Partial differential equations : methods, applications and theories / Harumi Hattori.

Includes bibliographical references (pages 369-370) and index.

Table Of Contents: Preface 1 First and Second Order Linear Equations - Preparation 1.1 Terminologies 1.2 Linearity 1.2.1 Superposition Principle 1.2.2 Linear Independence 1.3 First Order Linear Equations 1.3.1 Initial Value Problems 1.3.2 General Solutions 1.4 Classification of Second Order Linear Equations 1.5 Well-posedness 2 Heat Equation 2.1 Derivation of the Heat Equation 2.1.1 One-dimensional Case 2.1.2 Divergence Theorem 2.1.3 Multi-dimensional Case 2.2 Initial Boundary Value Problems 2.3 Homogeneous Boundary Conditions 2.3.1 Temperature is Fixed at Zero at Both Ends 2.3.2 Brief Discussion of the Fourier Series 2.3.3 Both Ends are Insulated 2.3.4 Temperature of One End is Zero and the Other End is Insulated 2.4 Non-homogeneous Boundary Conditions 2.4.1 Steady State Solutions 2.4.2 Non-homogeneous Boundary Conditions 2.5 Robin Boundary Conditions 2.6 Infinite Domain Problems 2.6.1 Initial Value Problems 2.6.2 Initial Value Problems via Fourier Transform 2.6.3 Semi-infinite Domains 2.7 Maximum Principle, Energy Method, and Uniqueness of Solutions 2.7.1 Maximum Principle 2.7.2 Energy Method 3 Wave Equation 3.1 Derivation of Wave Equation 3.1.1 One-dimensional Case 3.1.2 Multi-dimensional Case 3.2 Initial Value Problems 3.2.1 Homogeneous Wave Equation 3.2.2 Non-homogeneous Wave Equation 3.3 Wave Reflection Problems 3.3.1 Homogeneous Boundary Conditions 3.3.2 Non-homogeneous Boundary Conditions 3.4 Initial Boundary Value Problems 3.5 Energy Method 4 Laplace Equation 4.1 Motivations 4.2 Boundary Value Problems - Separation of Variables 4.2.1 Laplace Equation on a Rectangular Domain 4.2.2 Laplace Equation on a Circular Disk 4.3 Fundamental Solution 4.3.1 Green's Identity 4.3.2 Derivation of Fundamental Solution 4.3.3 Green's Identity and Fundamental Solution 4.4 Green's Function 4.4.1 Definition 4.4.2 Green's Function for a Half Space 4.4.3 Green's Function for a Ball 4.4.4 Symmetry of Green's Function 4.5 Properties of Harmonic Functions 4.5.1 Mean Value Property 4.5.2 The Maximum Principle and Uniqueness 4.6 Well-posedness Issues 4.6.1 Laplace Equation 4.6.2 Wave Equation 5 First Order Equations Revisited 5.1 First Order Quasilinear Equations 5.2 An Application of Quasilinear Equations 5.2.1 Scalar Conservation Law 5.2.2 Rankine-Hugoniot Condition 5.2.3 Weak Solutions 5.2.4 Entropy Condition and Admissibility Criterion 5.2.5 Traffic Flow Problem 5.3 First Order Nonlinear Equations 5.4 An Application of Nonlinear Equations - Optimal Control Problem 5.5 Systems of First Order Equations 5.5.1 2 x 2 System 5.5.2 n x n System 6 Fourier Series and Eigenvalue Problems 6.1 Even, Odd, and Periodic Functions 6.1.1 Even and Odd Functions 6.1.2 Periodic Functions 6.2 Fourier Series 6.2.1 Fourier Series 6.2.2 Fourier Sine and Cosine Series 6.3 Fourier Convergence Theorems 6.3.1 Mean-square Convergence 6.3.2 Pointwise Convergence 6.3.3 Uniform Convergence 6.4 Derivatives of Fourier Series 6.5 Eigenvalue Problems 6.5.1 The Sturm-Liouville Problems 6.5.2 Proofs 7 Separation of Variables in Higher Dimensions 7.1 Rectangular Domains 7.2 Eigenvalue Problems 7.2.1 Multidimensional Case 7.2.2 Gram-Schmidt Orthogonalization Procedure 7.2.3 Rayleigh Quotient 7.3 Eigenfunction Expansions 7.3.1 Non-homogeneous Boundary Conditions 7.3.2 Homogeneous Boundary Conditions 7.3.3 Hybrid Method 8 More Separation of Variables 8.1 Circular Domains 8.1.1 Initial Boundary Value Problems 8.1.2 Bessel and Modified Bessel Functions 8.2 Cylindrical Domains 8.2.1 Initial Boundary Value Problems 8.2.2 Laplace Equation 8.3 Spherical Domains 8.3.1 Initial Boundary Value Problems 8.3.2 Legendre Equation 8.3.3 Laplace Equation 9 Fourier Transform 9.1 Delta Functions 9.1.1 Classical Introduction 9.1.2 Modern Introduction 9.2 Fourier Transform 9.2.1 Complex Form of the Fourier Series 9.2.2 Fourier Transform and Inverse 9.3 Properties of Fourier Transform 9.3.1 Fourier Transform of Derivatives 9.3.2 Convolution 9.3.3 Plancherel Formula 9.4 Applications of Fourier Transform 9.4.1 Heat Equation 9.4.2 Wave Equation 9.4.3 Laplace Equation in a Half Space 9.4.4 Black-Scholes-Merton Equation 10 Laplace Transform 10.1 Laplace Transform and the Inverse 10.1.1 Laplace Transform 10.1.2 Inverse Transform 10.2 Properties of the Laplace Transform 10.2.1 Laplace Transform of Derivatives 10.2.2 Convolution Theorem 10.2.3 Relation with the Fourier Transform 10.3 Applications to Differential Equations 10.3.1 Applications to ODE's 10.3.2 Applications to PDE's 11 Higher Dimensional Problems - Other Approaches 11.1 Spherical Means and Method of Descent 11.1.1 Method of Spherical Means 11.1.2 The Method of Descent 11.2 Duhamel's Principle 11.2.1 Heat Equation 11.2.2 Wave Equation 12 Green's Functions 12.1 Green's Functions for the Laplace Equation 12.1.1 Eigenfunction Expansion 12.1.2 Modified Green's Function 12.2 Green's Functions for the Heat Equation 12.2.1 Initial Boundary Value Problems 12.2.2 Initial Value Problems 12.3 Green's Functions for the Wave Equation 12.3.1 Initial Boundary Value Problems 12.3.2 Initial Value Problems Appendices A.1 Exchanging the Order of Integration and Differentiation A.2 Infinite Series A.3 Useful Formulas in ODE's A.3.1 First Order Linear Equations A.3.2 Bernoulli Equations A.3.3 Second Order Linear Constant Coefficient Equations A.3.4 Variation of Parameters Formula A.4 Linear Algebra A.4.1 Solutions to Systems of Linear Equations A.4.2 Eigenvalues, Eigenvectors, and Diagonalization Hints and Solutions to Selected Exercises Bibliography Index