# Partial Differential Equations of Mathematical Physics

### Adiwes International Series in Mathematics

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Partial Differential Equations of Mathematical Physics emphasizes the study of second-order partial differential equations of mathematical physics, which is deemed as the foundation of investigations into waves, heat conduction, hydrodynamics, and other physical problems. The book discusses in detail a wide spectrum of topics related to partial differential equations, such as the theories of sets and of Lebesgue integration, integral equations, Green's function, and the proof of the Fourier method. Theoretical physicists, experimental physicists, mathematicians engaged in pure and applied mathematics, and researchers will benefit greatly from this book.

## Table of Contents

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

## Product details

- No. of pages: 438
- Language: English
- Copyright: © Pergamon 1964
- Published: January 1, 1964
- Imprint: Pergamon
- eBook ISBN: 9781483149165

## About the Author

### S. L. Sobolev

## About the Editor

### A.J. Lohwater

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