Advanced Mathematics for Engineering

Unit 59: Advanced Mathematics for Engineering

 

(QCF Level 5; Credit Value: 15)

 

 

Assignment 2 – Ordinary Differential Equations and Fourier Series

 

 

 

Grading Criteria
Pass	Achieved	Merit	Achieved	Distinction	Achieved
LO 1.1		M1		D1
LO 1.2		M2		D2
LO 1.3		M3a	N/A	D3	N/A
LO 1.4		M3b
LO 3.1		M3c	N/A
LO 3.2
LO 3.3
LO 3.4

 

 

Assignment Author	IV signature (brief)	IV signature (assessment)
Dr Ismail Farhan

 

 

ASSESSMENT FEEDBACK                                                                                                         

Note: the Notional Score is for formative feedback purposes only. 5 is allocated as sufficient for meeting the criteria, less than 5 is an indication of extra work required, more than 5 is an indication extra work included.

 

PASS grade must be achieved:

Outcomes	Learner has demonstrated the ability to:	Source of evidence	Tutors comments	Notional Score (10)
Outcome 1   Be able to analyse and model engineering situations and solve engineering problems using series and numerical methods for the solution of ordinary differential equations	LO1.1: determine power series values for common scientific and engineering functions	Task 1
	LO1.2: Solve ordinary differential equations using power series methods	Task 2
	LO1.3: Solve ordinary differential equations using numerical methods	Task 3
	LO1.4: Model engineering situations, formulate differential equations and determine solutions to these equations using power series and numerical methods	Task 4

 

 

Learning Outcome 3   Be able to analyse and model engineering situations and solve engineering problems using Fourier series	LO3.1: determine Fourier coefficients and represent periodic functions as infinite series	Task 5
	LO3.2: apply the Fourier series approach to the exponential form and model phasor behaviour	Task 6
	LO3.3: apply Fourier series to the analysis of engineering problems	Task 7
	LO3.4  use numerical integration methods to determine Fourier coefficients from tabulated data and solve engineering problems using numerical harmonic analysis	Task 7

 MERIT grade descriptors that may be achieved for this assignment:

Merit Grade Descriptors	Indicative Characteristics	Source of evidence	Tutors comments	Notional Score (10)
M1               Identify and apply strategies to find appropriate solutions	Effective judgements have been made   Complex problems with more than one variable have been explored	Tasks 1, 2 and 3
M2               Select/design and apply appropriate methods/techniques	Relevant theories and techniques have been applied   Complex information / data has been synthesised and processed	Tasks 4, 5 and 6   Task 7
M3b   Present and communicate appropriate findings	Throughout the report, the solutions are coherently presented using technical language appropriately and in a professional manner	All Tasks

 Distinction grade descriptors that may be achieved for this assignment:

Merit Grade Descriptors	Indicative Characteristics	Source of evidence	Tutors comments	Notional Score (10)
D1   Use critical reflection to evaluate own work and justify valid conclusions	Conclusions have been arrived at through synthesis of ideas and have been justified   The validity of results has been evaluated using defined criteria	Task 4         Task 7
D2   Take responsibility for managing and organising activities	Autonomy / independence has been demonstrated   Activities have been managed	All tasks     Tasks 5, 6 and 7
D3   Demonstrate convergent/ lateral/ creative thinking	N/A

General Information

 

All questions in the tasks must be completed correctly with sufficient detail to gain the pass criteria.

 

All submissions to be electronic in MS Word format with a minimum of 20 typed words. Also add footer to the document with your name. All answers must be clearly identified as to which task and question they refer to. All work must be submitted through Learnzone.

 

Task 1 – Learning Outcome 1.1

Determine power series values for common scientific and engineering functions

 

1. Obtain the Maclaurin series for the following functions. State the values of the x which the series converge.

Task 2 – Learning Outcome 1.2

Solve ordinary differential equations using power series methods.

 

2. Solve the following ordinary differential equation using Maclaurin series.

 

 

Task 3  – Learning Outcome 1.3

Solve ordinary differential equations using numerical methods.

 

3. Use the Euler and the improved Euler methods and comment on the two results. Use the step size shown to advance four steps from the given initial condition with the given differential equation:

 

 

4. Use the Runge-Kutta method with the step size shown to advance four steps from the given initial condition with the given differential equation:

 

 

Write a conclusion on the accuracy and the validity of the above methods used (for Distinction Only)

 

Task 4 – Learning Outcome 1.4

Model engineering situation, formulate differential equations and determine solutions to these equations using power series and numerical methods.

 

5. During the manufacture of steel component it is often necessary to quench them in a large bath of liquid in order to cool them down. This reduces the temperature of the components to the temperature of the liquid. If T is the temperature of the component in excess of the liquid temperature, the rate of change of the component temperature proportion to the temperature of the component. Take the proportional constant as K.  K depends upon the volume and surface area of the component, its specific heat capacity, and the heat transfer coefficient between the component and the liquid.

 

1. Formulate a differential equation for the above engineering situation

 

1. Determine a solution for the formulated formula using numerical method and power series, the initial condition that at t=0 the temperature excess is 250⁰

 

 

Task 5 – Learning Outcome 3.1

Determine Fourier coefficients and represent periodic functions as infinite series.

 

6. Find the Fourier coefficients of the following equation and write the function as infinite series.

 

 

 

 

Sketch a graph of the function within and outside of the given range, assuming the period is 2π.

Task 6 – Learning Outcome 3.2

Apply Fourier series approach to the exponential form and model of phasor behaviour.

7. Determine the complex Fourier series for the function defined by:

 

 

The function is a periodic outside the range of period 7

Task 7 – Learning Outcome 3.3 and 3.4

Apply Fourier series to the analysis of engineering problem

 

Use numerical integration methods to determine Fourier coefficients from tabulated data and solve   engineering problems using numerical harmonic analysis

 

8. In engineering wave analysis, the values of voltage over a complete cycle of a waveform are shown in the table below:

 

Angle (q) (Degree)	Voltage V (Volts)
0	0
30	-1.4
60	6.0
90	12.5
120	16.0
150	16.5
180	15.0
210	12.5
240	6.50
270	-4.00
300	-7.00
330	-7.50

 

Use a tabular method to determine the Fourier series for the waveform.

 

End of assessment brief