The Complete 18MAT31 | TRANSFORM CALCULUS, FOURIER SERIES AND NUMERICAL TECHNIQUES - Maths Notes
2018 Scheme | CSE Department
- Created by AcquireHowTo
- 5 Modules
Course Learning Objectives:
• To have an insight into Fourier series, Fourier transforms, Laplace transforms, Difference equations
• To develop proficiency in variational calculus and solving ODE’s arising in engineering
applications, using numerical methods.
At the end of the course the student will be able to:
• CO1: Use Laplace transform and inverse Laplace transform in solving differential/ integral equation
arising in network analysis, control systems and other fields of engineering.
• CO2: Demonstrate Fourier series to study the behaviour of periodic functions and their applications in
system communications, digital signal processing and field theory.
• CO3: Make use of Fourier transform and Z-transform to illustrate discrete/continuous function arising
in wave and heat propagation, signals and systems.
• CO4: Solve first and second order ordinary differential equations arising in engineering problems
using single step and multistep numerical methods.
• CO5:Determine the extremals of functionals using calculus of variations and solve problems
arising in dynamics of rigid bodies and vibrational analysis.
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What You’ll Learn
Laplace Transforms Definition and Laplace transform of elementary functions. Laplace transforms of
Periodic functions and unit-step function – problems.
Inverse Laplace Transforms: Inverse Laplace transform - problems, Convolution theorem to find the inverse
Laplace transform (without proof) and problems, solution of linear differential equations using Laplace
Fourier Series: Periodic functions, Dirichlet’s condition. Fourier series of periodic functions period 2 and
arbitrary period. Half range Fourier series. Practical harmonic analysis, examples from engineering field.
Fourier Transforms: Infinite Fourier transforms, Fourier sine and cosine transforms. Inverse Fourier
transforms. Simple problems.
Difference Equations and Z-Transforms: Difference equations, basic definition, z-transform-definition,
Standard z-transforms, Damping and shifting rules, initial value and final value theorems (without proof) and
problems, Inverse z-transform. Simple problems.
Numerical Solutions of Ordinary Differential Equations (ODE’s): Numerical solution of ODE’s of first
order and first degree- Taylor’s series method, Modified Euler’s method. Range - Kutta method of fourth
order, Milne’s and Adam-Bashforth predictor and corrector method (No derivations of formulae), Problems.
Numerical Solution of Second Order ODE’s: Runge -Kutta method and Milne’s predictor and corrector
method.(No derivations of formulae).
Calculus of Variations: Variation of function and functional, variational problems, Euler’s equation,
Geodesics, hanging chain, problems.
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