## Description

This book is designed for a one semester course on Transforms and Partial Differential Equations taught across all branches of engineering. This book has simple and clear presentation with wide variety of solved examples which enables the students to understand the course in depth. The author has taken care to maintain an optimum depth in covering all the topics, which fulfills the requirements of both students and faculty.

## Table of Content

**Chapter 1 Partial Differential Equations**

1.1 Introduction

1.2 Formation of PDE

1.2.1 Elimination of Arbitrary Constants

1.2.2 Elimination of Arbitrary Functions

1.3 Formation of First Order Partial Differential Equation

1.4 Solution of First Order PDE

1.5 Lagrange’s Equation

1.5.1 Method of Grouping

1.5.2 Method of Multiplier

1.6 First Order Partial Differential Equation

1.6.1 Type-I: f (p, q) = 0

1.6.2 Type-II: Clairut’s Type z = px + qy + f (p, q)

1.6.3 Type-III: First Order PDE ( z is missing)

1.6.4 Type-IV: F (z, p, q) = 0 (x and y are missing)

1.6.5 Type-V: First Order PDE

1.6.6 Type-VI: First Order PDE

1.7 Partial Differential Equations of Higher Order

1.7.1 Homogeneous Linear Partial Differential Equation

1.7.2 Non-homogeneous Partial Differential Equation

**Chapter 2 Fourier Series**

2.1 Introduction

2.2 Periodic Function

2.3 Fourier Series

2.4 Euler’s Formula (General Form)

2.5 Derivation of Euler’s Formula

2.6 Point of Discontinuity

2.7 Change of Interval

2.8 Even and Odd Function in Fourier Series

2.8.1 Even Function

2.8.2 Odd Function

2.8.3 Even and Odd Function in Change of Interval

2.9 Half-range Series

2.9.1 Half-range Cosine Series

2.9.2 Half-range Sine Series

2.10 Mean Value of a Function

2.11 Root Mean Square Value of a Function (RMS)

2.12 Parseval’s Identity for Fourier Series

2.13 Parseval’s Identity for Fourier Cosine Series

2.14 Parseval’s Identity for Fourier Sine Series

2.15 Complex Form of Fourier Series

2.16 Harmonic Analysis

2.17 Symmetry in Curves

**Chapter 3 Applications of Partial Differential Equations**

3.1 Classification

3.2 Method of Separation of Variable

3.3 Vibrations of a String

3.3.1 One-dimensional Wave Equation

3.3.2 Possible Solutions of Wave Equation

3.4 One-dimensional Heat Equation

3.5 Possible Solutions of One-dimensional Heat Equation

3.6 Steady State Solution

3.7 Fourier Law of Heat Conduction

3.8 Thermally Insulated Ends

3.9 Temperature Gradient

3.10 Two-dimensional Heat Equation

3.11 Possible Solutions of Two-dimensional Heat Equation

**Chapter 4 Fourier Transform**

4.1 Introduction

4.2 Fourier Integral Theorem

4.3 Fourier Sine and Cosine Integral

4.4 Fourier Transform (FT) and its Inverse (Complex Form)

4.5 Fourier Cosine Transform (FCT) and its Inverse

4.6 Fourier Sine Transform and its Inverse

4.7 Fourier Cosine Transform

4.8 Properties of Fourier Transform

4.9 Finite Fourier Transform

**Chapter 5 –Transforms and Difference Equations **

5.1 Introduction

5.2 Properties of -Transform

5.3 Inverse -Transform

5.3.1 Type-I: Partial Fraction Method

5.4 Unit-impulse Function

5.5 Power Series Method or Long Division Method

5.6 Difference Equation

5.7 Inverse -Transform Using Residue Technique

5.8 Initial Value Theorem

5.9 Final Value Theorem

5.10 Convolution

5.11 Convolution Theorem for -Transform

5.12 Time Dependent Sequence

• Solved University Question Papers

## About The Author

**Dr. R. Muthucumaraswamy** is Dean (Research) and Professor and Head, Department of Applied Mathematics at Sri Venkateswara College of Engineering, Chennai. He has 30 years of experience in teaching this course. He received his B.Sc. in Mathematics, from Gurunanank College, University of Madras in the year 1985, M.Sc. in Applied Mathematics from Madras Institute of Technology, Anna University in 1987, M.Phil. in Mathematics from Pachaiyappas’ College, University of Madras in 1991 and Ph.D. in Mathematics from Anna University in 2001.

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