems most commonly encountered in applications are formed from sets of differential equations, and these are described, including some practical aspects of their simulation. The most regular kinds of behaviour---equilibrium and periodic---are the most easy to analyze

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theoretically; linearization about such trajectories are discussed (for periodic behaviour this is done using the PoincarÃ© map.) Much more complex behaviours, including chaos, may be found; these are described by means of their attractors. The linearization approach can be extended to these, and leads to the concept of Lyapunov exponents. In applications it is often important to know how the observed behaviour changes with changes in the system parameters; such changes can often be sudden, but frequently conform to one of a relatively small number of scenarios: the study of these forms the subject of bifurcation theory. The simplest bifurcations are discussed. Learning outcomes     

On successful completion of this course unit students will understand the general concept of a dynamical system, and the significance of dynamical systems for modelling real world phenomena; be able to analyze simple dynamical systems to find and classify regular behaviour; appreciate some of the more complex behaviours (including chaotic), and understand some of the features of the attractors characterizing such behaviour; be familiar with some of the simpler bifurcation scenarios, and how they can be analyzed. Assessment methods

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Other - 25% Written exam - 75% Assessment Further Information

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Mid-semester coursework: 25% End of semester examination: three hours weighting 75% Syllabus

1. Basics. Basic concepts of dynamical systems: states, state spaces, dynamics. Discrete and continuous time systems. [1 lecture] Some motivating examples: (discrete): simple population models, numerical algorithms; (continuous): chemical and population kinetics, mechanical systems, electronic and biological oscillators. [1] 2. Basic features of dynamical systems. Trajectories, fixed points, periodic orbits, attractors and basins. Autonomous and non-autonomous systems. Phase portraits in the plane and higher dimensions; examples of phase portraits of 2-d and 3-d systems. [2] 3. Ordinary differential equations. Systems of first order ordinary differential equations; initial value problems, existence and uniqueness of solutions. Flows. [2]

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4. Equilibria and linearization. Fixed and equilibrium points and their linearization; classification and the Hartman-Grobman theorem; invariant manifolds; examples in 2-d and 3-d. Computing equilibrium points. Stability and Liapounov functions. [5] 5. Periodic orbits and linearization. Poincaré sections and the Poincaré map. Linearization and characteristic multipliers of periodic orbits, and stability; examples. Computing periodic orbits. [3] 6. Attractors and long-term behaviour. ω-limit sets and long term behaviour. Chaotic attractors; illustrative examples. Lyapunov exponents and their computation. Onedimensional maps and simple routes to chaos (unimodal maps and Lorenz maps). Crises, chaotic transients [8] 7. Bifurcations of flows. Local bifurcations and centre manifolds, global bifurcations; examples. Computing bifurcation diagrams by continuation.[8] Recommended reading     

Stephen~H. Strogatz, Nonlinear Dynamics and Chaos, Perseus Books, Cambridge, MA, USA, 1994. Kathleen T. Alligood, Tim D. Sauer and James A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer-Verlag, New York, NY, USA, 1996. Thomas S. Parker and Leon O. Chua, Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York, NY, USA, 1989. Stephen Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, NY, USA, second edition, 2003. Y.A. Kuznetsov Elements of Applied Bifurcation Theory, Springer Applied Math. Sci. 112, 1995. Feedback methods Tutorials will provide a place for student worked examples to be marked and discussed providing feedback on performance and understanding. Feedback is also provided via return of marked coursework. Study hours

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Lectures - 22 hours Tutorials - 11 hours Independent study hours - 117 hours Teaching staff Paul Glendinning - Unit coordinator Mathematical Methods (as MAGIC022)

Unit code:

MATH64051

8

Credit Rating:

15

Unit level:

Level 6

Teaching period(s):

Semester 1

Offered by



N

Requisites None Aims To provide training in a variety of mathematical methods and techniques in order that they may be applied to a wide range of problems in applied mathematics and numerical analysis. Overview This unit will teach a variety of powerful mathematical methods that can be used in order to solve mathematical problems that arise in numerous applications. This will include both exact and approximate solutions. In the case of the latter, asymptotic and perturbation methods will be developed to solve partial differential equations and determine approximations for integrals. Learning outcomes On successful completion of this course unit students will have developed an understanding of a variety of powerful mathematical methods, and in particular they will be able to     

Be able to non-dimensionalize a problem and understand the importance of nondimensional parameters Determine series solutions of ODEs Use transforms and complex variable theory in order to solve a variety of initial and boundary value problems Use asymptotic and perturbative methods in order to solve ordinary and partial differential equations approximately Approximate integrals asymptotically where appropriate

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Assessment methods  


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Mid-semester coursework as a take-home test: 25% End of semester examination: three hours weighting 75% Syllabus Topic 1-9 below are the same as the MAGIC022 course with 10-14 being the additional material on applications.

