- MST209_10Mathematical methods and models
Modelling with Fourier series
About this free course
This free course provides a sample of level 2 study in Mathematics: www.open.ac.uk/courses/find/mathematics.This version of the content may include video, images and interactive content that may not be optimised for your device. You can experience this free course as it was originally designed on OpenLearn, the home of free learning from The Open University: www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics-education/modelling-fourier-series/content-section-0.There you’ll also be able to track your progress via your activity record, which you can use to demonstrate your learning.The Open University, Walton Hall, Milton Keynes, MK7 6AA.Copyright © 2016 The Open University
Intellectual property
Unless otherwise stated, this resource is released under the terms of the Creative Commons Licence v4.0 http://creativecommons.org/licenses/by-nc-sa/4.0/deed.en_GB. Within that The Open University interprets this licence in the following way: www.open.edu/openlearn/about-openlearn/frequently-asked-questions-on-openlearn. Copyright and rights falling outside the terms of the Creative Commons Licence are retained or controlled by The Open University. Please read the full text before using any of the content. We believe the primary barrier to accessing high-quality educational experiences is cost, which is why we aim to publish as much free content as possible under an open licence. If it proves difficult to release content under our preferred Creative Commons licence (e.g. because we can’t afford or gain the clearances or find suitable alternatives), we will still release the materials for free under a personal end-user licence. This is because the learning experience will always be the same high quality offering and that should always be seen as positive – even if at times the licensing is different to Creative Commons. When using the content you must attribute us (The Open University) (the OU) and any identified author in accordance with the terms of the Creative Commons Licence.The Acknowledgements section is used to list, amongst other things, third party (Proprietary), licensed content which is not subject to Creative Commons licensing. Proprietary content must be used (retained) intact and in context to the content at all times.The Acknowledgements section is also used to bring to your attention any other Special Restrictions which may apply to the content. For example there may be times when the Creative Commons Non-Commercial Sharealike licence does not apply to any of the content even if owned by us (The Open University). In these instances, unless stated otherwise, the content may be used for personal and non-commercial use.We have also identified as Proprietary other material included in the content which is not subject to Creative Commons Licence. These are OU logos, trading names and may extend to certain photographic and video images and sound recordings and any other material as may be brought to your attention.Unauthorised use of any of the content may constitute a breach of the terms and conditions and/or intellectual property laws.We reserve the right to alter, amend or bring to an end any terms and conditions provided here without notice.All rights falling outside the terms of the Creative Commons licence are retained or controlled by The Open University.Head of Intellectual Property, The Open UniversityThe Open UniversityUnited Kingdom by Martins the Printers Ltd, Berwick-upon-Tweed, TD15 1RS978-1-4730-1560-9 (.kdl)
978-1-4730-0792-5 (.epub)IntroductionThis course shows how partial differential equations can be used to model phenomena such as waves and heat transfer. The prerequisite requirements to gain full advantage from this course are an understanding of ordinary differential equations and basic familiarity with partial differential equations.This OpenLearn course provides a sample of level 2 study in Mathematics.After studying this course, you should be able to:understand how the wave and diffusion partial differential equations can be used to model certain systems;determine appropriate simple boundary and initial conditions for such models;find families of solutions for the wave equation, damped wave equation, diffusion equation and similar homogeneous linear second-order partial differential equations, subject to simple boundary conditions, using the method of separating the variables;combine solutions of partial differential equations to satisfy given initial conditions by finding the coefficients of a Fourier series. 1 Modelling with Fourier seriesThe main teaching text of this course is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.Click the link below to open the workbook (PDF, 0.6 MB).WorkbookClick the link below to open the answerbook (PDF, 0.2 MB).Answer bookWorkbook contentsIntroduction1 Modelling using the wave equation
1.1 The taut string
1.2 Adding damping
2 Modelling using the diffusion equation
2.1 The insulated rod
2.2 The convecting rod
3 Separating the variables
3.1 Normal modes
3.2 The initial displacement
3.3 The damped plucked string
4 Solving the heat transfer problems
4.1 The insulated rod problem solved
4.2 The convecting rod problem solved
ConclusionThis free course provided an introduction to studying Mathematics. It took you through a series of exercises designed to develop your approach to study and learning at a distance and helped to improve your confidence as an independent learner.Keep on learning Study another free courseThere are more than 800 courses on OpenLearn for you to choose from on a range of subjects. Find out more about all our free courses. Take your studies furtherFind out more about studying with The Open University by visiting our online prospectus. If you are new to university study, you may be interested in our Access Courses or Certificates. What’s new from OpenLearn?
Sign up to our newsletter or view a sample. For reference, full URLs to pages listed above:OpenLearn – www.open.edu/openlearn/free-courses
Visiting our online prospectus – www.open.ac.uk/courses
Access Courses – www.open.ac.uk/courses/do-it/access
Certificates – www.open.ac.uk/courses/certificates-he
Newsletter – www.open.edu/openlearn/about-openlearn/subscribe-the-openlearn-newsletter
The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.Course image: Emlyn Stokes in Flickr made available under Creative Commons Attribution 2.0 Licence.All materials included in this course are derived from content originated at the Open University.Don't miss out:If reading this text has inspired you to learn more, you may be interested in joining the millions of people who discover our free learning resources and qualifications by visiting The Open University - www.open.edu/openlearn/free-courses
Discussion
2015081400