Transforms and Partial Differential Equations 3e

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

Exercises

Additional Solved Problems

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

Exercises

Additional Solved Problems

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

3.12 Two-dimensional Heat Equation in Polar Form

3.13 Solution of Polar Form of Laplace Equation

3.14 Steady-state Temperature Distribution in a Circular Annulus

Exercises

Chapter 4 Fourier Transforms

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

Exercises

Additional Solved Problems

Chapter 5 –ZZ-Transforms and Difference Equations

5.1 Introduction

5.2 Properties of–Z -Transform

5.3 Inverse –Z-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 –Z-Transform Using Residue Technique

5.8 Initial Value Theorem

5.9 Final Value Theorem

5.10 Convolution

5.11 Convolution Theorem for–Z -Transform

5.12 Time Dependent Sequence

Exercises

Solved University Question Papers

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