APM3701 - Unisa Study Notes...APM3701 -18-Y1-S1 is hosted by your e-tutor and contains the tutorials and online assignments. The site APM3701 -18-Y1 contains the main study materials

BAR CODE

Define tomorrow. universityof south africa

Tutorial Letter 101/3/2018

PARTIAL DIFFERENTIAL EQUATIONS

APM3701

Semesters 1 & 2

Department of Mathematical Sciences

IMPORTANT INFORMATION:

This tutorial letter contains importantinformation about your module.

APM3701/101/3/2018

CONTENTS

Page

1 INTRODUCTION ................................................................................................................ 4

2 PURPOSE OF OUTCOMES FOR THE MODULE.............................................................. 5

2.1 Purpose of the module:........................................................................................................ 5

2.2 Specific Outcomes and Assessment Criteria:...................................................................... 5

3 LECTURER(S) AND CONTACT DETAILS ........................................................................ 9

3.1 Lecturer(s) .......................................................................................................................... 9

3.2 Department ....................................................................................................................... 10

3.3 University .......................................................................................................................... 10

4 RESOURCES ................................................................................................................... 10

4.1 Prescribed Book................................................................................................................ 10

4.2 Recommended Books....................................................................................................... 10

4.3 Electronic Reserves (e-Reserves)..................................................................................... 10

5 Library services and resources information ................................................................ 11

5.1 STUDENT SUPPORT SERVICES .................................................................................... 11

6 STUDY PLAN ................................................................................................................... 12

7 PRACTICAL WORK AND WORK-INTEGRATED LEARNING ........................................ 12

8 ASSESSMENT ................................................................................................................. 12

8.1 Assessment criteria .......................................................................................................... 12

8.2 Assessment plan .............................................................................................................. 12

8.3 Very Important: Authentic Work......................................................................................... 14

8.4 Assignment numbers ........................................................................................................ 14

8.4.1 General assignment numbers ........................................................................................... 14

8.4.2 Unique assignment numbers ............................................................................................ 14

8.5 Assignments due dates .................................................................................................... 14

8.6 Submission of assignments .............................................................................................. 14

8.7 The assignments .............................................................................................................. 15

8.8 Assignments Semester 1/2018 ......................................................................................... 17

8.9 Assignments Semester 2/2018 ......................................................................................... 22

8.10 Other assessment methods .............................................................................................. 26

2

APM3701/101

9 The EXAMINATIONS........................................................................................................ 26

9.1 Examination admission, Examination period and Examination paper............................... 26

9.2 Moderation of Exam........................................................................................................... 26

9.3 Examination Period............................................................................................................ 26

9.4 Previous examination papers............................................................................................. 27

9.5 Book work for the examination........................................................................................... 27

10 FREQUENTLY ASKED QUESTIONS .............................................................................. 28

11 SOURCES CONSULTED ................................................................................................. 28

12 IN CLOSING ..................................................................................................................... 28

3

1 INTRODUCTION

Dear Student

Welcome to the APM3701 module in the department of Mathematical Sciences at Unisa. We trustthat you will find this module both interesting and rewarding.

Some of this tutorial matter may not be available when you register. Tutorial matter that is not avail-able when you register will be posted to you as soon as possible, but is also available on myUnisa.

Please read this tutorial letter in detail as it contains vital information.

You must be registered on myUnisa (http://my.unisa.ac.za) to be able to submit assign-ments online, gain access to the library functions and various learning resources, download studymaterial, “chat” to your lecturers and fellow students about your studies and the challenges youencounter, and participate in online discussion forums. myUnisa provides additional opportunitiesto take part in activities and discussions of relevance to your module topics, assignments, marksand examinations.

A tutorial letter is our way of communicating with you about teaching, learning and assessment.You will receive a number of tutorial letters during the course of the module.

This particular tutorial letter contains important information about the scheme of work, resources,assignments, the assessment criteria, admission requirements for the examination as well as in-structions on the preparation and submission of assignments for this module. We urge you to readit and read all following subsequent tutorial letters carefully and to keep it at hand when workingthrough the study material, preparing and submitting the assignments, preparing for the exam-ination and addressing queries that you may have about the course (course content, textbook,worked examples and exercises, theorems and their applications in your assignments, tutorial andtextbook problems, etc.) to your APM3701 lecturers. More general and detailed information and anorientation to your studies at Unisa is contained in the Study @ Unisa brochure which is includedin your study package.

Please note that this is a special module in that it is supported through Extended Science Pathway(ESP). The program provides additional learning support in different forms, which is designed tomake your studies easier and to help you succeed. The additional support is largely in the form oftutorials, and is offered to you free of charge. It is very important that you take advantage of thesupport to ensure your own success.

The main course material for this module is presented in the form of a study guide. Other studymaterials include Tutorial Letter 102, which contains the Tutorial Resource for the module. Tu-torial Letters 201 and 202 contain solutions to assignments 1 and 2. A Tutorial letter containingthe examination guidelines to help you to prepare for the examination will be made available in duecourse. Lastly, Tutorial Letter 301 contains general but important information on foundation supportinterventions in modules. Some of this study material may not have been available to you whenyou registered. Study material that was not available when you registered will be posted to you assoon as possible, and will also be available on myUnisa.

4

http://my.unisa.ac.za

APM3701/101

Both the study guide and the tutorial resource are divided into study or tutorial units in whichthere are different learning activities which you should complete at specific periods during the year.Please take time to complete the activities in these study materials. The activities have beendesigned to help you to understand the difficult concepts, and to supplement the theoretical knowl-edge with practical experience.

