Laplace and Fourier Transforms

Course Code (Credit):

CUTM1002(2-0-1)

Course Objectives:

  • To describe the ideas of Fourier and Laplace Transforms and indicate their applications in the fields such as application of PDE, Digital Signal Processing, Image Processing, Theory of wave equations, Differential Equations and many others.
  • To use Fourier series for solving boundary value problems appearing in scientific & engineering problems.

Learning Outcomes:

  • Upon successful completion of this course, students will be able to:•Solve differential equations with initial conditions using Laplace transform.• Evaluate the Fourier transform of a continuous function and be familiar with its basic properties.

Course Syllabus:

Module I: (T-3 h + Pj-2 h)

Laplace Transforms, Properties of Laplace transforms, Unit step function.

Project 1:Make a short draft of properties of Laplace transform from memory. Then compare your notes with the text and write a report of 2-3 pages on these operations and their significance in applications.

Module II: (T-2 h + Pj-2 h)

Second shifting theorem, Laplace transforms of Derivatives and Integrals.

Project 2:Find the Laplace transform of the following functions(sinat,cos at,sinh at,cosh at,e^at,t^n,etc).

Module III: (T-3 h + Pj-2 h)

Derivatives and Integrals of Transforms, Inverse Laplace transform.

Project 3:Application of Unit step function (RC- Circuit to a single square wave).

Module IV: (T-2 h + Pj-2 h)

Solution of differential equations by using Laplace transform.

Project 4:Find the solution of differential equation by using Laplace Transform.

Module V: (T-4 h + Pj-2 h)

Periodic function, Fourier series, Fourier series expansion of an arbitrary period, Half range expansions.

Project 5:Find the Fourier series expansion of a 2-pi periodic function.

Module VI: (T-3 h + Pj-2 h)

Complex form of Fourier series, Fourier Integrals, Different forms of Fourier Integral.

Project 6Find the Fourier sine and cosine integral of the following functions.

Module VII: (T-3 h )

Fourier Transforms, Fourier sine and cosine Transforms.

Text Books:

  1. E. Kreyszig , Advanced Engineering Mathematics, Johnwilley & Sons Inc-8th Edition.Chapters:5(5.1 to 5.4(without Dirac's delta function ) ),10(10.1,10.4 and 10.7-10.9(definitions only , no proofs))
  2. Higher Engineering Mathematics by B.V.Ramana, Tata McGraw-Hill Education India, Inc-8th Edition

Reference Books:

  1. Advanced Engineering Mathematics by P.V.O’ Neil Publisher: Thomson
  2. Mathematical Methods by Potter & Goldberg ; Publisher : PHI

Session Plan:

Session Topic Reference Link (if any)
Session 1 Introduction to Laplace Transform lec-1 pdf video-1
Session 2 Properties of Laplace Transform (Shifting Property) Shifting Property (Slideshare) video-2
Session 3 Unit Step Function lec-3 pdf video-3
Session 4 & 5 Project-1: Report on Laplace Properties Make a short draft and compare with text (2–3 pages)
Session 6 Second Shifting Properties lec-4 pdf video-4
Session 7 Transforms of Derivatives and Integrals lec-5 pdf video-5
Session 8 & 9 Project-2: Laplace Transform of Functions Find Laplace transform of given functions
Session 10 Derivatives and Integrals of Transforms lec-6 pdf video-6
Session 11 Inverse Laplace Transform lec-7 pdf video-11
Session 12 Properties of Inverse Laplace Transform let-8 pdf video-12
Session 13 & 14 Project-3: RC-Circuit (Unit Step Function Application) Application of Unit step function to square wave
Session 15 Solution of Differential Equation by Laplace lec-9 pdf video-15
Session 16 Solution of Initial Value Problem lec-10 pdf video-16
Session 17 & 18 Project-4: Solve DEs using Laplace Solve DEs using Laplace Transform
Session 19 Periodic Function Fourier Series (Slideshare) video-19
Session 20 Fourier Series Expansion Fourier series expansion of periodic function video-20
Session 21 Fourier Expansion (Any Periods 2L) Fourier series expansion of period 2L video-21
Session 22 Half Range Expansion Half Range Expansion of Fourier series video-22
Session 23 & 24 Project-5: Fourier Series of 2π-periodic Function Find Fourier Series Expansion
Session 25 Complex Form of Fourier Series Complex form of Fourier series and Fourier integral video-25
Session 26 Fourier Integral Transform Fourier Integral video-26
Session 27 Forms of Fourier Integral Theorems Fourier Integral video-27
Session 28 & 29 Project-6: Fourier Sine & Cosine Integral Find Fourier sine and cosine integral
Session 30 Fourier Sine and Cosine Transforms Fourier Sine and Transform video-24
Session 31 Fourier and Infinite Fourier Transforms lec-26 video-26
Session 32 First Fourier Transforms Fourier Transform video-27