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| Preface | |
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| Introduction | |
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| Preliminaries | |
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| Classification | |
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| Differential operators and the superposition principle | |
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| Differential equations as mathematical models | |
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| Associated conditions | |
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| Simple examples | |
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| Exercises | |
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| First-order equations | |
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| Introduction | |
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| Quasilinear equations | |
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| The method of characteristics | |
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| Examples of the characteristics method | |
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| The existence and uniqueness theorem | |
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| The Lagrange method | |
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| Conservation laws and shock waves | |
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| The eikonal equation | |
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| General nonlinear equations | |
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| Exercises | |
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| Second-order linear equations in two indenpendent variables | |
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| Introduction | |
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| Classification | |
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| Canonical form of hyperbolic equations | |
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| Canonical form of parabolic equations | |
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| Canonical form of elliptic equations | |
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| Exercises | |
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| The one-dimensional wave equation | |
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| Introduction | |
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| Canonical form and general solution | |
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| The Cauchy problem and d' Alembert's formula | |
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| Domain of dependence and region of influence | |
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| The Cauchy problem for the nonhomogeneous wave equation | |
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| Exercises | |
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| The method of separation of variables | |
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| Introduction | |
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| Heat equation: homogeneous boundary condition | |
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| Separation of variables for the wave equation | |
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| Separation of variables for nonhomogeneous equations | |
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| The energy method and uniqueness | |
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| Further applications of the heat equation | |
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| Exercises | |
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| Sturm-Liouville problems and eigenfunction expansions | |
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| Introduction | |
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| The Sturm-Liouville problem | |
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| Inner product spaces and orthonormal systems | |
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| The basic properties of Sturm-Liouville eigenfunctions and eigenvalues | |
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| Nonhomogeneous equations | |
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| Nonhomogeneous boundary conditions | |
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| Exercises | |
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| Elliptic equations | |
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| Introduction | |
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| Basic properties of elliptic problems | |
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| The maximum principle | |
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| Applications of the maximum principle | |
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| Green's identities | |
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| The maximum principle for the heat equation | |
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| Separation of variables for elliptic problems | |
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| Poisson's formula | |
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| Exercises | |
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| Green's functions and integral representations | |
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| Introduction | |
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| Green's function for Dirichlet problem in the plane | |
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| Neumann's function in the plane | |
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| The heat kernel | |
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| Exercises | |
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| Equations in high dimensions | |
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| Introduction | |
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| First-order equations | |
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| Classification of second-order equations | |
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| The wave equation in R[superscript 2] and R[superscript 3] | |
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| The eigenvalue problem for the Laplace equation | |
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| Separation of variables for the heat equation | |
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| Separation of variables for the wave equation | |
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| Separation of variables for the Laplace equation | |
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| Schrodinger equation for the hydrogen atom | |
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| Musical instruments | |
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| Green's functions in higher dimensions | |
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| Heat kernel in higher dimensions | |
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| Exercises | |
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| Variational methods | |
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| Calculus of variations | |
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| Function spaces and weak formulation | |
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| Exercises | |
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| Numerical methods | |
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| Introduction | |
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| Finite differences | |
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| The heat equation: explicit and implicit schemes, stability, consistency and convergence | |
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| Laplace equation | |
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| The wave equation | |
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| Numerical solutions of large linear algebraic systems | |
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| The finite elements method | |
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| Exercises | |
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| Solutions of odd-numbered problems | |
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| Trigonometric formulas | |
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| Integration formulas | |
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| Elementary ODEs | |
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| Differential operators in polar coordinates | |
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| Differential operators in spherical coordinates | |
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| References | |
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| Index | |