Outline: Solving nonlinear equations: Bisection, Regula Falsi, Newton-Raphson. Optimisation: Fermat condition, Newton’s method, Lagrange and KKT multipliers, curve fitting. Solving systems of linear equations: Gaussian elimination with pivoting, Gauss-Seidel. Solving ordinary differential equations, initial value problems: Euler, Runge-Kutta, Adams-Bashforth-Moulton. Numerical Quadrature: Newton-Cotes, Gauss Quadrature. Solving ordinary differential equations, boundary value problems: finite differences.

Learning Outcomes Rationale:

LO1: Many engineering, scientific, economic, etc problems give rise to the mathematical problem of solving a nonlinear equation or a system thereof. Few nonlinear equations permit analytic solution and one must perforce resort to numerical methods.

LO2: Likewise many engineering, scientific, economic, etc problems approximately give rise to the mathematical problem of solving a system of linear equations. Whereas such systems may be solved analytically the task exceeds human computation capability if the number of equations is large, as it commonly is in practice. In these circumstances we resort to numerical methods and seek great efficiency.

LO3: Very many engineering, scientific, economic, etc problems give rise to the mathematical problem of solving a system of first order ordinary differential equations subject to given initial conditions. The great majority of these equations cannot be solved analytically and numerical methods must be employed. Whereas tables of data or graphs can of course be used to present the results, 3D graphics and animation make for a far more accessible and engaging presentation. Whereas MATLAB cannot claim to be a state of the art it can produce somewhat acceptable 3D graphics and animation.

LO4: Boundary value problems describe some aspects of the physical world. In practice professional finite-element modelling packages are commonly employed, but a familiarity with the difficulties of numerical solution is valuable.

LO5: With the advance in available computing power it is now appropriate to present simulation results for physical systems in the form of 3D animations if possible, as this is how such systems manifest in reality.

LO6: Problems tend to be large and intimidating initially. It is critical to understand the methodology of breaking large problems down into collections of smaller, more manageable problems. Taking the view that one learns by doing the module offers opportunities to become familiar with this methodology by presenting problems where its application is required for their solution. As well as writing reports for the team minor project, team members give a multimedia presentation of their work, affording them an opportunity to practice creating and giving such presentations.

LO7: Assessment of MATLAB code created presses for the development of good coding practice again affording the opportunity to develop such good practice by doing.

Module Plan:

Section 1: Nonlinear equations

Week 1:

Lectures: 2 lectures on solving single and systems of nonlinear equations, covering bisection, Regula Falsi and Newton-Raphson algorithms.

Laboratories: offerings 1, 2 of Introduction to MATLAB, loops, function m-files, 3D graphics. 2 contemporaneous offerings of courtesy Introduction to MATLAB, general commands for MATLAB novices.

Week 2a:

Lectures: 1 lecture on solving single and systems of nonlinear equations, covering bisection, Regula Falsi and Newton-Raphson algorithms.

Laboratories: offering 3 of Introduction to MATLAB, loops, function m-files, 3D graphics. 1 contemporaneous offering of courtesy Introduction to MATLAB, general commands for MATLAB novices.

Week 2b:

Lectures: 1 lecture on examples of solving nonlinear equations and sample reporting for module assignments.

Laboratories: offering 4 of Introduction to MATLAB, loops, function m-files, 3D graphics. 1 contemporaneous offering of courtesy Introduction to MATLAB, general commands for MATLAB novices.

Section 2: Optimisation

Week 3:

Lectures: 2 lectures on optimisation with or without constraints, covering Fermat’s necessary condition, Newton’s method, Lagrange multipliers and KKT multipliers, curve fitting.

Laboratories: offerings 1, 2 of MS1: solving nonlinear equations.

Week 4a:

Lectures: 1 lecture on optimisation with or without constraints, covering Fermat’s necessary condition, Newton’s method, Lagrange multipliers and KKT multipliers, curve fitting.

Laboratories: offering 3 of MS1: solving nonlinear equations.

Section 3: Systems of linear equations

Week 4b:

Lectures: 1 lecture on Gaussian elimination with pivoting.

Laboratories: offering 4 of MS1: solving nonlinear equations.

Week 5:

Lectures: 2 lectures on Gaussian elimination with pivoting.

Minor Project (first session): offerings 1, 2 of MP1: static equations.

Week 6a:

Lectures: 1 lecture on Gaussian elimination with pivoting.

Minor Project (first session): offering 3 of MP1: static equations.

Week 6b:

Lectures: 1 lecture on iterative methods, Gauss-Seidel.