Remember that your lecturer and tutor are always available to assist you with your studies, butthe responsibility to contact us if you experience any difficulties lies with you. Please feel free tocontact us during office hours. You will find our contact details in Tutorial Letter 102. During theyear, your lecturer will be communicating with you by means of sms and through myUnisa.

You are strongly advised to spend as much time as you can on learning sites on the internet towhich you may referred to, and to myUnisa. On myUnisa, besides the mentoring by a tutor, youhave a variety of opportunities to interact with other students to enrich your learning experience.To register on myUnisa, log onto the Unisa website myUnisa (http://my.unisa.ac.za) andfollow the relevant links. On myUnisa, you will see two sites, the main lecturer’s teaching sitewith the code APM3701 -18-Y1, and the tutoring site with the code APM3701 -18-Y1-S1. The siteAPM3701 -18-Y1-S1 is hosted by your e-tutor and contains the tutorials and online assignments.The site APM3701 -18-Y1 contains the main study materials for the module. It is also the site fromwhich your lecturer will communicate to you about various learning activities during the year, andfrom which your lecturer will control the learning activities, including your tutorials.

We hope that you will enjoy APM3701 and we wish you all the best in your studies at Unisa!

2 PURPOSE OF OUTCOMES FOR THE MODULE

2.1 Purpose of the module:

Students completing this module are introduced to Partial Differential equations and boundaryvalue problems, including Fourier series, Bessel series, Fourier transforms, the Storm Liouvilleproblem. This module is designed for students who are interested in pursuing an academic andresearch career in mathematics, applied mathematics, physics and engineering.

2.2 Specific Outcomes and Assessment Criteria:

Specific outcome 1:Define partial differential equationsRange: This includes but is not limited to equations in mathematical physics such as Laplace, thewave, and the heat or diffusion equation.Assessment criteria

1.1 Definitions of a partial differential equation, the order of a partial differential equation, a so-lution of a partial differential equation, linear partial differential equation, and homogeneouspartial differential equations are given.

1.2 The principle of superposition is proven and its failure stated.

1.3 The initial-boundary value problem is formally defined.

5

http://my.unisa.ac.za

1.4 Well-posed and ill-posed problems are also defined.

1.5 Examples of some important linear partial differential equations are given.

Specific outcome 2:Classify second order linear partial differential equations and find their solutions.Assessment criteria

2.1 Second order linear partial differential equations are classified as elliptic, parabolic, or hyper-bolic according to the formula B2 − AC < 0,= 0 or > 0, respectively.

2.2 General solutions of linear partial differential equation are found by direct integration.

2.3 Partial differential equations are reduced to canonical forms and solutions are obtained forparabolic, elliptic and hyperbolic equations.

2.4 For the wave equation, d’Alembert solution is obtained.

Specific outcome 3:Calculate the Fourier coefficients and Fourier series of a given function.Obtain eigenvalues and eigenfunctions of an eigenvalue problem.Assessment criteria

3.1 Formal definitions of orthogonal functions, orthogonal sequence, and properties of orthogo-nal systems are reviewed.

3.2 The notion of the inner product is recalled.

3.3 Fourier series and Fourier coefficients are formally defined.

3.4 The fundamental Convergence theorem for Fourier series is proven.

3.5 Cosine, sine and complex Fourier series are defined.

3.6 The Sturm-Liouville theory is stated.

3.7 Properties of eigenvalues and eigenfunctions of the regular Sturm-Liouville problem aregiven.

Specific outcome 4:Derive and find solutions for the heat, wave and Laplace equations.Range: The heat and wave equations are studied in one dimensions and in rectangular coordi-nates. Whereas the Laplace equation is considered in two dimensions and in rectangular coordi-nates.Assessment criteria

4.1 The heat equation is derived by assuming that heat is transferred by conduction.

4.2 Different initial and boundary conditions for the heat equation are considered.

Range: This includes boundary conditions of first, second and third kind, and dynamicalboundary conditions.

6

APM3701/101

4.3 Obtain the solution of the one dimensional heat equation by method of separation of variables.

4.4 Solutions of the heat equation are obtained:

Range: These solutions are obtained with the following boundary conditions:

Zero temperature at finite ends, non-homogeneous boundary conditions, insulated end(s),and radiating end (or boundary conditions of the third kind).

4.5 The heat equation is solved in a thin circular ring: the heat equation with periodic boundaryconditions.

4.6 The wave equation is derived by determining the vibrations of a stretched string.

4.7 The following initial and boundary conditions for the wave equation are derived: fixed end withzero displacement, varying end, spring-mass system boundary conditions, free end boundaryconditions, dynamic boundary conditions.

4.8 The method of separation of variables applying to Fourier series is applied to solve the waveequation.

4.9 The solution of the wave equation with fixed ends is obtained by the method of separation ofvariables.

4.10 Laplace equation is derived and its solution is obtained inside a rectangle by the method ofseparation of variables.

Specific outcome 5Derive partial differential equations by laws of conservation and by the energy method.Assessment criteria

5.1 Laws of conservation, in particular the law of conservation of Energy are reviewed.

5.2 The Heat equation is derived by the law of conservation by assuming that there are nosources or sinks which release or absorb heat: heat flows in and out of the rod only atthe end points of the rod being studied.

