The Complete 18ec31 | Engineering Mathematics Iii Notes

Description

VTU 18EC36 - Engineering Mathematics III: Course Summary

Course Code: 18EC31

Course Title: Engineering Mathematics III- Transform Calculus, Fourier Series, and Numerical Techniques

University: Visvesvaraya Technological University (VTU)

Summary:

Transform Calculus, Fourier Series, and Numerical Techniques are fundamental concepts in mathematics and engineering with diverse applications in various fields.

1. Transform Calculus:

Transform calculus is a branch of mathematics that deals with mathematical transformations used to simplify complex problems. It includes techniques like the Laplace transform and the Fourier transform, which help convert differential equations or time-domain signals into simpler algebraic equations or frequency-domain representations. Transform calculus plays a pivotal role in engineering, physics, and signal processing, aiding in the analysis and solution of complex problems involving dynamic systems and signals.

2. Fourier Series:

Fourier series is a powerful mathematical tool that allows the decomposition of complex periodic functions into a sum of simpler sinusoidal functions. This concept is essential in signal processing, image analysis, and various fields of science and engineering. It enables the representation and analysis of complex waveforms, making it indispensable in tasks such as data compression, audio processing, and the study of periodic phenomena.

3. Numerical Techniques:

Numerical techniques refer to a set of methods used to solve mathematical problems approximately using numerical calculations. These methods are particularly valuable when analytical solutions are difficult or impossible to obtain. Numerical techniques encompass a wide range of algorithms and approaches, including finite element analysis, numerical integration, and iterative solvers. They find applications in fields like engineering, physics, finance, and computer science, where they are used to simulate, analyze, and solve complex real-world problems.

In summary, Transform Calculus, Fourier Series, and Numerical Techniques are indispensable tools in mathematics and engineering, enabling the simplification of complex problems, the analysis of intricate data, and the numerical solution of a wide range of practical challenges. Their versatility and applicability make them essential for researchers, engineers, and scientists across various disciplines.

What will you 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.