Partial Differential Equations of Mathematical Physics

Translation Editor's Preface

Author's Prefaces to the First and Third Editions

Lecture 1. Derivation of the Fundamental Equations

§ 1. Ostrogradski's Formula

§ 2. Equation for Vibrations of a String

§ 3. Equation for Vibrations of a Membrane

§ 4. Equation of Continuity for Motion of a Fluid. Laplace's Equation

§ 5. Equation of Heat Conduction

§ 6. Sound Waves

Lecture 2. The Formulation of Problems of Mathematical Physics. Hadamard's Example

§ 1. Initial Conditions and Boundary Conditions

§ 2. The Dependence of the Solution on the Boundary Conditions. Hadamard's Example

Lecture 3. The Classification of Linear Equations of the Second Order

§ 1. Linear Equations and Quadratic Forms. Canonical Form of an Equation

§ 2. Canonical Form of Equations in Two Independent Variables

§ 3. Second Canonical Form of Hyperbolic Equations in Two Independent Variables

§ 4. Characteristics

Lecture 4. The Equation for a Vibrating String and its Solution by d'Alembert's Method

§ 1. D'Alembert's Formula. Infinite String

§ 2. String with Two Fixed Ends

§ 3. Solution of the Problem for a Non-Homogeneous Equation and for More General Boundary Conditions

Lecture 5. Riemann's Method

§ 1. The Boundary-Value Problem of the First Kind for Hyperbolic Equations

§ 2. Adjoint Differential Operators

§ 3. Riemann's Method

§ 4. Riemann's Function for the Adjoint Equation

§ 5. Some Qualitative Consequences of Riemann's Formula

Lecture 6. Multiple Integrals: Lebesgue Integration

§ 1. Closed and Open Sets of Points

§ 2. Integrals of Continuous Functions on Open Sets

§ 3. Integrals of Continuous Functions on Bounded Closed Sets

§ 4. Summable Functions

§ 5. The Indefinite Integral of a Function of One Variable. Examples

§ 6. Measurable Sets. Egorov's Theorem

§ 7. Convergence in the Mean of Summable Functions

§ 8. The Lebesgue-Fubini Theorem

Lecture 7. Integrals Dependent on a Parameter

§ 1. Integrals which are Uniformly Convergent for a Given Value of Parameter

§ 2. The Derivative of an Improper Integral with respect to a Parameter

Lecture 8. The Equation of Heat Conduction

§ 1. Principal Solution

§ 2. The Solution of Cauchy's Problem

Lecture 9. Laplace's Equation and Poisson's Equation

§ 1. The Theorem of the Maximum

§ 2. The Principal Solution. Green's Formula

§ 3. The Potential due to a Volume, to a Single Layer, and to a Double Layer

Lecture 10. Some General Consequences of Green's Formula

§ 1. The Mean-Value Theorem for a Harmonic Function

§ 2. Behavior of a Harmonic Function near a Singular Point

§ 3. Behavior of a Harmonic Function at Infinity. Inverse Points

Lecture 11. Poisson's Equation in an Unbounded Medium. Newtonian Potential

Lecture 12. The Solution of the Dirichlet Problem for a Sphere

Lecture 13. The Dirichlet Problem and the Neumann Problem for a Half-Space

Lecture 14. The Wave Equation and the Retarded Potential

§ 1. The Characteristics of the Wave Equation

§ 2. Kirchhoff's Method of Solution of Cauchy's Problem

Lecture 15. Properties of the Potentials of Single and Double Layers

§ 1. General Remarks

§ 2. Properties of the Potential of a Double Layer

§ 3. Properties of the Potential of a Single Layer

§ 4. Regular Normal Derivative

§ 5. Normal Derivative of the Potential of a Double Layer

§ 6. Behavior of the Potentials at Infinity

Lecture 16. Reduction of the Dirichlet Problem and the Neumann Problem to Integral Equations

§ 1. Formulation of the Problems and the Uniqueness of their Solutions

§ 2. The Integral Equations for the Formulated Problems

Lecture 17. Laplace's Equation and Poisson's Equation in a Plane

§ 1. The Principal Solution

§ 2. The Basic Problems

§ 3. The Logarithmic Potential

Lecture 18. The Theory of Integral Equations

§ 1. General Remarks

§ 2. The Method of Successive Approximations

§ 3. Volterra Equations

§ 4. Equations with Degenerate Kernel

§ 5. A Kernel of Special Type. Fredholm's Theorems

§ 6. Generalization of the Results

§ 7. Equations with Unbounded Kernels of a Special Form

Lecture 19. Application of the Theory of Fredholm Equations to the Solution of the Dirichlet and Neumann Problems

§ 1. Derivation of the Properties of Integral Equations

§ 2. Investigation of the Equations

Lecture 20. Green's Function

§ 1. The Differential Operator with One Independent Variable

§ 2. Adjoint Operators and Adjoint Families

§ 3. The Fundamental Lemma on the Integrals of Adjoint Equations

§ 4. The Influence Function

§ 5. Definition and Construction of Green's Function

§ 6. The Generalized Green's Function for a Linear Second-Order Equation

§ 7. Examples

Lecture 21. Green's Function for the Laplace Operator

§ 1. Green's Function for the Dirichlet Problem

§ 2. The Concept of Green's Function for the Neumann Problem

Lecture 22. Correctness of Formulation of the Boundary-Value Problems of Mathematical Physics

§ 1. The Equation of Heat Conduction

§ 2. The Concept of the Generalized Solution

§ 3. The Wave Equation

§ 4. The Generalized Solution of the Wave Equation

§ 5. A Property of Generalized Solutions of Homogeneous Equations

§ 6. Bunyakovski's Inequality and Minkovski's Inequality

§ 7. The Riesz-Fischer Theorem

Lecture 23. Fourier's Method

§ 1. Separation of the Variables

§ 2. The Analogy between the Problems of Vibrations of a Continuous Medium and Vibrations of Mechanical Systems with a Finite Number of Degrees of Freedom

§ 3. The Inhomogeneous Equation

§ 4. Longitudinal Vibrations of a Bar

Lecture 24. Integral Equations with Real, Symmetric Kernels

§ 1. Elementary Properties. Completely Continuous Operators

§ 2. Proof of the Existence of an Eigenvalue

Lecture 25. The Bilinear Formula and the Hilbert-Schmidt Theorem

§ 1. The Bilinear Formula

§ 2. The Hilbert-Schmidt Theorem

§ 3. Proof of the Fourier Method for the Solution of the Boundary-Value Problems of Mathematical Physics

§ 4. An Application of the Theory of Integral Equations with Symmetric Kernel

Lecture 26. The Inhomogeneous Integral Equation with a Symmetric Kernel

§ 1. Expansion of the Resolvent

§ 2. Representation of the Solution by means of Analytical Functions

Lecture 27. Vibrations of a Rectangular Parallelepiped

Lecture 28. Laplace's Equation in Curvilinear Coordinates. Examples of the Use of Fourier's Method

§ 1. Laplace's Equation in Curvilinear Coordinates

§ 2. Bessel Functions

§ 3. Complete Separation of the Variables in the Equation Δ2 u = in Polar Coordinates

Lecture 29. Harmonic Polynomials and Spherical Functions

§ 1. Definition of Spherical Functions

§ 2. Approximation by means of Spherical Harmonics

§ 3. The Dirichlet Problem for a Sphere

§ 4. The Differential Equations for Spherical Functions

Lecture 30. Some Elementary Properties of Spherical Functions

§ 1. Legendre Polynomials

§ 2. The Generating Function

§ 3. Laplace's Formula

Index