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Non-dimensionalization and scaling.[1] Advanced differential equations. Series solution, classification of singularities. Properties near ordinary and regular singular points. Approximate behaviour near irregular singular points. Method of dominant balance. Airy, Gamma and Bessel functions. [3] Asymptotic methods. Boundary layer theory. Regular and singular perturbation problems. Uniform approximations. Interior layers. LG approximation, WKBJ method. Multiple scales. [3] Generalized functions. Basic definitions and properties. [2]. Complex analysis. Revision of background material. Laurent expansions. Singularities, Cauchy?s theorem. Residue calculus. Branch points and cuts. Plemelj formulae. [3] Transform methods. Fourier transform, FT of generalized functions. Laplace transform. Properties of Gamma function. Mellin transform. Analytic continuation of Mellin transforms. .[3] Similarity solutions of pdes [1] Asymptotic expansion of integrals Laplace?s method. Watson?s Lemma. Methods of stationary phase and steepest descent. Estimation using Mellin transform technique.[2] Conformal mappings [2] The MSc course will then showcase these methods with applications to 2-3 of the following areas:[10] - Greens functions for ODEs - Thin aerofoil theory - Application of steepest descent to a classical problem, e.g. water waves - Application of multiple scales to homogenization - Transforms to solve BVPs and use of complex analysis to invert - Use of WKB for e.g. geometric optics, caustics, etc

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Recommended reading      

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 Study hours

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Lectures - 30 hours Tutorials - 6 hours Independent study hours - 114 hours Teaching staff Jitesh Gajjar - Unit coordinator

Continuum Mechanics Unit code:

MATH65061

Credit Rating:

15

Unit level:

Level 6

Teaching period(s):

Semester 1

Offered by



N

Requisites None

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Aims The course unit concerns the formulation and solution of problems in continuum mechanics (solid and fluid mechanics) from a modern unified perspective. The aims are (i) to introduce students to the general analytic machinery of tensor calculus, variational principles and conservation laws in order to formulate governing equations; and (ii) to be aware of exact, approximate and numerical methods to solve the resulting equations. Overview This unit describes the fundamental theory of continuum mechanics in a unified mathematical framework. The unit will cover theories of nonlinear and linear elasticity together with those of compressible and incompressible fluid mechanics. Learning outcomes On successful completion of this course unit students will    

Be able to formulate governing equations for a variety of problems in continuum mechanics. Understand the relationship between the general theory and its specialisation to the equations of linear elasticity and incompressible Newtonian fluid mechanics. Solve simple problems in continuum mechanics analytically. Be aware of certain numerical techniques that can be applied to problems in continuum mechanics. Assessment methods

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Foundations and Fundamentals [3]. Eulerian and Lagrangian coordinates, vectors and tensors, material/convected derivatives, integral theorems Kinematics: Deformation and Flow [3]. Measures of strain, stretch, rotation, compatibility, applications of convective derivative, velocity and vorticity. Stress [3]. The continuum hypothesis, Cauchy's stress principle, measures of stress, objectivity. Fundamental laws, governing equations and thermodynamics [4]. Conservation laws (mass, linear and angular momentum, energy), Cauchy's equations, variational principles, thermodynamics, constitutive modelling.

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Linear elasticity [3]. Hooke's law, Navier-Lame equations, elastostatics, elastodynamics, Airy stress function. Incompressible Newtonian Fluids [3]. Viscosity, potential flow, slow flow, high-speed flow, boundary layers Nonlinear elasticity [4]. Hyperelasticity and constitutive models, simple exact solutions, principle of virtual work and basic finite elements. Complex fluids [4]. Non-Newtonian behaviour, shear thinning/thickening, viscoelasticity. Further advanced topics[3]. Interfaces and Fluid-structure interaction. Recommended reading

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Spencer, A.J.M, "Continuum Mechanics", Dover Gonzalez, O. and Stuart, A.M., "A first course in continuum mechanics", CUP Irgens, F., "Continuum Mechanics", Springer Feedback methods Tutorials will provide an opportunity for students’ work to be discussed and provide feedback on their understanding. Study hours

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Lectures - 22 hours Tutorials - 11 hours Independent study hours - 117 hours Teaching staff Andrew Hazel - Unit coordinator Numerical Linear Algebra Unit code:

MATH66101

Credit Rating:

15

Unit level:

Level 6

Teaching period(s):

Semester 1

Offered by



N

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Requisites None Aims To develop understanding of modern methods of numerical linear algebra for solving linear systems, least squares problems and the eigenvalue problem. Overview This module treats the main classes of problems in numerical linear algebra: linear systems, least square problems, and eigenvalue problems, covering both dense and sparse matrices. It provides analysis of the problems along with algorithms for their solution. It also introduces MATLAB as tool for expressing and implementing algorithms and describes some of the key ideas used in developing highperformance linear algebra codes (blocking, BLAS). Applications will be introduced throughout the module. Learning outcomes On successful completion of this course unit students will     

understand the concepts of efficiency and stability of algorithms in numerical linear algebra; understand the importance of matrix factorizations, and know how to construct some key factorizations using elementary transformations; be familiar with some important methods for solving linear systems, least squares problems, and the eigenvalue problem; appreciate the issues involved in the choice of algorithm for particular problems (sparsity, structure, etc.); appreciate the basic concepts involved in the efficient implementation of algorithms in a high-level language. Assessment methods

