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| Preface | |
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| Mathematical Models, Conservation and Constitutive Laws | |
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| Basic Notions | |
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| Evolution Conservation Law | |
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| Stationary Conservation Law | |
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| Conservation Law in One Dimension | |
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| Constitutive Laws | |
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| Exercises | |
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| Classification, Types of Equations, Boundary and Initial Conditions | |
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| Basic Types of Equations, Boundary and Initial Conditions | |
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| Classification of Linear Equations of the Second Order | |
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| Exercises | |
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| Linear Partial Differential Equations of the First Order | |
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| Convection and Transport Equation | |
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| Equations with Constant Coefficients | |
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| Equations with Non-Constant Coefficients | |
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| Exercises | |
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| Wave Equation in One Spatial Variable-Cauchy Problem in R | |
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| String Vibrations and Wave Equation in One Dimension | |
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| Cauchy Problem on the Real Line | |
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| Wave Equation with Sources | |
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| Exercises | |
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| Diffusion Equation in One Spatial Variable-Cauchy Problem in R | |
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| Diffusion and Heat Equations in One Dimension | |
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| Cauchy Problem on the Real Line | |
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| Diffusion Equation with Sources | |
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| Exercises | |
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| Laplace and Poisson Equations in Two Dimensions | |
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| Steady States and Laplace and Poisson Equations | |
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| Invariance of the Laplace Operator, Its Transformation into Polar Coordinates | |
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| Solution of Laplace and Poisson Equations in R[superscript 2] | |
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| Exercises | |
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| Solutions of Initial Boundary Value Problems for Evolution Equations | |
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| Initial Boundary Value Problems on Half-Line | |
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| Initial Boundary Value Problem on Finite Interval, Fourier Method | |
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| Fourier Method for Nonhomogeneous Problems | |
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| Transformation to Simpler Problems | |
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| Exercises | |
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| Solutions of Boundary Value Problems for Stationary Equations | |
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| Laplace Equation on Rectangle | |
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| Laplace Equation on Disc | |
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| Poisson Formula | |
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| Exercises | |
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| Methods of Integral Transforms | |
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| Laplace Transform | |
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| Fourier Transform | |
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| Exercises | |
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| General Principles | |
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| Principle of Causality (Wave Equation) | |
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| Energy Conservation Law (Wave Equation) | |
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| Ill-Posed Problem (Diffusion Equation for Negative t) | |
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| Maximum Principle (Heat Equation) | |
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| Energy Method (Diffusion Equation) | |
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| Maximum Principle (Laplace Equation) | |
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| Consequences of Poisson Formula (Laplace Equation) | |
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| Comparison of Wave, Diffusion and Laplace Equations | |
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| Exercises | |
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| Laplace and Poisson equations in Higher Dimensions | |
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| Invariance of the Laplace Operator | |
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| Green's First Identity | |
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| Properties of Harmonic Functions | |
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| Green's Second Identity and Representation Formula | |
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| Boundary Value Problems and Green's Function | |
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| Dirichlet Problem on Half-Space and on Ball | |
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| Exercises | |
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| Diffusion Equation in Higher Dimensions | |
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| Heat Equation in Three Dimensions | |
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| Cauchy Problem in R[superscript 3] | |
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| Diffusion on Bounded Domains, Fourier Method | |
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| Exercises | |
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| Wave Equation in Higher Dimensions | |
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| Membrane Vibrations and Wave Equation in Two Dimensions | |
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| Cauchy Problem in R[superscript 3]-Kirchhoff's Formula | |
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| Cauchy problem in R[superscript 2] | |
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| Wave with sources in R[superscript 3] | |
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| Characteristics, Singularities, Energy and Principle of Causality | |
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| Wave on Bounded Domains, Fourier Method | |
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| Exercises | |
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| Appendix | |
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| Sturm-Liouville problem | |
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| Bessel Functions | |
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| Some Typical Problems Considered in This Book | |
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| Notation | |
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| Bibliography | |
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| Index | |