 | April 24, 2015

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Unit 59: Advanced Mathematics for Engineering

Assignment 2 – Ordinary Differential Equations and Fourier Series

 Grading Criteria Pass Achieved Merit Achieved Distinction Achieved L01.1 2 M1 2,3 D1 2 L01.2 2 M2 1,3 D2 1 L01.3 2 M3a 1, D3 3 L01.4 2 M3b 2 L02.1 1 M3c 3 L02.2 1 L02.3 1 L03.1 2 L03.2 2 L03.3 2 L03.4 2 L04.1 3 L04.2 3 L04.3 3

 Assignment Author IV signature JAHAYES ThanujaGoonetilleke

PASS grade must be achieved:

 Outcomes Learner has demonstrated the ability to: Source of evidence 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 Contextualisation Source of evidence M1                    Identify and apply strategies to find appropriate solutions Complex problems with more than one variable have been explored The results of both methods have been compared and valid conclusions on the accuracy given. Reasons for the difference have been identified. Task 3 Q3 M3b   Present and communicate appropriate findings Appropriate structure and approach has been used 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 Contextualisation Source of evidence D1   Use critical reflection to evaluate own work and justify valid conclusions The validity of results has been evaluated using defined criteria The complex Fourier solution has been developed to show the equivalent Fourier series expressed in sine and cosine terms Task 3 Q7

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.

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

Task 3  – Learning Outcome 1.3

Solve ordinary differential equations using numerical methods.

1. 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:

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

Task 4 – Learning Outcome 1.4

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

1. During the manufacture of steel components it is often necessary to quench them in a large bath of liquid. This reduces the temperature of the components to the temperature of the liquid .The rate of change of the component temperature is proportional to the difference in temperature between the component and the liquid.

1. Formulate a differential equation for the above engineering situation

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

Task 5 – Learning Outcome 3.1

Determine Fourier coefficients and represent periodic functions as infinite series.

1. Determine the Fourier series for the periodic function defined by:

Sketch a graph of the function within and outside of the given range for two periods.

Task 6 – Learning Outcome 3.2

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

1. 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

1. 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.

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