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Introduction. Summary/recap of basic concepts from linear algebra and numerical analysis: matrices, operation counts. [1 lecture]. Introduction to MATLAB. [2]. Matrix norms. Linear system sensitivity. [2]

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Matrix factorizations. Cholesky factorization. QR factorization by Householder matrices and by Givens rotations. [5]. LU factorization and Gaussian elimination; partial pivoting. Error analysis. [2]. Block algorithms and their suitability for modern machine architectures. [1]. The BLAS and LAPACK. [1] Linear systems. Solving triangular systems by substitution. Solving full systems by factorization. Application: Newton's method for nonlinear systems. [1] Sparse and banded linear systems and iterative methods. LU factorization for banded and sparse matrices. Storage schemes. [1]. Iterative methods: Jacobi, Gauss-Seidel and SOR iterations. Krylov subspace methods, conjugate gradient method. Preconditioning. Application: differential equations. [4] Linear least squares problem. Basic theory using singular value decomposition (SVD) and pseudoinverse. Perturbation theory. Numerical solution: normal equations. SVD and rank deficiency. Application: image deblurring. [5] Eigenvalue problem. Basic theory, including perturbation results. Power method, inverse iteration. Similarity reduction. QR algorithm. Application: Google PageRank. [5] Recommended reading

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[1] Timothy A. Davis, Direct Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2006, ISBN 0-89871613-6, xii+217pp. [2] James W. Demmel, Applied Numerical Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1997, ISBN 0-89871-389-7, xi+419pp. [3] Gene H. Golub and Charles F. Van~Loan, Matrix Computations Johns Hopkins University Press, Baltimore, MD, USA, third edition, 1996, ISBN 0-8018-5413-X (hardback), 0-8018-5414-8 (paperback), xxvii+694pp. [4] Desmond J. Higham and Nicholas J. Higham, MATLAB Guide, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, second edition, 2005, ISBN 0-89871-578-4, xxiii+382pp. [5] Nicholas J. Higham, Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, second edition, 2002, ISBN 0-89871-521-0, xxx+680pp. [6] G. W. Stewart, Introduction to Matrix Computations, Academic Press, New York, 1973, ISBN 0-12-670350-7, xiii+441pp. [7] G. W. Stewart, Matrix Algorithms, Volume I: Basic Decompositions, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1998, ISBN 0-89871414-1, xx+458pp. [8] G. W. Stewart, Matrix Algorithms, Volume II: Eigensystems, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2001, ISBN 0-89871-503-2, xix+469pp. [9] Lloyd N. Trefethen and David Bau III, Numerical Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1997, ISBN 0-89871361-7, xii+361pp. [10] David S. Watkins, Fundamentals of Matrix Computations, Wiley, New York, second edition, 2002, ISBN 0-471-21394-2, xiii+618pp.

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[11] Per Christian Hansen, James G. Nagy, and Dianne P. O'Leary, Deblurring Images: Matrices, Spectra, and Filtering, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2006, ISBN 0-89871-618-7, xiv+130pp. [12] C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1995, ISBN 0-89871352-8, xiii+165pp. [13] Amy N. Langville and Carl D. Meyer, Google's PageRank and Beyond: The Science of Search Engine Rankings, Princeton University Press, Princeton, NJ, USA, 2006, ISBN 0-691-12202-4, x+224 pp. Books [1] - [10] cover the core material, while [11] - [13] cover the applications. Feedback methods Tutorials will provide an opportunity for students’ work to be discussed and provide feedback on their understanding. Study hours

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Lectures - 22 hours Tutorials - 11 hours Independent study hours - 117 hours Teaching staff Nicholas Higham - Unit coordinator Francoise Tisseur - Unit coordinator

Scientific Computing Unit code:

MATH69111

Credit Rating:

15

Unit level:

Level 6

Teaching period(s):

Semester 1

Offered by



N

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Requisites None Aims To develop the knowledge required to solve mathematical and scientific problems by writing computer programs in C++. Overview This course covers the techniques required to develop C++ programs that solve mathematical and scientific problems. As well as covering the rudiments of C++ (which will be taught with no assumed prior knowledge) the course will also outline the basic techniques used in scientific programming, such as discretisation of equations, numerical error and code validation. The material is examined primarily through two programming projects, chosen from a list of mathematical topics, which investigate particular algorithms or techniques in more depth. The projects will be assessed by a written report and a demonstration/oral description of the code. Much of this course is taught in practical computer labs, which limits the number of places available. Learning outcomes On successful completion of this module, students will be able to formulate and write C++ programs to solve mathematical problems. They will understand how to:   

choose appropriate algorithms for the problem, develop a suitable program architecture and implement it in C++, debug the code and validate its results in the context of the mathematical problem. Assessment Further Information