5.3 The Burgers and elastic beam equations in one dimension are derived by laws of conserva-tion.

Specific outcome 6Formulate the maximum-minimum principle for the heat and Laplace equation and prove unique-ness and stability results for the heat, wave, and Laplace equations.Assessment criteria

6.1 The definition of well-posed and ill-posed problems is given.

6.2 The maximum principle for the heat and Laplace equations is proven.

6.3 The maximum principle is physically interpreted.

6.4 Stability of the solution of the heat and Laplace equation are stated and proven.

7

6.5 An energy method is used to prove uniqueness for the Dirichlet problem for the wave equa-tion.

6.6 Posteriori existence, uniqueness, and stability for the Dirichlet problem for the Heat equationare established.

Specific outcome 7Apply the Fourier transform method in order to solve standard partial differential equations.Assessment criteria

7.1 The Fourier integral theorem is proven.

7.2 Formal definitions of the Fourier transform and inverse Transform are given.

7.3 Fourier sine and cosine transforms are analysed.

7.4 Properties of the Fourier transform are stated and proven.

7.5 The Convolution theorem is stated.

7.6 Fourier sine and cosine transforms and their applications are applied to boundary-value prob-lems in semi-infinite and infinite regions.

7.7 Fourier transforms are applied to solve boundary-value problems in semi-infinite and infiniteregions.

7.8 The solution of boundary value problems for infinite domains is expressed as convolutionsinvolving the boundary values or initial data.

Specific outcome 8Calculate Bessel and Legendre series and apply them to solve boundary-value problems in othercoordinate systems.Range: The boundary-value problems describing the heat, wave and Laplace equations in polarand cylindrical coordinates.Assessment criteria

8.1 Bessel series and coefficient of Bessel series of a function are formulated and calculated.

8.2 Solution of Laplace’s equation in polar coordinates is obtained.

8.3 Heat, Wave and Laplace problems are solved in polar and cylindrical coordinates by applyingparametric Bessel series.

8.4 Solutions of problems in spherical coordinates are obtained by applying Bessel series.

8

APM3701/101

3 LECTURER(S) AND CONTACT DETAILS

3.1 Lecturer(s)

The contact details for the lecturer responsible for this module is

Postal address: The APM3701 LecturesDepartment of Mathematical SciencesPrivate Bag X6Florida1709South Africa

LoadLevel

No daylight saving time In Daylight saving timeMonday-Saturday

Sunday &holidays

Tuesday-Sunday

Sunday &holidays

Light 00:10-06:08 00:05-16:0922:07-23:22 00:04-06:5900:03-17:0923:05-23:05

Medium 07:03-17:5621:02-23:54 17:00-21:5207:01-18:5022:00-23:53 18:20-22:55

Heavy 18:20-20:57 19:10-21:58

Information concerning lecturers responsible for this module will be included in Tutorial Letter 102,which you will receive shortly. If you need to contact a lecturer before you receive Tutorial Letter102 you may phone the secretary of the department at 011 670 9147, i.e. the Department of Math-ematical Sciences. When you contact the secretary, please say which module you are enquiringabout. She will put you through to an available lecturer. Please have your study material with youwhen you contact us. When you speak to a lecturer, it is helpful to be very specific about yourproblem.

It may help you explain your problem more clearly over the telephone if you first write it down your-self. Make sure you have paper and a pen available when you phone, so that you can write downthe explanation. Please remember, if you want to visit a lecturer you must make an appointment.If you do not make an appointment beforehand there may not be anyone available to help you.To make an appointment, phone the secretary at (011) 670 9147. She will put you through to therelevant lecturer. Not all the lecturers are always available, but there will usually (except, for exam-ple, when all the lecturers have to attend a meeting) be someone to see you between 08:00 and13:00. When you phone to make an appointment, please give the lecturer your name and studentnumber, the code of the module for which you want to make an appointment, a contact telephonenumber (if possible).

If for some reason you cannot keep the appointment, please let the lecturer know. When you havemade an appointment to see someone, please bring with you your initial attempts at solving theproblem. It is usually more helpful if we can show you where you have misunderstood a particularconcept, or applied a method incorrectly, and then suggest how to carry on from there.

9

All queries that are not of a purely administrative nature but are about the content of this moduleshould be directed to your lecturer(s). Tutorial letter 301 will provide additional contact details foryour lecturer. Please have your study material with you when you contact your lecturer by tele-phone. If you are unable to reach us, leave a message with the departmental secretary. Provideyour name, the time of the telephone call and contact details. If you have problems with questionsthat you are unable to solve, please send your own attempts so that the lecturers can determinewhere the fault lies.

Please note: Letters to lecturers may not be enclosed with or inserted into assignments.

3.2 Department

Fax number: 011 670 9171 (RSA) +27 11 670 9171 (International)Departmental Secretary: 011 670 9147 (RSA) +27 11 670 9147 (International)

3.3 University

If you need to contact the University about matters not related to the content of this module, pleaseconsult the publication Study @ Unisa that you received with your study material. This bookletcontains information on how to contact the University (e.g. to whom you can write for differentqueries, important telephone and fax numbers, addresses and details of the times certain facilitiesare open). Always have your student number at hand when you contact the University.

4 RESOURCES

For library service request procedures (listed below), please consult the Study @ Unisa brochure.

4.1 Prescribed Book

The prescribed textbook for this module for this semester is:

Nakhlé H. AsmarPartial Differential Equations with Fourier and Boundary Value Problems,

Second Edition, Pearson Hall, Upper Saddle River, 2005

It is essential that you have this book. You may start working on the assignments without theprescribed book as the Study Guide is 95% self-contained, it is necessary to have the PrescribedBook since many exercises are referred to it. Please consult the list of official booksellers and theiraddresses listed in Unisa: My studies @Unisa.

4.2 Recommended Books

There are no recommended books for this module.

4.3 Electronic Reserves (e-Reserves)

There are no e-Reserves for this module.

10

APM3701/101

5 Library services and resources information

For brief information go to http://www.unisa.ac.za/brochures/studies

For detailed information, go to http://www.unisa.ac.za/library. For research support andservices of personal librarians, click on ”Research support”.

