Unit Description and General Aims

This unit builds on the fundamentals discussed in Mathematics units 1 and 2 by providing the student with a sound understanding of advanced engineering mathematical concepts involving solving differential equations, advanced matrix applications, Laplace, z, Fourier transforms, and conformal mapping. Students will be able to solve problems related to engineering applications by applying these techniques. The topics in the unit are so structured that the student is able to achieve proficiency in all three phases of problem-solving viz. modelling, solving the model by applying a suitable mathematical model, and interpreting the results.

Learning Outcomes

1. Solving differential equations by assessing and applying various analytical and numerical methods.
   Bloom's Level 4
2. Apply matrices in solving linear systems.
   Bloom's Level 3
3. Apply Laplace, Fourier and z transforms, and evaluate their applications in engineering contexts.
   Bloom's Level 4
4. Conduct conformal mappings and make use of them in engineering.
   Bloom's Level 3

Student Assessment

Overall Requirements: Students must achieve a result of 50% or above in the exam itself to pass the exam and must pass the exam to be able to pass the unit. An overall final unit score of 50% or above must be achieved to pass the unit once all assessment, including the exam, has been completed.

Prescribed and Recommended Readings

Textbook

• P. O'Neil, Advanced Engineering Mathematics, SI Edition, 8th Edition. Cengage, 2018. ISBN 9781337274524

Second Textbook

• J. Bird, Higher Engineering Mathematics, 7th Edition. Routledge, 2014. ISBN-13: 978-0415662826

Reference

• A. Kreyszig, Advanced Engineering Mathematics Student Solutions Manual, 10th Edition. John Wiley & Sons, 2012. ISBN-13: 978-1118007402

Journal, website

• http://www.elsevier.com/physical-sciences/mathematics/mathematics-journals
• Notes and Reference texts
• Open Textbook Library: http://open.umn.edu/opentextbooks/
• Knovel library: http://app.knovel.com
• IDC Technologies
• Other material advised during the lectures

Unit Content

Topic 1

Differential Equations -1

• Definition
• The Solutions of equations of some simple forms
• Homegeneous first-order differential equations and their solutions
• Linear first-order differential equations and their solutions

Topic 2

Differential Equations -2

• Second-order differential equations
• The Solutions of homogeneous second-order differential equations
• The Solutions of non-homogeneous second-order differential equations
• An introduction to partial differential equations

Topic 3

Differential Equations -3

• Power series Solutions
• Frobenius Solutions
• Euler Lagrange equation
• Lagrangian equation

Topic 4

Applications of Matrices

• Eigenvalues and eigenvectors
• Diagonalization
• Special Matrices
• Linear systems and their solutions using matrices

Topic 5

Laplace Transforms 1

• Laplace Transform and inverse Laplace Transforms
• Laplace Transforms of Elementary functions
• Transforms of derivatives and integrals
• Initial and final value theorems
• Laplace transform in a solution of initial value problems

Topic 6

Laplace Transforms 2

• Unit step function
• Dirac's delta function
• Shifting theorems
• Convolution
• Differential equations with polynomial coefficients

Topic 7

z-transforms

• Definition
• Some properties of z-transforms
• Inverse z-transforms
• Using z-transforms to solve difference equations

Topic 8

Fourier Series, Integrals and Transforms 1

• Fourier series
• Functions having points of discontinuity
• Even and Odd functions
• Convergence of Fourier series

Topic 9

Fourier Series, Integrals and Transforms 2

• Fourier cosine and sine series
• Integration and differentiation of Fourier series
• Phase angle form of Fourier series
• Complex Fourier series

Topic 10

Fourier Series, Integrals and Transforms 3

• Fourier integral
• Complex Fourier integral
• Fourier transform
• Discrete and Fast Fourier transforms
• Fourier Transform in signal analysis using engineering tools

Topic 11

Conformal Mappings

• Definition
• Bilinear transformation
• Special bilinear transformations
• Construction of conformal mappings

Topic 12

Exam Revision

• Mathematical induction proofs
• Unit revision

Software/Hardware Used

Software

• Software: Python Jupyter Notebook or Google Colab 

• Version: N/A

• Instructions:  N/A

• Additional resources or files: N/A

Hardware

• N/A

Unit Changes Based on Student Feedback