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Weekly courseworks: 10% Project 1: 40% Project 2: 50% Syllabus Introduction to C++ programming language: - statements, expressions, control flow, functions, types - standard C++ library: streams and file i/o, strings, containers, algorithms - use of external libraries

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Code structure and object-oriented programming: - methods, member data, constructors, destructors, access specifiers - inheritance, virtual methods and run-time polymorphism. - operator overloading Fundamental concepts and techniques: - numerical error - discretisation - writing efficient code (algorithm complexity, optimisation, parallelism) - communication and visualisation of numerical results - common algorithms (covered in coursework, and in lectures as time permits) such as numerical linear algebra, root finding, quadrature, sorting, BVPs, PDEs Debugging and validation: - error handling - testing and test-driven development - debugging - validation of numerical results Recommended reading     

S.B. Lippman, J. Lajoie, B. Moo. C++ Primer (Fourth edition). Addison Wesley, 2005. (Available as an e-book from the university library) W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery. Numerical Recipes: The Art of Scientific Computing (Third edition). Cambridge University Press, 2007. B. Stroustrup. The C++ Programming Language (Third edition). Addison-Wesley, 1997 S. Meyers. Effective C++: 55 specific ways to improve your programs and designs (Third edition). Addison-Wesley, 2005. D. Yang. C++ and object-oriented numeric computing for scientists and engineers. Springer, 2000. Feedback methods Tutorials will provide an opportunity for students’ work to be discussed and provide feedback on their understanding. Study hours

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Lectures - 11 hours Practical classes & workshops - 33 hours Independent study hours - 106 hours Teaching staff Christopher Johnson - Unit coordinator

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PDEs: Theory and Practice (MAGIC058) Unit code:

MATH64062

Credit Rating:

15

Unit level:

Level 6

Teaching period(s):

Semester 2

Offered by



N

Requisites None Aims To provide an introduction to the theory of partial differential equations (pdes) by considering the twin questions of existence and uniqueness of solutions of pdes and systems of pdes. The modern theory is inextricably linked to the methods of functional analysis and an introduction to this important subject will be given. The practice of pdes is exemplified by methods for the numerical approximation to solutions and the methods are closely associated with the type of the pde or system. An introduction to these methods is presented. Overview This course unit develops certain aspects of the theory and practice of pdes and systems of pdes, in particular existence and uniqueness of solutions together with discretisation methods for approximating such solutions. The course will introduce basic definitions and concepts pertaining to pdes and develop both classical and modern methods of proving existence and uniqueness results. In particular analytic function theory will be used to prove the classical Cauchy-Kowalevskaya theorem and functional analysis (including the Lebesgue integral) will form the basis for proving theorems for second and higher order elliptic equations via the Riesz representation theorem. The basic methods of discretisation of partial derivatives will be presented for various types of pde, for example, elliptic, parabolic and hyperbolic equations. Examples will be presented of boundary and initial value problems of interest in

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Applied Mathematics, Industrial Mathematics, Numerical Analysis, various branches of Engineering and Physics (including geophysics). Learning outcomes Learning Outcomes On successful completion of this course unit students will    

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understand the theorems governing the existence and uniqueness of solutions to various types of pde and systems of pdes; be able to use these theorems for particular pdes that the student may come across, understand the elements of functional analysis that underpin the modern theory of pdes; understand the discretisation of pdes and systems of pdes that allow numerical approximations to solutions to be calculated for various types of pde and systems of pdes, be able to apply these methods to various problems that arise in Applied Mathematics, Industrial Mathematics, Numerical Analysis, Engineering and Physics. Assessment methods

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Mid-semester coursework: 25% End of semester examination: three hours weighting 75% Syllabus 1.Introductory concepts [2] Systems of first order pdes and single pdes of higher order, examples from continuum mechanics 2.Well-posedness [5] Symbol of a pde and of systems; characteristics; existence, uniqueness and continuous dependence on the data; well- and ill-posedness (including the CauchyKowalevskaya thorem). 3.Introductory Functional Analysis [2] Brief exposition of necessary functional analysis, e.g. operator theory, distributions, Sobolev spaces. 4.Weak Formulation[2] Weak and strong solutions.

