## Description

This book offers a clear exposition of advanced mathematical methods. It is designed as a text book for this subject taught across PG courses related to Civil Engineering offered at Anna University, Chennai and affiliated colleges. Basic concepts are explained briefly. Each topic is illustrated through a number of solved examples. Problems from the recent question papers of Anna University are presented either as solved examples or exercises with answers. Detailed hints and solutions of the problems are given for the benefit of the students.

## Table of Content

**Chapter 1 LAPLACE TRANSFORM TECHNIQUES FOR PARTIAL**

DIFFERENTIAL EQUATIONS

1.1 Introduction

1.2 Integral Transform

1.3 Laplace Transform

1.3.1 Laplace Transform for Standard Functions

1.3.2 Properties

1.3.3 Initial Value Theorem (IVT) and Final Value Theorem (FVT)

1.3.4 Laplace Transform of Derivatives

1.3.5 Laplace Transform of Unit Step Function (Heaviside’s Unit Function)

1.3.6 Laplace Transform of Dirac Delta Function (Unit Impulse Function)

1.3.7 The Error Function

1.3.8 Laplace Transform of Bessel’s Function

1.4 Inverse Laplace Transform

1.4.1 Convolution Theorem

1.5 Application of Laplace Transform to IBVP

1.5.1 One Dimensional Wave Equation (ODWE)

1.5.2 Application of Laplace Transform to ODWE

1.6 Laplace Transform Method to Solve ODHE (One Dimension Heat Equation)

**Chapter 2 FOURIER TRANSFORM TECHNIQUES FOR PARTIAL**

DIFFERENTIAL EQUATIONS

2.1 Fourier Transform

2.1.1 Properties of Fourier Transform

2.2 Infinite Fourier Sine and Cosine Transform

2.2.1 Fourier Sine Transform

2.2.2 Fourier Cosine Transform

2.3 Fourier Transform of Partial Derivatives

2.4 Fourier Transform Application to Laplace Equation and Poisson Equation

**Chapter 3 CALCULUS OF VARIATIONS**

3.1 Introduction

3.2 Theorem−Derivation of Euler’s Equation

3.3 Solutions of Euler’s Equation

3.4 Type I−Functional Involving x, y, y

3.5 Type II−Euler’s Equation for Several Dependent Variables

3.6 Type III−Functions Involving Higher Derivatives

3.7 Geodesics

3.8 Variational Problems with Moving Boundaries

3.9 Rayleigh−Ritz Method (Approximate Solution of BVP)

3.10 Kantrovich Method

**Chapter 4 CONFORMAL MAPPING AND APPLICATIONS**

4.1 Conformal Mapping

4.1.1 Isogonal Transformation−Definition

4.1.2 Critical Point

4.2 Bilinear Transformation

4.3 The Schwarz−Cristoffel Transformation

4.4 Problems on Conformal Mapping

4.5 Problems on Schwarz−Christoffel Transformation

4.6 Physical Application−Fluid Flow and Heat Flow

**Chapter 5 TENSOR ANALYSIS**

5.1 Introduction

5.2 Summation Convention

5.3 N−Dimensional Space

5.3.1 Transformation of Co-ordinates

5.3.2 Scalars or Invariants

5.4 Contravariant Vectors

5.5 Covariant Vectors

5.5.1 Kronecker Delta

5.5.2 Contraction of a Tensor

5.5.3 Outer Product of Two Tensors

5.5.4 Inner Product of Two Tensors

5.5.5 Quotient Law

5.5.6 Christoffel Symbols

5.5.7 Covariant Differentiation of a Covariant Vector

Appendix − University Question Papers

## About The Authors

**Dr. M. B. K. Moorthy** was Professor and Head at Department of Mathematics, Institute of Road and Transport Technology, Erode which is an automobile research oriented institute. He has obtained his M.Sc., M.Phil., and Ph.D. in Applied Mathematics from Anna University, Chennai, and also holds an M.B.A. in HRM and M.Tech. in Computer Science and Engineering. He has over 36 years of experience in teaching at engineering colleges for both UG and PG students. He was a visiting professor at ISTE, New Delhi. He has published many research papers in national and international journals. He is a recognized guide for M.S/Ph.D. programs of Anna University. Six students have successfully completed their Ph.D. programs under his guidance.

**A Manikandan **is Assistant Professor at Department of Mathematics, SRM University, Chennai. He is currently pursuing his Ph.D. in mathematics. He is having 8 years of teaching experience.

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