**Series, Partial Differential Equations and Fourier Transforms**

The first part of the module follows on first year Calculus, and covers infinite sequences and infinite series as well as the theorem of Taylor and the representation of functions as infinite series. This is followed by the expansion of periodic functions in terms of Fourier series, with applications to the solution of partial differential equations. The module concludes with an introduction to the Fourier transform.

**Module Information**

- 38571 242 (8) Engineering Mathematics 242
- Second year, second semester of the Programs in Engineering (except Civil Engineering).
- Two lectures and one tutorial of 1 hour per week.
- Prerequisite pass modules (PP³50): Engineering Mathematics 145 or Engineering Mathematics 214; Prerequisite module (PP³40): Engineering Mathematics 214.
- Language specification : A & E

**Lecturer**

- Prof A Fransman (Afrikaans); Office 3006, Bedryfsielkunde building (email: af@sun.ac.za)
- Prof Z Janelidze (English); Office 1023C, Bedryfsielkunde building (e-mail: zurab@sun.ac.za)

**Learning Material**

Prescribed textbooks:

- [1] J Stewart:
*Calculus*(7th Edition), Brooks/Cole Publishing Company, 2012. - [2] DG Zill and /WS Wright:
*Advanced Engineering Mathematics*(5th Edition), Jones and Bartlett Publishers, 2014.

**Module Content**

- [1] Chapter 11: Infinite sequences and infinite series: 11.1-11.10
- [2] Chapter 12, Sections 12.2 to 12.4: Fourier series.
- [2] Chapter 13, Sections 13.1, 13.3: Partial differential equations with application the heat equation.
- [2] Chapter 15, Sections 15.4 : Fourier Transform with application the heat equation.

**Learning Opportunities**

- The course material is dealt with in detail during the lectures. Tutorials provide the opportunity to work on problems and to discuss coursework under the supervision of the lecturer.

**Assessment**

- This module uses the “Flexible Assessment” method, as applied in the Engineering Faculty. For details, please refer to the Faculty’s Assessment Rules, which is available on the Moodle pages offered by the Engineering Faculty (in block titled “General Programme Information” on the left-hand side of the screen, just below the “Navigation” block).
- The pass requirement for this module is final mark FM≥50%.
- A semester mark (SM) is required by writing tutorial tests during weekly tutorials.
- There are three formal schedules assessment opportunities: A1 (mid-semester test), A2 (first examination) and A3 (second examination).
- The following weighting factors are used to determine the FM: wSM=0.1; wA1=0.4; wA2=0.5.
- The date and time of the semester test are determined by the Engineering Faculty (www.eng.sun.ac.za) and those of A2 and A3 are published on the University’s website. For any further information on examinations and promotions, see the University Calendar, Parts 1 and 11.

**General Information**

- Please note the following regarding the tutorials: (a) Attendance of all tutorials is compulsory. (b) No other appointments (academic or otherwise) may be observed during tutorial periods.
- The solutions of tutorial problems will be made available on Sun Learn after the tutorial.

**Study Hints**

- It is important that you understand the basic concepts very well, so that you are able to apply the theory.
- To determine whether you have mastered the course material, you must work a variety of problems from the
*Exercises*in the textbook. The lecturer will sometimes mention a number of these problems during the lectures. These problems will not be marked and remains your own responsibility. - You are welcome to consult your lecturer if there is anything which you do not understand, or if you get stuck with a problem. The lecturer will be available after each lecture, as well as in his/her office (preferably by appointment).
- Revise each section as soon as it has been discussed in class. It is very important that you do not fall behind, since it is very difficult to catch up.

**Rationale**

The module is presented as part of the Programs in Chemical and Mechanical Engineering and offers basic training in mathematics which is necesary for the successful completion of other modules within this Program. The module supports the Program outcome that graduates should be able to use their knowledge of Mathematics to solve engineering problems.

**Learning Outcomes**

After completing this course, students should be able to:

- Understand the basic concepts of infinite sequences and infinite series; determine limits of sequences; apply convergence tests to determine the convergence or divergence of infinite series (including power series); know the theorem of Taylor and the Taylor series expansions of the elementary functions.
- Understand the basic concepts about periodic functions and are able to calculate the Fourier expansions of simple periodic functions, including half-range expansions.
- Solve boundary value problems for the basic linear partial differential equations using separation of variables and Fourier expansions of non-homogeneous boundary values.
- Know the basic principles/applications of Fourier Transforms.