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5.Linear elliptic equations [5] Linear elliptic pde's, coercivity/energy estimates; Lax-Milgram lemma, Garding's inequality, existence and uniqueness of weak solutions. 6.Linear Parabolic equations [4] Evolutionary pde's - abstract parabolic initial value problems, energy methods, uniqueness and existence. Introduction to semi-group methods. 7.Numerical Approximation to Solutions[10] The link between theory and practice. Finite differences for first and second order derivatives. Elliptic, parabolic and hyperbolic equations: discretisation of bvps and ivps. Explicit and implicit methods. Stability. Recommended reading  

An introduction to partial differential equations, 2nd. edition, M. Renardy and R.C. Rogers, Springer Texts in Applied Mathematics, New York, 2004. Numerical solution of partial differential equations, 2nd. edition, K.W. Morton and D.F. Mayers, Cambridge University Press, Cambridge, 2005. Feedback methods Tutorials will provide an opportunity for students’ work to be discussed and provide feedback on their understanding. Study hours

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Lectures - 30 hours Tutorials - 6 hours Independent study hours - 114 hours Teaching staff Alice Thompson - Unit coordinator

Transport Phenomena and Conservation Laws Unit code:

MATH65122

Credit Rating:

15

Unit level:

Level 6

21

Teaching period(s):

Semester 2

Offered by



N

Requisites None Aims The aim of the course is to introduce some ideas associated with transport equations and conservation laws, including linear and nonlinear wave propagation, wave steepening, shock formation, diffusion, dispersion and solitons. Overview Transport phenomena and conservation laws are ubiquitous and are a crucial element in all models of physical systems. As material is transported waves may form and the course starts with some examples of linear wave propagation. The effects of non-linearity are very important and lead to wave expansion, wave steepening and shock formation, at which there are discontinuous jumps in the solutions. It is shown how these can be modelled using jump conditions at propagating discontinuities and how diffusion and/or dispersion competes against wave steepening to create diffuse non-linear wave fronts and solitons. The formation of non-linear waves will be demonstrated using simple experiments during the lectures. The mathematical theory underlying these systems will be explained in detail, using many examples ranging from gas dynamics and shallow water flows, to granular, two-phase and traffic flows. Learning outcomes On successful completion of the course unit students will be able to 

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understand the differences in the solution properties and physics of physical systems governed by the kinematic wave equation, the K - dV equation, Burger's equation and segregation equation, as well as the avalanche and the shallow water equations; solve ut + c(u)ux = 0 for given initial data and be able to identify the formation of shocks; understand how breaking waves in two-dimensions can be represented in terms of shock waves; solve the gas dynamic and shallow water equations using the method of characteristics for simple flows;

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perform a phase plane analysis for the K - dV and related equations to identify travelling wave solutions, solitary wave solutions. Assessment methods

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Mid-semester coursework: weighting 25% End of semester examination: three hours weighting 75% Syllabus 1.The hyperbolic wave utt = c02Î'2u, ut + c0ux = 0; wave forms; Fourier synthesis; dispersion; C(k) = dw/dk, group velocity; diffusion, e.g. Burger's linear equation ut + c0ux = vuxx. 2.First order wave equation ut + c(u)ux = 0; characteristics; conservation ideas; conservation forms; granular and traffic flow models. Waves in other physical systems. 3.First order equations in two-dimensions; breaking size segregation waves and their representation in terms of shocks. 4.Shallow water wave theory; the nonlinear equations; wave breaking, dam break problems, via characteristics; normal and oblique shocks in granular flows, linearisation and check against linear theory, and linear irrotational theory. 5.Irrotational water wave theory to obtain the Boussinesq equations; steady solutions of the Boussinesq equations; derivation of the Korteweg-de Vries equation from Boussinesq equations; conservation laws for K - dV; analytical solution of K dV equation; the soliton. Recommended reading

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P.G. Drazin and R.S. Johnson, Solitons, An Introduction, CUP 1989. G.B. Whitham, Linear and Non-linear Waves, Wiley 1974. J. Stoker, Water Waves, Wiley Interscience 1957. L. Debnath, Nonlinear Water Waves, Academic Press 1994. Feedback methods Tutorials will provide a place for student worked examples to be marked and discussed providing feedback on performance and understanding. Feedback is also provided via return of marked coursework.

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Study hours   

Lectures - 22 hours Tutorials - 11 hours Independent study hours - 117 hours Teaching staff Julien Landel - Unit coordinator

Stability Theory Unit code:

MATH65132

Credit Rating:

15

Unit level:

Level 6

Teaching period(s):

Semester 2

Offered by



N

Requisites None Additional Requirements Students are not permitted to take, for credit, MATH45132 in an undergraduate programme and then MATH65132 in a postgraduate programme at the University of Manchester, as the courses are identical.

This course is largely self-contained. Aims The aim of this course unit is to introduce students to the basic concepts and techniques of modern stability theory, through case studies in fluid mechanics and transport phenomena.