Minor Project (first session): offering 4 of MP1: static equations.

Section 4: Solving ordinary differential equations, initial value problems

Week 7:

Lectures: 2 lectures on solving ODEs, initial value problems, Euler, Runge-Kutta algorithms.

Minor Project (second session): offerings 1, 2 of MP1: static equations.

Week 8:

Lectures: 2 lectures on solving ODEs, initial value problems, multi-step, Adams-Bashforth-Moulton algorithms.

Minor Project (second session): offerings 3, 4 of MP1: static equations.

Week 9a:

Lectures: 1 lecture on solving ODEs, initial value problems, multi-step, Adams-Bashforth-Moulton algorithms.

Team Minor Project (first session): offering 1 of MP1: dynamic equations.

Section 5: Numerical quadrature

Week 9b:

Lectures: 1 lecture on numerical quadrature, Newton-Cotes, Gauss quadrature.

Team Minor Project (first session): offering 2 of MP1: dynamic equations.

Section 6: Boundary Value Problems

Week 10a:

Lectures: 1 lecture on solving ordinary differential equations, boundary value problems, finite differences.

Team Minor Project (first session): offering 3 of MP2: dynamic equations.

Week 10b:

Lectures: 1 lecture wrap up and additional commentary upon reporting errors.

Team Minor Project (first session): offerings 4 of MP2: dynamic equations.

Week 11:

Team Minor Project (second session): offerings 1, 2 of MP2: dynamic equations.

Week 12:

Team Minor Project (second session): offerings 3, 4 of MP2: dynamic equations.

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*Curricular information is subject to change*

Learning Outcomes:

On successful completion of this module the student will be able to:

1. Approximately numerically solve a given algebraic or transcendental equation or system of such equations, appreciate the level of approximation involved and understand some of the more common problems which can occur and how to remedy them. Leverage this ability to approximately numerically identify the local maxima and minima of functions, both subject to constraints and otherwise.

2. Efficiently numerically solve a system of linear equations and/or invert a matrix, even when the number of variables becomes relatively large. Have an appreciation of ill-conditioning and of sparsity.

3. Efficiently numerically solve a system of first order differential equations subject to given initial conditions. Be able to use this ability to determine the actual performance of a significant class of engineering systems. Again appreciate the approximation involved, the problems which can occur, how to identify them and what to do if they arise. Present results in a manner more engaging than a simple table or graph.

4. Approximately numerically solve boundary value problems. The algorithm presented is widely recognised as being the simplest, but not the best.

5. Produce a 3D graphical representation of an object and animate in MATLAB. Recognise the importance of quality of presentation of data/reports and improve personal effectiveness in this regard.

6. Break down an engineering problem into solvable sub-problems both as an individual and as part of a team. Make an engaging multi-media presentation of results.

7. Develop a MATLAB function m-file to a professional standard.

Student Effort Hours:

Student Effort Type | Hours |
---|---|

Lectures | 24 |

Laboratories | 12 |

Specified Learning Activities | 48 |

Autonomous Student Learning | 36 |

Total | 120 |

Approaches to Teaching and Learning:

Project-based continuous assessment. Emphasis on practical application.

Project-based continuous assessment. Emphasis on practical application.

Requirements, Exclusions and Recommendations

Differential and Integral Calculus to advanced level 1 or better.

Differential Equations to advanced level 1 or better.

Mechanics to advanced level 1 or better.

Matrix Theory to advanced level 1 or better.

Module Requisites and Incompatibles

BSEN30530 -

Resit In | Terminal Exam |
---|---|

Autumn | No |

Feedback Strategy/Strategies

• Feedback individually to students, post-assessment

• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Graded reports available for collection after full collection of reports graded. Associated grading scheme and common errors document available to class through Brightspace.

Lecture | Offering 1 | Week(s) - 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32 | Thurs 09:00 - 09:50 |

Lecture | Offering 1 | Week(s) - 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32 | Tues 09:00 - 09:50 |

Laboratory | Offering 1 | Week(s) - 19, 21, 23, 25, 29, 31 | Tues 11:00 - 12:50 |

Laboratory | Offering 2 | Week(s) - 20, 22, 24, 28, 30, 32 | Tues 11:00 - 12:50 |

Laboratory | Offering 3 | Week(s) - 19, 21, 23, 25, 29, 31 | Thurs 11:00 - 12:50 |

Laboratory | Offering 4 | Week(s) - 20, 22, 24, 28, 30, 31 | Mon 15:00 - 16:50 |