VTU Notes

# The Complete 18MAT31 | TRANSFORM CALCULUS, FOURIER SERIES AND NUMERICAL TECHNIQUES - Maths Notes

• 4.9
• 2018 Scheme | CSE Department ### Description

#### Course Learning Objectives:

• To have an insight into Fourier series, Fourier transforms, Laplace transforms, Difference equations

and Z-transforms.

• To develop proficiency in variational calculus and solving ODE’s arising in engineering

applications, using numerical methods.

### Course Outcomes:

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.

### Module-1

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

transform.

### Module-2

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.

### Module-3

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.

### Module-4

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.

### Module-5

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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3rd
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3rd Sem CSE Department VTU Notes