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Overview Many physical systems can become unstable in the sense that small disturbances superimposed on their basic state can amplify and significantly alter their initial state. In this course we introduce the basic theoretical and physical methodology required to understand and predict instability in a variety of situations with focus on hydrodynamic instabilities and on instabilities in reaction-diffusion systems. Learning outcomes Learning Outcomes On successful completion of this course unit students will    

derive linearised stability equations for a given basic state; perform a normal-mode analysis leading to an eigenvalue problems; use the ideas of weakly non-linear stability theory in simple systems; appreciate the different physical mechanisms leading to instability.

Assessment methods  


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Mid-semester coursework: 25% End of semester examination: two and a half hour weighting 75% Syllabus Assuming general mechanics and fluid mechanics in particular (viscous/inviscid), as well as some aspects of dynamical systems as prerequisites for course. 1. Introduction to stability Nonlinear dynamics. Linear instability versus nonlinear instability. Outline of the basic procedure involved in a linear stability analysis: dispersion relation, marginal stability curve. Role of weakly nonlinear theory, e.g. normal form for pitchfork bifurcation. 2. Linear stability analysis: a case study of Rayleigh-Benard convection Introduction to physical system, Boussinesq equations, dimensional analysis, Basic state, linear theory, normal modes, marginal stability curve: Analytical approach for idealised boundary conditions. 3. Interfacial instabilities Examples: Rayleigh-Taylor and capillary instabilities.

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4. Shear flow instabilities Inviscid/viscous, Squire's theorem. Rayleigh's equation, Rayleigh's inflexion point criterion, Howard's semi-circle theorem, Orr-Sommerfeld equation. Examples: plane Couette flow, plane Poiseuille flow, pipe flow, Taylor-Couette flow. 5. Stability in reaction diffusion systems Stability of propagating fronts. 6. Bifurcation theory Local bifurcations, normal forms, structural stability. 7. Nonlinear stability theory Weakly nonlinear theory, derivation of Stuart-Landau equation, Ginzburg-Landau equation. 8. Introduction to pattern formation (if time allows) Stripes, squares and hexagons, three-wave interactions, role of symmetry, longwave instabilities of patterns: Eckhaus. Recommended reading   

P.G. Drazin, Introduction to hydrodynamic stability. Cambridge University Press (2002) F. Charru, Hydrodynamic Instabilities. Cambridge University Press (2011) P. Manneville, Instabilities, chaos and turbulence. Imperial College Press (2004) Feedback methods Tutorials will provide an opportunity for students’ work to be discussed and provide feedback on their understanding. Study hours

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Lectures - 27 hours Tutorials - 6 hours Independent study hours - 117 hours Teaching staff Joel Daou - Unit coordinator

Approximation Theory and Finite Element Analysis

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Unit code:

MATH66052

Credit Rating:

15

Unit level:

Level 6

Teaching period(s):

Semester 2

Offered by



N

Requisites None Aims To give an understanding of the fundamental methods and theoretical basis of approximation. To provide students with the technical tools enabling them to solve practical elliptic PDE problems using the finite element method. Overview This course unit covers the theory of approximation and applications to the numerical solution of linear elliptic partial differential equations (PDEs) using finite element approximation methods. Such methods are universally used to solve practical problems associated with physical phenomena in complex geometries. The emphasis is on assessing the accuracy of the approximation using a priori and a posteriori error estimation techniques. Practical issues will be illustrated with MATLAB using the IFISS software toolbox. Learning outcomes Learning Outcomes On successful completion of this course unit students will   

understand notions of best approximation in different norms and be able to find the best approximation; understand the concepts of weak and classical solutions of elliptic boundary value problems; understand the concept of piecewise polynomial approximation in two dimensions, and have an appreciation for the underlying error analysis;

27 

have an appreciation of the computational issues that arise when solving convection-diffusion problems; Future topics requiring this course unit None Assessment methods

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Mid-semester coursework: 25% End of semester examination: three hours weighting 75% Syllabus 1.Basics. Review of basic functional analysis concepts: norms, inner-products. Sobolev spaces. Weak derivatives. Lax-Milgram lemma. [3] 2.Linear approximation. Best approximation in Lp norms. Existence and uniqueness. Choice of norm in practical curve fitting. Least squares approximation and normal equations. Orthogonal basis functions. Overview of the cases p=1 and p=â'ž. Choice of linear approximating functions. Polynomials, orthogonal polynomials, Chebyshev polynomials, Spline functions. Surface fitting by polynomials and splines, including the thin plate spline and radial basis functions.[8] 3.Finite element methods for the diffusion equation. Affine mappings. Linear, bilinear, quadratic and biquadratic approximation. Finite element assembly process. Properties of the discrete equation system. A priori error bounds: best approximation in energy, H1 error bounds. H2 regularity and singular problems. A posteriori error bounds. Local error estimators. Self adaptive refinement strategies. [14] 4.Finite element methods for the convection-diffusion equation. Well-posedness. Weak formulation. Galerkin approximation. The streamline-diffusion method. A priori and a posteriori error bounds. Self-adaptive refinement strategies for resolving layers. [5] Recommended reading

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Michael J. D. Powell, Approximation Theory and Methods, ISBN 978-0-521-295149 (pbk) Cambridge University Press, Cambridge, 1981. Howard Elman, David Silvester and Andy Wathen, Finite Elements and Fast Iterative Solvers, ISBN 0-19-852868-X (pbk) Oxford University Press, Oxford 2005 Dietrich Braess, Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics, ISBN 978-0-521-70518-9 (pbk) Cambridge University Press, Cambridge, third edition, 2007.

