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# engineering mathematics iiI

VTU
Subject code
18MAT31
Semester
3rd Sem

## TRANSFORM CALCULUS, FOURIER SERIES AND NUMERICAL TECHNIQUES

### Module-1

Laplace Transform: Definition and Laplace transforms of elementary functions (statements only). Laplace transforms of Periodic functions (statement only) and unit-step function – problems.
Inverse Laplace Transform: Definition and problem s, Convolution theorem to find the inverse Laplace transforms (without Proof) and problems. Solution of linear differential equations using Laplace transforms.

### Module-2

Fourier Series: Periodic functions, Dirichlet’s condition. Fourier series of periodic functions period 2Π and arbitrary period. Half range Fourier series. Practical harmonic analysis.

### Module-3

Fourier Transforms: Infinite Fourier transforms, Fourier sine and cosine transforms. Inverse Fourier transforms. 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 and applications to solve difference equations.

### 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. Runge – Kutta method of fourth order, Milne’s and Adam-Bash forth 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.