Chapter
§ 5. Equation of Heat Conduction
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
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
§ 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
§ 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
§ 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. Behaviour of a Harmonic Function near a Singular Point
§ 3. Behaviour 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
§ 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. Behaviour 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 EQUATON AND POISSON'S EQUATION IN A PLANE
§ 1. The Principal Solution
§ 3. The Logarithmic Potential
LECTURE 18. THE THEORY OF INTEGRAL EQUATIONS
§ 2. The Method of Successive Approximations
§ 4. Equations with Degenerate Kernel
§ 5. A Kernel of Special Type. Fredhohn'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 Difíerential 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
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
§ 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 EQUATONS WTIH 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 WTTH 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 EQUATON IN CURVILINEAR COORDINATES. EXAMPLES OF THE USE OF FOURIER'S METHOD
§ 1. Laplace's Equation in Curvilinear Coordinates
§ 3. Complete Separation of the Variables in the Equation V2u= O 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
OTHER VOLUMES IN THIS SERIES
Partial Differential Equations of Mathematical Physics
：International Series of Monographs in Pure and Applied Mathematics
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