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Feedback methods Tutorials will provide an opportunity for students’ work to be discussed and provide feedback on their understanding. Study hours   

Lectures - 30 hours Tutorials - 6 hours Independent study hours - 114 hours Teaching staff David Silvester - Unit coordinator

Numerical Optimization and Inverse Problems Unit code:

MATH66132

Credit Rating:

15

Unit level:

Level 6

Teaching period(s):

Semester 2

Offered by



N

Requisites None Aims To provide a mathematical understanding of modern numerical optimization, and to discuss practical aspects of implementation and available software. To introduce inverse and ill-posed problems and their application to industrial, geophysical and medical imaging problems. Overview This course unit develops the theory and practice of numerical methods for nonlinear optimization with and without constraints, covering methods suitable for both small and large-scale problems and discussing available software. The subject

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is of considerable importance in scientific, industrial and economic problems, and the ideas and methods of solution have been stimulated by practical applications. Optimization problems as discussed in this course are encountered by many practicians working in financial mathematics, industrial mathematics or in mathematical economics. For example, one often needs to infer the material properties of an object from some physical measurement. Often the measurements are taken exterior to the object and we wish to identify variable material properties on the inside: in industrial and medical imaging, one applies ultra sound, X-rays or an electromagnetic field to an object, makes measurements on the outside and then attempts to form an image of the inside. These problems are examples of so-called 'inverse problems' and are typically ill-posed in the sense that the mapping taking data to images is discontinuous, and numerically reconstruction algorithms tend to be unstable unless one makes sufficient assumptions, such as smoothness, about the image. As discretized inverse problems are ill-conditioned, we have to constrain the solution using extra (a priori) information and the usual compromise results in an optimization problem that trades off fitting the data against satisfying the constraints given by the a priori information. Numerical optimisation techniques are often used to solve such problems. The course will be illustrated by practical examples including visits to experimental groups in the University, and will include numerical examples illustrated by MATLAB programs. Learning outcomes On successful completion of this course unit students will      

be able to understand the nature and setup of optimization problems in typical practical applications; understand the mathematical conditions characterizing the solutions of these optimization problems; have developed their knowledge of matrices, analysis and geometry in n dimensions; be familiar with several differing types of optimization problems and some methods of their numerical solution; appreciate the issues involved in the choice of algorithm for particular optimization problems. understand the basic theory of regularization for ill-posed problems, and its application to a number of medical, geophysical and industrial problems. Future topics requiring this course unit None. Assessment methods

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Other - 25%

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Written exam - 75% Assessment Further Information

 

Mid-semester coursework: 25% End of semester examination: three hours weighting 75% Syllabus 1. Introduction. [2] Motivating examples. Continuous vs discrete optimization. Local vs Global optimization. 2. Unconstrained Optimization. [10] Optimality conditions. Convexity results. Local and Global solutions. Convergence of steepest descent algorithm. Search directions: nonlinear conjugate gradient method, Newton's method; Quasi-Newton methods. Sums of squares problems: GaussNewton and Levenberg-Marquardt methods. 3. Constrained Optimization. [6] Optimality conditions. Linear algebra for solving KKT systems. Overview of methods for constrained optimisation. 4. Inverse Problems [12] Introduction to inverse problems. Tikhonov regularization of linear ill-posed problems. Total Variation regularization in image deblurring/denoising. Solving the Euler Lagrange equations. Adjoint formulation and seismic imaging. The Radon Transform and X-Ray computerized Tomography. Recommended reading

1. J. T. Betts, Practical Methods for Optimal Control using Nonlinear Programming, SIAM 2001. 2. R. Fletcher, Practical Methods of Optimization, Wiley, 1987. 3. Philip E. Gill, Walter Murray, and Margaret H. Wright, Practical Optimization, Academic Press, 1981. 4. Jorge Nocedal and Stephen J. Wright, Numerical Optimization, Springer-Verlag, 1999. 5. N. Gould and S. Leyffler, An introduction to algorithms for nonlinear optimization; in Frontiers in Numerical Analysis (Durham, 2002), Springer, 2003. 6. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, Taylor & Francis, 1998. 7. P. C. Hansen, Rank-deficient and Discrete Ill-posed Problems, SIAM, 1987. 8. R. Aster, B. Borchers and C. Thurber, Parameter Estimation and Inverse Problems. Academic Press, 2012. 9. A. Tarantola, Inverse Problem Theory, SIAM, 2005.