The library has compiled a number of library guides:

• finding recommended reading in the print collection and e-reserves –http://libguides.unisa.ac.za/request/undergrad

• request material – http://libguides.unisa.ac.za/request/request

• postgraduate information services – http://libguides.unisa.ac.za/request/postgrad

• finding, obtaining and using library resources and tools to assist in doing research –http://libguides.unisa.ac.za/Research_Skills

• how to contact the Library/finding us on social media/frequently asked questions –http://libguides.unisa.ac.za/ask

5.1 STUDENT SUPPORT SERVICES

Extended Science Pathway

Science Pathway provides additional learning support so that you have a better chance of passingthe module. Your learning is supported through special tutorials which are designed to ensure thatyou master the basic concepts in the module first, and to help you through the concepts that areknown to be typically difficult for students. The tutorials are also designed to help you to developyour reading, writing and study skills, and to understand your curriculum in relation to your careerchoices. To achieve these goals, the tutorials provide ample time for you to constantly interact witha tutor and with your fellow students to clarify difficult questions and concepts.

The tutorials are important because your formative assessment is inbuilt in the tutoring, wherebytutors mark your assignments, give you self-assessment tasks to do, and provide you with feed-back on the learning activities. Tutors are also tasked to prepare you for the final examination. Youhave the choice of attending face-face tutorials at a learning centre, and/or on-line on myUnisa.Upon registration, an E-tutor is automatically allocated to you. For face-face tutorials, you have toregister at the learning center nearest to you. The tutorials are contained in Tutorial Letter 102, theTutorial Resource for the Module APM3701 . You will receive the printed version of the tutorialresource before the tutorials start. By that time, you will also be able to access the electronic tuto-rial resource among the official study materials on the module site, and the on-line version on thetutorial site, on myUnisa.

You are strongly advised to prepare for the tutorials by reading through the tutorial resource beforeyou start your tutorials. It is important to understand the rules and your responsibilities with regardto the tutorials and to your assessment. Remember, the tutorial resource does not replace your

11

http://www.unisa.ac.za/brochures/studieshttp://www.unisa.ac.za/libraryhttp://libguides.unisa.ac.za/request/undergradhttp://libguides.unisa.ac.za/request/requesthttp://libguides.unisa.ac.za/request/postgradhttp://libguides.unisa.ac.za/Research_Skillshttp://libguides.unisa.ac.za/ask

study guide, but complements it with more basic content and interactive learning activities thattarget the difficult concepts. The tutorial resource must be used with reference to the study guidewhich is your main learning resource.

As a word of caution, please note that in distance learning, the fact that students are enrolled in self-study is frequently the reason for failure. This is because in the distance education environment,minimal yet vital interaction takes place among students, and between students and mentors suchas tutors and lecturers. It is therefore your responsibility to take full advantage of the tutorials bysparing as much time as you can for tutorials, and by aggressively engaging with the tutor andwith other students, to deepen your own understanding, which will enrich your learning experience.Details on other student support are contained in Tutorial Letter 301 and in the Study @ Unisabrochure.

6 STUDY PLAN

The due dates of the assignments set the pace at which you should work through the content. (SeeSection 8.5).

Study plan Semester 1 Semester 2Outcomes 1 to 4 to be achieved by 12 March 28 AugustOutcomes 5 to 8 to be achieved by 9 April 24 SeptemberRevision of Assignments 20 April 10 OctoberRevision of the Study Guide 30 April 20 October

Draw up your own study schedule and keep to it!

See the brochure Study @ Unisa for general time management and planning skills.

7 PRACTICAL WORK AND WORK-INTEGRATED LEARNING

There are no practicals for this module.

8 ASSESSMENT

8.1 Assessment criteria

There are no assessment criteria for this module.

8.2 Assessment plan

Assignments are seen as part of the learning material and assessment for this module. As youdo the assignments, discuss with fellow students or tutor, you are actively engaged in learning.It is therefore important that you complete all the assignments. This tutorial letter contains the

assignments for this year’s semesters. With respect to assignments, I want to emphasize thefollowing points:

12

APM3701/101

8.2.1 Please take note that: there are two assignments for each semester, ONLY DO THEASSIGNMENTS FOR THE SEMESTER YOU ARE REGISTERED FOR. Each assign-ment covers a new section of work. The questions for Assignments are attached at theend of this tutorial letter.

8.2.2 The work you submit for marking should be your best version, an assignment should notjust be neat, but also logical. An example of sloppy presentation is to simply write downa series of statements without any indication of connection between them

8.2.3 The semester mark for APM3701 in 2018 is 20% of the average mark obtained in As-signment 01 and 02. Your final mark will be the sum of your semester mark and 80%of the exam mark (For example, if you get 60% in Assignment 01, 50% in Assignment02, and 64% in Exam; your average semester mark will be (60+50)/2=55% and yoursemester mark will be 20% of 55 which is 55/5=11, the contribution of the exam to-ward final mark will be 80% of 65 which is 65*80/100=52. Thus, your final mark willbe 11+52=63%. Please apply the following formula to calculate your final mark, whereAS1 and AS2 are marks (in percentage) obtained in Assignment 01 and Assignment 02respectively, EX is your exam mark (in percentage), and FM your final mark (in percent-

age): FM =AS1 + AS2

10+

4EX

5.

Please, see also section 9 on how the examination system works.

8.2.4 These minimum requirements must not be regarded as sufficient preparation for successin the examination. It is important that you complete all the assignments.

8.2.5 Your assignments must be correctly numbered, i.e. the number must correspond withthe number given in the Tutorial Letter. Even though Assignment 02 may be the firstassignment done by you, it must be numbered 02 not 01.

