Partial Differential Equations : An Introductory Treatment With Applications

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This book presents comprehensive coverage of the fundamental concepts and applications of partial differential equations (PDEs). It is designed for the undergraduate [BA/BSc(Hons.)] and postgraduate (MA/MSc) students of mathematics, and conforms to the course curriculum prescribed by UGC.

This book is recommended in Manipur University, Manipur, Gauhati University, Assam, D.M. College of Science, Manipur.

The text is broadly organized into two parts. The first part (Lessons 1 to 15) mostly covers the first-order equations in two variables. In these lessons, the mathematical importance of PDEs of first order in physics and applied sciences has also been highlighted. The other part (Lessons 16 to 50) deals with the various properties of second-order and first-order PDEs.

The book emphasizes the applications of PDEs and covers various important topics such as the HamiltonJacobi equation, Conservation laws, Similarity solution, Asymptotics and Power series solution and many more. The graded problems, the techniques for solving them, and a large number of exercises with hints and answers help students gain the necessary skill and confidence in handling the subject.

Key Features :

1. Presents self-contained topics in a cohesive style.

2. Includes about 300 worked-out examples to enable students to understand the theory and inherent aspects of PDEs.

3. Provides around 450 unsolved problems with hints and answers to help students assess their comprehension of the subject.

Preface
0 Preliminaries
1 Formulation of Partial Differential Equation
2 Lagrange Equations
3 Cauchy Initial Value Problem for Linear First Order Equations
4 Total Differential Equations (Pfaffian Equations)
5 Generating Integral Surfaces of Lagrange Equation
6 Orthogonal Surface to a Given System of Surfaces
7 Compatible System of First Order Equations
8 Classification of the Solution of First Order Equation
9 Nonlinear First Order Equations
10 Integrals of Certain Nonlinear Equations (Standard Forms)
11 Cauchy Initial Value Problem for Nonlinear First Order Equations
12 System of First Order Equations and Legendre Transformation
13 Linear Equations with Constant Coefficients
14 Discontinuous Solutions of Conservation Laws
15 Transport Problems and Hamiltonian Dynamics
16 Second Order Equations
17 Special Types of Second Order Equations
18 Classification of Second Order Linear Equations
19 Canonical Forms of Second Order Linear Equations
20 Cauchy Initial Value Problem for Second Order Equations
21 Linear Hyperbolic Equations (Riemanns Method)
22 Nonlinear Second Order Equations (Monges Method)
23 Separation of Variables Method
24 Method of Integral Transforms
25 Method of Greens Function
26 Potential Equations
27 Equipotential Surfaces
28 Elliptic Boundary Value Problems
29 Potential Problems in Polar Coordinates
30 Potential Problems by Hankel Transform
31 Potential Problems by Green Function
32 Potential Problems in Polar Coordinates by Green Function
33 Potential Problems in Two-dimensional Case by Green Function
34 Potential Problems in Polar Coordinates (Continued)
35 Diffusion Equations
36 Diffusion Problems by Separation of Variables Method
37 Diffusion Problems by Integral Transforms
38 Diffusion Problems by Duhamel Principle and Burgers Equation
39 Diffusion Problems by Green Function
40 Diffusion Problems for Finite Boundary by Green Function
41 Diffusion Problems in One Dimension by Green Function
42 Diffusion Problems for Semi-infinite Domain by Green Function
43 Wave Equations
44 Wave Problems by Separation of Variables Method
45 Wave Problems by Integral Transforms
46 Wave Problems by Duhamel Principle
47 Wave Problems by Green Function
48 Wave Problems in Two Dimension by Green Function
49 Nonlinear Effects in Wave Propagation
50 Perturbation Methods
Typical IBVPs in Rectangular Regions
Some Well-known Mathematicians
References Index