Partial Differential Equations: Classical Theory with a Modern Touch (Cambridge IISc Series)

Contents
List of Illustrations
Preface
Acknowledgments
Notations
1 Introduction
	1.1 General Nature of PDE
	1.2 Two Examples
	1.3 Description of the Contents
2 Preliminaries
	2.1 Multivariable Calculus
		2.1.1 Introduction
		2.1.2 Partial, Directional and Frechét Derivatives
		2.1.3 Inverse Function Theorem
		2.1.4 Implicit Function Theorem
	2.2 Multiple Integrals and Divergence Theorem
		2.2.1 Multiple Integrals
		2.2.2 Green’s Theorem
		2.3 Systems of First-Order Ordinary Differential Equations: Existence and Uniqueness Results
	2.4 Fourier Transform, Convolution and Mollifiers
		2.4.1 Convolution
		2.4.2 Mollifiers
3 First-Order Partial Differential Equations: Method of Characteristics
	3.1 Introduction
	3.2 Linear Equations
	3.3 Quasilinear Equations
	3.4 General First-Order Equation in Two Variables
	3.5 First-Order Equation in Several Variables
		3.5.1 Linear First-Order Equation in Several Variables
		3.5.2 Quasilinear Equation in Several Variables
		3.5.3 General Non-linear Equation in Several Variables
	3.6 Hamilton–Jacobi Equation
	3.7 Notes
	3.8 Exercises
4 Hamilton–Jacobi Equation
	4.1 Hamilton–Jacobi Equation
	4.2 Hopf–Lax Formula
	4.3 Euler–Lagrange Equations
	4.4 Legendre Transformation
	4.5 Notes
	4.6 Exercises
5 Conservation Laws
	5.1 Introduction
	5.2 Generalized Solution and Rankine–Hugoniot (R–H) Condition
	5.3 Lax–Oleinik Formula
	5.4 Generalized Solution and Uniqueness
	5.5 Riemann Problem
	5.6 Notes
	5.7 Exercises
6 Classification of Second-Order Equations
	6.1 Introduction
	6.2 Cauchy Problem
		6.2.1 Non-characteristic Cauchy Problem
	6.3 Classification of Linear Equations
		6.3.1 Second-Order Equations in Two Variables
	6.4 Higher-Order Linear Equations
	6.5 Notes
	6.6 Exercises
7 Laplace and Poisson Equations
	7.1 Introduction
		7.1.1 Physical Interpretation
	7.2 Fundamental Solution, Mean Value Formula and Maximum Principles
		7.2.1 Mean Value Formula
		7.2.2 Maximum and Minimum Principles
		7.2.3 Uniqueness and Regularity of the Dirichlet Problem
		7.2.4 Green’s Function and Representation Formula
		7.2.5 MVP Implies Harmonicity
	7.3 Existence of Solution of Dirichlet Problem (Perron’s Method)
	7.4 Poisson Equation and Newtonian Potential
		7.4.1 Hölder Continuous Functions
	7.5 Hilbert Space Method: Weak Solutions
		7.5.1 Fourier Method
	7.6 Notes
	7.7 Exercises
8 Heat Equation
	8.1 Introduction
		8.1.1 Derivation of One-Dimensional Heat Equation
	8.2 Heat Transfer in an Unbounded Rod
		8.2.1 Solution in Higher Dimensions
		8.2.2 Uniqueness
		8.2.3 Inhomogeneous Equation
	8.3 Maximum and Minimum Principles
	8.4 Heat Equation on a Finite Interval: Fourier Method
		8.4.1 Prescribed Non-zero Boundary Conditions
		8.4.2 Free Exchange of Heat at the Ends
	8.5 Notes
	8.6 Exercises
9 One-Dimensional Wave Equation
	9.1 Introduction
	9.2 Cauchy Problem on the Line
		9.2.1 Inhomogeneous Equation: Duhamel’s Principle
		9.2.2 Characteristic Parallelogram
	9.3 Cauchy Problem in a Quadrant (Semi-infinite String)
	9.4 Wave Equation in a Finite Interval
	9.5 Notion of a Weak Solution
	9.6 General Second-Order Equations
		9.6.1 An Example
	9.7 Notes
	9.8 Exercises
10 Wave Equation in Higher Dimensions
	10.1 Introduction
	10.2 Three-Dimensional Wave Equation: Method of Spherical Means
		10.2.1 Characteristic Cone: Second Method
	10.3 Two-Dimensional Wave Equation: Method of Descent
		10.3.1 Telegraph Equation
	10.4 Wave Equation for General n
		10.4.1 Solution Formula via Euler–Poisson–Darboux Equation
		10.4.2 An Inversion Method
	10.5 Mixed or Initial Boundary Value Problem
	10.6 General Hyperbolic Equations and Systems
	10.7 Notes: Quasilinear Equations
	10.8 Exercises
11 Cauchy–Kovalevsky Theorem and Its Generalization
	11.1 Introduction
	11.2 Cauchy–Kovalevsky Theorem
		11.2.1 Real Analytic Functions
		11.2.2 Non-characteristic Cauchy Problem
	11.3 A Generalization: Application to First-Order Systems
	11.4 Holmgren’s Uniqueness Theorem
	11.5 Notes
12 A Peep into Weak Derivatives, Sobolev Spaces and Weak Formulation
	12.1 Weak Derivatives
	12.2 Existence of an L2 Weak Solution
		12.2.1 Constant Coefficient Operators
	12.3 Sobolev Spaces
	12.4 Notes
References
Index