8.2.6 The assignments are somewhat long, please avoid repeating the proof of formulae al-ready done in the Study Guide and Prescribed Book, apply them directly instead.No mark will be awarded if you copy solution from past assignments and examsolutions or repeat proof of formulae already done in the Study Guide and Pre-scribed Book.

8.2.7 Please take heed of the closing dates for the assignments. In order to obtain full creditfor your work, you must see to it that your assignments reach me on or before the closingdates. You will receive the solution for both assignments, only if you submit the relevantassignment. No solution will be posted to you if the relevant assignment is not received.These solutions will be available about two weeks after the closing date of the relevantassignment.

13

8.3 Very Important: Authentic Work

PLEASE NOTE: Although students may work together when preparing assignments, eachCOPYING student must write and submit his or her own individual assignment. In otherAND words, you must submit your own ideas in your own words, sometimesPLAGIARISM interspersing relevant short quotations that are properly referenced. It is

unacceptable for students to submit identical assignments on the basis thatthey worked together. That is copying (a form of plagiarism) and none of theseassignments will be marked. Furthermore, you may be penalised or subjectedto disciplinary proceedings by the University. This also applies to solutionstaken from books and previous years assignments and exams.

8.4 Assignment numbers

8.4.1 General assignment numbers

Each of your assignments has a general identification number which is assigned consecutivelystarting from 01 to 02 for each semester.

8.4.2 Unique assignment numbers

Each assignment also has a unique 6-digit assignment number (e.g. 102717).

SEMESTER ANDASSIGNMENT NUMBER UNIQUE NUMBER

SEMESTER 1 / ASSIGNMENT 01 709317SEMESTER 1 / ASSIGNMENT 02 894835SEMESTER 2 / ASSIGNMENT 01 746621SEMESTER 2 / ASSIGNMENT 02 766401

8.5 Assignments due dates

This module has two assignments for each semester, with the following closing dates:

ASSIGNMENT 01 ASSIGNMENT 02SEMESTER 1 19 MARCH 2018 16 APRIL 2018SEMESTER 2 31 AUGUST 2018 1 OCTOBER 2018

Solution to all assignments will be available on myUnisa website about 2 weeks after closing dates.

8.6 Submission of assignments

You can submit your assignments in different ways, depending on the type of assignment as follows;

SEMESTER ANDASSIGNMENT NUMBER FORMAT METHOD OF SUBMISSION

SEMESTER 1 / ASSIGNMENT 01 Written Postal or electronically via myUnisaSEMESTER 1 / ASSIGNMENT 02 Written Postal or electronically via myUnisaSEMESTER 2 / ASSIGNMENT 01 Written Postal or electronically via myUnisaSEMESTER 2 / ASSIGNMENT 02 Written Postal or electronically via myUnisa

EXAM PAPER Written In venue and handed in

14

APM3701/101

For detailed information on assignments, please refer to the Study @ Unisabrochure which you received with your study package.

To submit an assignment via myUnisa:

• Go to myUnisa.

• Log in with your student number and password.

• Select the module.

• Click on “Assignments” in the menu on the left-hand side of the screen.

• Click on the assignment number you wish to submit.

• Follow the instructions.

8.7 The assignments

Before you proceed with the assignments please refer to this brief list of DO’S and DON’TS.DO

• use the assignments to plan your work schedule

• work regularly at the assignment once you have begun

• read each question carefully

• check your answers and presentation for errors in logic as well as careless calculation errors

• discuss difficult questions with other students if possible

• read the comments carefully when marked assignments are returned to you, and wherepossible apply these comments in future

• compare the solutions you receive with your own answers.

DON’T

• rush through questions

• leave out questions if you think your answer is wrong – we can help you more effectively if wesee where you have made mistakes

• scribble changes over other work – rather begin again

• write down someone else’s answer, hence possibly duplicating someone else’s mistakes

• be discouraged when you make mistakes and get low marks.

15

In each written assignment, clearly show all workings, calculations, possible diagramsand reasoning used in determining your answers. Note that marks will be deducted forunsatisfactory presentation of answers.Do not use of a calculator. Doing numerical calculations mentally or by using the definitionsand properties of the different functions will be good exercise in view of the exam at the endof the year. The assignments for 2018 follow below.NOTE: All numbers and sections in brackets refer to the Study GuidePlease avoid repeating proofs of formulae already done in the Study Guide and Prescribed Book,use or apply them directly instead.Very Important: Only do the assignment for the semester you registered for.

16

APM3701/101

8.8 Assignments Semester 1/2018

ONLY FOR SEMESTER 1ASSIGNMENT 01

CHAPTER 1 – CHAPTER 5 OF STUDY GUIDECLOSING DATE: 19 MARCH 2018

Unique Number: 709317

PLEASE DO ONLY THE ASSIGNMENTS FOR THE SEMESTER YOU ARE REGISTERED FOR.TAKE NOTE OF THE FOLLOWING:

• All numbers and sections in bracket refer to the Study Guide (SG) and to the Prescribe Book(PB), unless specified otherwise.

• Please avoid repeating proofs of formulae and theorems already done in the Study Guideand Prescribed Book, use or apply them directly instead.

• No mark will be awarded if you copy solution from past assignments and exam solutions orrepeat proof of formulae already done in the Study Guide and Prescribed Book

QUESTION 1

(a) [Based on Chapter 1 of SG ]

Given the equation∂u

∂y− ∂u∂x

= 0.

(a1) Find its general solution by using following change of variables

α = ax+ by, β = cx+ dy.

(5 Marks)

(a2) Find a solution which is equal to1

x2 + x+ 1along the x− axis. (8 Marks)

(b) Use the method of characteristic to solve the equation

∂u

∂x+

1

x

∂u

∂y= 0.

