The Complete 18mat31 - Engineering Mathematics 3 Notes

Description

VTU | 18MAT31 | TRANSFORM CALCULUS, FOURIER SERIES AND NUMERICAL TECHNIQUES

Transform Calculus:

Transform calculus is a branch of mathematics that deals with the study of mathematical transformations applied to functions or signals. It involves techniques like Laplace, Fourier, and Z-transforms, which help in simplifying complex differential or integral equations into simpler algebraic forms. Transform calculus is widely used in engineering, physics, and other fields to solve problems involving dynamic systems, differential equations, and signal processing.

Fourier Series:

Fourier series is a mathematical tool used to represent a periodic function as an infinite sum of sinusoidal functions (sines and cosines). This representation is valuable in analyzing and approximating various types of periodic phenomena, such as oscillations and waveforms. Fourier series has applications in signal processing, harmonic analysis, image compression, and solving partial differential equations.

Numerical Techniques:

Numerical techniques involve methods for solving mathematical problems using numerical approximations and computations on computers. These techniques are used when analytical solutions are difficult or impossible to obtain. Numerical integration, differentiation, root finding, and solving differential equations are common applications. Methods like the finite difference method, finite element method, and numerical optimization play a crucial role in solving real-world problems in engineering, physics, economics, and many other disciplines.

In summary, transform calculus provides powerful tools to simplify complex mathematical problems, Fourier series helps analyze periodic phenomena through sinusoidal approximations, and numerical techniques enable the practical computation and solution of a wide range of mathematical and scientific challenges. These concepts are fundamental in various fields, contributing to advancements in technology and our understanding of the natural world.

What you will learn?

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.