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A more comprehensive list of textbooks will be distributed at the beginning of the course. Feedback methods Tutorials will provide an opportunity for students’ work to be discussed and provide feedback on their understanding. Study hours   

Lectures - 22 hours Tutorials - 11 hours Independent study hours - 117 hours Teaching staff Oliver Dorn - Unit coordinator

Transferable Skills for Applied Mathematicians Unit code:

MATH65740

Credit Rating:

15

Unit level:

Level 6

Teaching period(s):

Full year

Offered by



N

Requisites None Aims To provide the soft skills that are useful and necessary in the working environment both in industry and academia. In particular presentation, writing, communication and teamwork skills. Mathematical modelling skills and practical skills will be gained by working on a variety of applied mathematical problems, experiments and computational problems.

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Overview Note that this course takes place over TWO semesters. Hours shown below are contact hours; note that significant work is required outside of these contact hours in order to complete the necessary assignments, etc. Initially students will be taught some essential skills deemed necessary for applied mathematicians. The typesetting language LaTeX will be described and a short course on the programming language Matlab will be taught and discussed with students carrying out a practical example. A significant proportion of the module will involve working in groups on mathematical modelling problems. Typically these problems will involve one lecture of background material and description of the problem to be modelled. Contact time is then used to work in groups in order to attempt to formulate the problem in mathematical language, do background reading, and then solve the problem by whatever techniques are necessary. In the final session, groups will present their work via a short 15-20 minute presentation. Three modelling problems will be worked on over the duration of the course and students will be assessed on their presentations for each modelling problem. In addition, students will prepare a poster describing their favourite problem. They will present this poster and describe its contents in Semester 2. Speakers invited from industrial collaborators will give lectures focusing on specific aspects of importance to them in their work. Past topics have included the Fast Fourier Transform, eigenvalue problems, pre-conditioners, non-Newtonian fluids, etc. Students will have an opportunity to discuss the mathematics used with the industrial contacts. At the end of each semester, students will write a short abstract of an industrial lecture of the unit coordinator's choosing, summarising the content in a form suitable for a general audience. After students have chosen their research dissertation in the early part of semester 2, their supervisor will provide details of one or two important papers relevant to their that dissertation. The student should then study these in detail and write a report on their contents, giving for example a summary of a paper, describing their understanding of the work, describing necessary background material, putting the work in context, describing applications, discussing some simpler examples (e.g. by reducing the dimension of the problem). Codes could also be written, deriving appropriate results and giving examples. In some cases ideas for further work, possible extensions could be proposed. Learning outcomes On successful completion of this course unit students will

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Be able to write mathematics effectively in LaTeX as both reports and in poster form. Be able to formulate mathematical models for a variety of problems that arise in the real world. Have an understanding and appreciation of how mathematics can be applied and used in academia and also the wider scientific world, including industry. Assessment Further Information

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Individual Poster presentation of one modelling problem: 30% Paper/literature report: 25% Modelling group talks: 30% "How to write mathematics" assignment: 5% Matlab short project: 5% Short lecture abstracts: 5% Syllabus

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How to write mathematics [4] Lectures on how to present and write mathematics effectvely. Students will work through a practical example in LaTeX. Matlab modelling classes [4] Practical classes introducing the programming language Matlab, with an assignment set at the end. What is mathematical modelling? [1] Describes the concept of mathematical modelling. Formulation of a problem in terms of mathematical language, writing down equations, what to neglect, incorporate? Modelling Problem 1 [7] First modelling problem. Lecture given on background material followed by splitting into groups working on formulating the problem (or some aspect of the problem) mathematically and then trying to solve. Modelling Problem 2 [7] Modelling Problem 3 [7] Invited industry lectures [6] (From various industrial collaborators). Lectures given by invited speakers on a topic of importance to them. Recommended reading

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Handbook of Writing for the Mathematical Sciences, Second Edition, Nicholas J. Higham, SIAM, 1998 Learning LaTeX, David F. Griffiths and Desmond J. Higham, SIAM, 1997 LaTeX: A Document Preparation System, Second Edition, Leslie Lamport, AddisonWesley Professional, 1994 Mathematical Modelling, Jagat Narain Kapur, New Age International Publishers, 1997 Mathematical modelling: classroom notes in applied mathematics, Murray S. Klamkin, SIAM, 1987 Mathematical modelling, John S. Berry and Ken Houston, Edward Arnold, 1995

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Feedback methods Tutorials will provide a place for student worked examples to be marked and discussed providing feedback on performance and understanding. Feedback is also provided via return of marked coursework and presentation/poster evaluations. Study hours  

Lectures - 36 hours Independent study hours - 114 hours Teaching staff Gareth Jones - Unit coordinator