Check that your answer satisfies the the differential equation. (7 Marks)

[20 Marks]

17

QUESTION 2

[Based on Chapter 1 of SG ]An uninsulated electrical wire runs along the x− axis, the leakage occurs along the entire lengthof the wire. If V (x, t) and I (x, t) are the voltage and current at a point x in the wire at time trespectively. V and G satisfy each the following differential equations

∂V

∂x= −L∂I

∂t−RI and ∂I

∂x= −C∂V

∂t−GV,

where L,R,C and G are the inductance, the resistance, the capacitance and the leakage to theground respectively. Show that V and I each satisfy the telegraph equation (1)

∂2u

∂x2= LC

∂2u

∂t2+ (RC + LG)

∂u

∂t+RGu (1)

[20 Marks]

QUESTION 3

[Based on Chapter 3 of SG ]Given the graph below of the function f.

x

y

K−K

k

O

L−L

a) Derive the Fourier series of f (17 Marks)

b) Determine its values at the points of discontinuity. (3 Marks)

[20 Marks]

QUESTION 4

(Based on §5.4 and 5.5 of SG)In the question, you will illustrate the failure of the principle of maximum/minimum principle, andexplain why the principle fails.Consider a following heat problem with an internal source of heat

ut = uxx + 2 (t+ 1) + x (1− x) , 0 < x < 1, t > 0u (0, t) = 0, u (1, t) = 0, t > 0

u (x, 0) = x (1− x) , 0 < x < 1.

1. Verify that u (x, t) = (t− 1)x (1− x) is a solution of the above heat equation. (10 Marks)

18

APM3701/101

2. Denote by M and m the maximum and minimum values of the initial and boundary conditions.Calculate M and m. (5 Marks)

3. Show that the principle of maximum principle does not hold, by finding x̄, 0 < x̄ < 1 and t̄ > 0,such that u (x̄, t̄) > M. (3 Marks)

4. Explain why did the principle of maximum principle fail in part c. (2 Marks)

[20 Marks]

QUESTION 5

The vertical vibration at position x and at time t of a stretched homogeneous and infinitely longstring is determined by the function u = u (x, y). Suppose that when the string is straight, it has alinear density of 2 and the tension at any given point of the string is 8.

a. Derive the partial differential equation satisfied by u . [Do not repeat the derivation done in theStudy Guide, you may use the appropriate formula and explain the constants and variablesused.] (3 Marks)

b. If the initial position is 1 − x and the initial velocity is e−x2 write down the initial–value problemwhich models the above situation. (3 Marks)

c. Use the appropriate formula to solve the initial value problem found in (b). (14 Marks)

[20 Marks]

TOTAL: 100 Marks

19


CHAPTER 5 – CHAPTER 7 OF STUDY GUIDECLOSING DATE: 16 APRIL 2018






QUESTION 1

(Base on 5.4.3 of Study Guide and 3.11 of PB) Consider the following heat equation

ut = uxx, 0 < x < L, t > 0, (2)u (0, t) = g1 (t) , u (L, t) = g2 (t) , t > 0, (3)u (x, 0) = f (x) , 0 < x < L. (4)

Suppose that f, g1, g2 are bounded, there exist constants m and M such that for all 0 ≤ x ≤ L, andall t ≥ 0, we have

m ≤ f (x) ≤M ;m ≤ g1 (x) ≤M ;m ≤ g2 (x) ≤M. (5)Then the solution of (2)–(4) satisfies the inequalities

m ≤ u (x, t) ≤M ; 0 ≤ x ≤ L;x ≥ 0. (6)

a) Show that the solution of the heat problem (2)–(4) is unique. [10 MARKS]

b) Suppose that u1 (x, t) and u2 (x, t) are solution of the heat problem (2)–(4) such that u1 (0, t) ≤u2 (0, t) , u1 (L, t) ≤ u2 (L, t) , and u1 (x, 0) ≤ u2 (x, 0) . Show that u1 (x, t) ≤ u2 (x, t) for all0 ≤ x ≤ L and all t ≥ 0. [10 MARKS]

[20 MARKS]

20

APM3701/101

QUESTION 2

The vertical vibrations u (x, t) of a stretched homogeneous string of length L with its ends fixed onthe x− axis at x = 0 and x = L, are determined by the following initial-boundary value problem

∂2u

∂t2= c2

∂2u

∂x2, 0 < x < L, t > 0

u(0, t) = 0 = u(L, t),u(x, 0) = f (x) , ut (x, 0) = g (x)

The energy at time t of the vibrating string described above is given by

E (t) =1

2

∫ L0

(u2t + c

2u2x)dx.

Prove that the energy during the free vibrations of the string described in question 4 is constant forall time. In other words, the principle of conservation of energy is satisfied.

[Hint: Prove thatdE

dt= 0, by showing first that

dE

dt= c2

∫ L0

(uxut)x dx, and ut (0, L) = 0 = ut (L, t),for all t > 0]. [20 Marks]

QUESTION 3

Use an appropriate Fourier transform to find the temperature u (x, t) in an infinite rod if the initialtemperature is constant (not zero) in (−1, 1), and zero outside (−1, 1). [20 Marks]

QUESTION 4

When there is heat transfer from the lateral side of an infinitely long cylinder of radius a into asurrounding medium, the temperature inside the cylinder depends upon the time t and the distancer from its longitudinal axis (i.e. the axis through the centre and parallel to the lateral side.)

a. Write down the partial differential equation that models this problem. (2 Marks)

b. Suppose that the surrounding medium is at temperature zero and the initial temperature isconstant at every point, derive the initial and boundary conditions.

