Partial Differential Equations: Classical Theory with a Modern Touch

Description

Suitable for both senior undergraduate and graduate students, this is a self-contained book dealing with the classical theory of the partial differential equations through a modern approach; requiring minimal previous knowledge. It represents the solutions to three important equations of mathematical physics – Laplace and Poisson equations, Heat or diffusion equation, and wave equations in one and more space dimensions. Keen readers will benefit from more advanced topics and many references cited at the end of each chapter. In addition, the book covers advanced topics such as Conservation Laws and Hamilton-Jacobi Equation. Numerous real-life applications are interspersed throughout the book to retain readers' interest. 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