BENG501 

Module Name Engineering Mathematics 3 
 Module Code BENG501  
Level Five 
 Credits 15 

  

Module Description and General Aims 

This module builds on the fundamentals discussed in Mathematics modules 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 module 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 

On successful completion of this Module, students are expected to be able to: 

  1. Solving differential equations by assessing and applying various analytical and numerical methods. 

Bloom’s Level 4 

  1. Apply matrices in solving linear systems 

Bloom’s Level 3 

  1. Apply Laplace, Fourier and z transforms, and evaluate their applications in engineering contexts 

Bloom’s Level 4 

  1. Conduct conformal mappings and make use of them in engineering   

Bloom’s Level 3 

  

Student Assessment 

Assessment Type When assessed Weighting (% of total module marks) Learning Outcomes Assessed 
Assessment 1 Type: Multi-choice test / Group work / Short answer questions Students may complete a quiz with MCQ type answers and solve some simple equations to demonstrate a good understanding of the fundamental concepts. Due after Topic 3 10% 1 
Assessment 2 – mid-semester test Type: Multi-choice test / Group work / Short answer questions / Practical Students will provide solutions to problems on differential equations and matrices to show evidence of their understanding of the concepts involved or complete a practical. Due after Topic 6 30% 1, 2 
Assessment 3 Type: Multi-choice test / Group work / Short answer questions / Practical Students will provide solutions to problems related to Laplace, z- transforms and Fourier series  Due after Topic 9 10% 3 
Assessment 4 Type: Examination An examination where the student will complete a quiz with MCQ type answers and perform calculations and provide solutions to mathematical problems to be completed in 3 hours Final Week 45% 1 to 4 
Attendance / Tutorial Participation Example: Presentation, discussion, group work, exercises, self-assessment/reflection, case study analysis, application. Continuous 5% – 

  

Overall Requirements: Students must achieve a result of 40% or above in the exam itself to pass the exam and must pass the exam to be able to pass the module. An overall final module score of 50% or above must be achieved to pass the module 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 

  • Notes and Reference texts 
  • IDC Technologies 
  • Other material advised during the lectures 

  

Module Content 

   

Topic 1: Differential Equations -1  

  1. Definition  
  1. The Solutions of equations of some simple forms 
  1. Homogeneous first-order differential equations and their solutions 
  1. Linear first-order differential equations and their solutions 

Topic 2: Differential Equations -2 

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

  

Topic 3: Differential Equations -3  

  1. Power series Solutions 
  1. Frobenius Solutions 
  1. Euler Lagrange equation 
  1. Lagrangian equation 

  

Topic 4: Applications of Matrices 

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

Topic 5: Laplace Transforms 1 

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

  

Topic 6: Laplace Transforms 2 

  1. Unit step function 
  1. Dirac’s delta function 
  1. Shifting theorems 
  1. Convolution 
  1. Differential equations with polynomial coefficients 
     

Topic 7: z-transforms 

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

  

Topic 8: Fourier Series, Integrals and Transforms 1 

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

Topic 9: Fourier Series, Integrals and Transforms 2 

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

  

Topic 10: Fourier Series, Integrals and Transforms 3 

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

  

Topic 11: Conformal Mappings 

  1. Definition 
  1. Bilinear transformation 
  1. Special bilinear transformations 
  1. Construction of conformal mappings  

  

Topic 12: Exam Revision 

  1. Mathematical induction proofs 
  1. Module 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