(a) [Hint: For the boundary condition use Newton’s law of cooling.] (4 Marks)

c. Solve the initial boundary value problem obtained in (a) and (b). (14 Marks)

[20 Marks]

QUESTION 5

(Based on Chapter 7 of SG).Determine the steady temperature inside a 90◦ sector of a circular annulus

(0 < a ≤ r ≤ b, 0 ≤ θ ≤ π

2

),

if its circular edges and the straight edge θ = 0, are in contact with ice (at 0oC, and the straightedge θ =

π

2is a given function of r. [20 Marks]

TOTAL: 100 Marks

21

8.9 Assignments Semester 2/2018


CHAPTER 1 – CHAPTER 5 OF STUDY GUIDECLOSING DATE: 31 AUGUST 2018






QUESTION 1

[Based on Chapter 1 of SG ]

(a) Find the solution of the equation (7) which is equal to1

2ye−2y

2 on the y− axis

∂u

∂y+ 3

∂u

∂x= 0. (7)

(13 Marks)

(b) Use the method of characteristic to find the general solution of the equation (7). Check thatyour answer satisfies the the differential equation. (7 Marks)

[20 Marks]

QUESTION 2

Consider the situation where a solute (i.e a substance dissolved in another substance, known assolvent, e.g in salted water, salt is the solute and water is the solvent), in a thin pipe of length L istransfer due to differences in concentration. Let us denote by c (x, t), the concentration at x at timet and let ϕ (x, t) be the flux (mass per second) at x at time t. If the solute is pumped in the pipe ata rate of f (x, t) , use the law of conservation to derive the diffusion equation for c. [20 Marks]

22

APM3701/101

QUESTION 3

Show that the Neumann problem

∇2u = 0 in Ω,∂u

∂n= g on ∂Ω.

has a unique solution only up to an additive constant, i.e. if u1 and u2 are two solutions, thenu1 − u2 = C.Hint: Use the identity

div (u∇u) = |∇u|2 + u∇2u,

where |∇u|2 =(∂u

∂x

)2+

(∂u

∂y

)2+

(∂u

∂z

)2. And then apply the Gauss’ Divergence Theorem to the

vector field u∇u. [20 Marks]

QUESTION 4

(Based on §5.4 and 5.5 of SG)The vertical vibration at position x and at time t of a stretched homogeneous and infinitely longstring is determined by the function u = u (x, y). Suyppose that when the string is straight, it has alinear density of 2 and the tension at any given point of the string is 8.

a. Derive the partial differential equation satisfied by u . [Do not repeat the derivation done in theStudy Guide, you may use the appropriate formula and explain the constants and variablesused.] (3 Marks)

b. If the initial position is 1 − x and the initial velocity is e−x2 write down the initial–value problemwhich models the above situation. (4 Marks)

c. Use the appropriate formula to solve the initial value problem found in (b). (13 Marks)

[20 Marks]

QUESTION 5

Consider the temperature u (x, t) at time t at each point x of a rod length 5, if one of its end is incontact with ice at 0◦ and the other end is exposed to warm air at 35◦. Assume that initially the rodis submerged in boiling water at 100o.

a) Derive the initial-boundary value problem of the problem described above. Explain all the steps,do not repeat the derivations done in the study Guide and Prescribed book. (8 MARKS).

b) Show that the solution of the initial-boundary value problem obtained in part (a) has a uniquesolution.[Hint: apply the maximum principle to the heat equation.] (12 MARKS)

[20 MARKS]

TOTAL: 100 Marks

23


CHAPTER 5 – CHAPTER 7 OF STUDY GUIDECLOSING DATE: 1 OCTOBER 2018






QUESTION 1

(Based on Chapters 4.3–4.5 of SG).Consider the heat flow in one–dimensional rod of unit length

(a) Assume that both ends of the rod are insulated. Derive the boundary conditions at both ends.(4 Marks)

b) If the initial temperature at each point of the rod is three times the distance of the point to theright end of the rod. Write down the initial–boundary value problem that models the aboveproblem. (5 Marks)

c) Solve the initial–boundary value problem obtained in (b). (11 Marks)

[20 Marks]

QUESTION 2

Show that the Neumann problem

∇2u = 0 in Ω,∂u

∂n= g on ∂Ω.

has a unique solution only up to an additive constant, i.e. if u1 and u2 are two solutions, thenu1 − u2 = C.Hint: Use the identity

div (u∇u) = |∇u|2 + u∇2u,

where |∇u|2 =(∂u

∂x

)2+

(∂u

∂y

)2+

(∂u

∂z

)2. And then apply the Gauss’ Divergence Theorem to the

vector field u∇u. [20 Marks]

24

APM3701/101

QUESTION 3

Find the displacement u (x, t) of a vibrating semi–infinite string, if the finite end is fixed, the initialvelocity is zero and the initial displacement is xe−x at every point x of the string. [20 Marks]

QUESTION 4

Consider Laplace’s equation inside a 90◦ sector of a circular annulus(0 < a ≤ r ≤ b, 0 ≤ θ ≤ π

2

)subject to the boundary conditions

u (r, 0) = 0, u(r,π

2

)= f (r) , u (a, θ) = 0, u (b, θ) = 0.

Solve completely the above Laplace equation, and show that the solution is

u (r, θ) =∞∑n=1

An sinhnπθ

ln ba

sin

(nπ

ln ba

lnr

a

)

where

An =2

ln ba

sinh nπ2

2 ln ba

∫ ba

f (r) sinnπ ln r

a

ln ba

dr

r.

[20 MARKS]

QUESTION 5

Find the temperature in a circular plate of radius c, if the edge of the plate is insulated and the initialtemperature is f (r) . [20 Marks]

TOTAL: [100 Marks]

25

8.10 Other assessment methods

There are no other assessment methods for this module.

9 The EXAMINATIONS

For general information and requirements as far as assignments are concerned, see the brochureUnisa: My studies @Unisa which you received with your study material.

