TY  - BOOK
AU  - Hattori,Harumi
TI  - Partial differential equations: methods, applications and theories 
SN  - 9789814407564
AV  - QA374 .H38 2013
PY  - 2013///
CY  - Singapore
PB  - World Scientific Publishing
KW  - Differential equations, Partial
N1  - 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
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