9.1 Examination admission, Examination period and Examination paper

The examination consists of one two–hour paper, and the procedure for gaining admission to thisexamination is as follows:

1. You are automatically admitted to the exam on the submission of Assignment 01 by a specificdate, see Section 8.5. Please note that the lecturers are not responsible for exam admission,and ALL enquiries about exam admission should be directed by email to [email protected].

If you are registered for the first semester, you will write the examination in May/June 2018and the supplementary examination will be written in October/November 2018. If you areregistered for the second semester you will write the examination in October/November 2018and the supplementary examination will be written in May/June 0.

During the relevant semester, the Examination Section will provide you with information re-garding the examination in general, examination venues, examination dates and examinationtimes.

The exam consists of a two hour paper.

2. The module will be assessed by means of: the two assignments, and a 2-hour written ex-amination. The weighting of the two components will be: Assignments(semester mark): 20%and Examination: 80%

9.2 Moderation of Exam

The exam paper will be set and marked by the first examiner (your 2018 Lecturer), and moderatedby a second examiner (from the Department) and an external examiner (from another University),their names will be displayed on the exam paper.

9.3 Examination Period

Registered for . . . Examination period Supplementary examination periodSemester 1 May/June 2018 October/November 2018Semester 2 October/November 2018 May/June 2019

26

APM3701/101

9.4 Previous examination papers

Previous examination papers are available to students on myUnisa. Please note that the papersare not posted by the lecturer and, NO SOLUTION is available to students. You will have to querywith your lecturer, if you have difficulties answering the exam paper. I am not prepared to discuss indetails and/or answer questions concerning past exam papers. I will help you to solve the questionsof past exam papers, only if you ask the questions in the general context of the module. Questionssuch that, please send me the memorandum of May/June 2012 exam paper, or I do not know howto solve question 5 of October/November 2013 question paper, will not be entertained.

9.5 Book work for the examination

To help you in your preparation for the examination, you will receive a tutorial letter that will explainthe format of the examination paper, give you examples of questions that you may expect and setout clearly what material you have to study for examination purposes. In the meantime you shouldknow the following book work for examination purposes: (PB = Prescribed book, SG = StudyGuide, TL 101/2018 = This Tutorial Letter)

• The classification of Partial Differential Equations according to B2 − AC ≥ 0, B2 − AC = 0,B2 − AC ≤ 0, (Section 2.3 of SG). Direct integration of PDEs with boundary conditions(section 2.2.1), General Solution of the equation Auxx + 2Buxy + Cuyy = 0 (Section 2.4.2 ofSG.)

• D’Alembert’s solution for the Wave Equation (Section 3.4 of PB, Section 2.5 of SG)

• Orthogonal functions (Section 3.4 of SG)

• Definitions pertaining to Fourier series (all kinds) (Section 3.5 of SG)

• Sturm–Liouville problem, (Eigenvalues and eigenfunctions) (Section 3.6 of SG.)

• Derivation of the Heat, Wave and Laplace Equations with various initial and boundary condi-tions, (Sections 4.4.1, 4.6.1 and 4.8.1 of SG.)

• Separation of variables (Sections 4.5, 4.7, 4.8.2 of SG)

• Maximum and minimum principle (Section 5.4.1 of SG)

• Uniqueness and stability (Section 5.5.2, 5.3 of SG)

• Law of conservation and Energy method (Section 4.9 of SG.)

• The convolution theorem for Fourier transforms, relationships between Fourier cosine (sine)transforms of a function and that of its derivatives. Obviously, definitions of the differentFourier transforms and their inverses must be known. (Chapter 6 of SG.).

• You should be able to solve boundary–value problems in other coordinate systems. Theforms of the standard equations in polar, cylindrical and spherical coordinates will not begiven to you in exam, you must know them as good as you know them in cartesian coordinates(Chapter 7 of SG).

27

• You should also know the general forms of the Bessel and Legendre equations and un-derstand their different properties. The formulae for the coefficient of Fourier–Legendre,Fourier–Bessel series will be given to you, as well as the recurrence relations (Chapter 7 ofSG).

We hope that you will enjoy this module and we wish you success with your studies.

10 FREQUENTLY ASKED QUESTIONS

The Study @ Unisa brochure contains an A–Z guide of the most relevant study information.

11 SOURCES CONSULTED

None

12 IN CLOSING

Remember, you are important to us and we are very willing and available to assist you with yourcourse content related problems.

Our best wishes.Your APM3701 lecturers.

28

INTRODUCTION PURPOSE OF OUTCOMES FOR THE MODULEPurpose of the module:Specific Outcomes and Assessment Criteria:

LECTURER(S) AND CONTACT DETAILS Lecturer(s) Department University

RESOURCES Prescribed BookRecommended BooksElectronic Reserves (e-Reserves)

Library services and resources information STUDENT SUPPORT SERVICES

STUDY PLAN PRACTICAL WORK AND WORK-INTEGRATED LEARNING ASSESSMENT Assessment criteria Assessment plan Very Important: Authentic WorkAssignment numbers General assignment numbers Unique assignment numbers

Assignments due dates Submission of assignments The assignments Assignments Semester 1/2018 Assignments Semester 2/2018 Other assessment methods

The EXAMINATIONSExamination admission, Examination period and Examination paperModeration of ExamExamination PeriodPrevious examination papersBook work for the examination

FREQUENTLY ASKED QUESTIONS SOURCES CONSULTED IN CLOSING

Documents

APM3701 - Unisa Study Notes...APM3701 -18-Y1-S1 is hosted by your e-tutor and contains the tutorials and online assignments. The site APM3701 -18-Y1 contains the main study materials