Boundary and eigenvalue problems in mathematical physics / by Hans Sagan.
Editor: New York : Dover Publications, 1989Descripción: xviii, 381 p. : ill. ; 22 cmISBN: 0486661326 :Tema(s): Mathematical physics | Boundary value problems | EigenvaluesOtra clasificación: *CODIGO* Recursos en línea: Publisher descriptionCONTENTS Chapter I HAMILTON’S PRINCIPLE AND THE THEORY OF THE FIRST VARIATION [1] §1. VARIATIONAL PROBLEMS IN ONE INDEPENDENT VARIABLE, [1] 1. Newton’s Equations of Motion, [1] 2. The Euler-Lagrange Equation, [7] §2. VARIATIONAL PROBLEMS IN TWO AND MORE INDEPENDENT VARIABLES, [15] 1. Vibrations of a Stretched String, [15] 2. The Euler-Lagrange Equation for the Two-Dimensional Problem, [16] §3. THE ISOPERIMETRIC PROBLEM, [22] 1. The Problem of Dido, [22] 2. The Euler-Lagrange Equation for the Isoperimetric Problem in One Independent Variable, [23] §4. NATURAL BOUNDARY CONDITIONS, [28] 1. A Problem of Zermelo in Modified Form, [28] 2. Natural Boundary Conditions for the One-Dimensional Problem, [29] 3. Natural Boundary Condition for the Two-Dimensional Problem, [30] REPRESENTATION OF SOME PHYSICAL PHENOMENA BY PARTIAL DIFFERENTIAL EQUATIONS §1. THE VIBRATING STRING, [34] 1. Vibrations of a Stretched String—Vectorial Approach, [34] 2. An Attempt to Solve a Specific String Problem, [37] 3. Boundary and Initial Conditions in Differential Equations, [42] 4. Boundary and Initial Value Problem for the String, [45] 5. The Uniqueness of the Solution of the String Equation, [46] §2. THE VIBRATING MEMBRANE, [49] 1. Vibrations of a Stretched Membrane, [49] 2. The Vibrations of a Membrane as a Variational Problem, [53] 3. The Boundary and Initial Value Problem for the Membrane, [55] 4. An Attempt to Solve a Specific Membrane Problem, [58] 5. The Uniqueness of the Solution of the Membrane Equation, [62] §3.iTHE EQUATION OF HEAT CONDUCTION AND THE POTENTIAL EQUATION, [66] 1. Heat Conduction without Convection, 66 2. The Potential Equation, [69] 3. The Initial and Boundary Value Problem for Heat Conduction, [72] 4. Stationary Temperature Distribution Generated by a Spherical Stove, [74] 5. The Uniqueness of the Solution of the Potential Equation, [77] 6 The Uniqueness of the Solution of the Heat Equation, [79] [34] THEOREMS RELATED TO PARTIAL DIFFERENTIAL EQUATIONS AND THEIR SOLUTIONS [82] §1. GENERAL REMARKS ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS, [82] 1. The Problem of Minimal Surfaces, [82] 2. The Problem of Cauchy-Kowalewski, [85] §2. INFINITE SERIES AS SOLUTIONS OF LINEAR HOMOGENEOUS PARTIAL DIFFERENTIAL EQUATIONS WITH HOMOGENEOUS BOUNDARY OR INITIAL CONDITIONS, [88] 1. Bernoulli’s Separation Method, [88] 2. Solution of the Homogeneous Boundary Value Problem by Infinite Series, [91] Chapter IV FOURIER SERIES [95] §1. THE FORMAL APPROACH, [95] 1. Introductory Remarks, [95] 2. Approximation of a Given Function by a Trigonometric Polynomial, [96] 3. The Trigonometric Functions as an Orthogonal System, [97] 4. Approximations by Trigonometric Polynomials, Continued, [101] 5. Formal Expansion of a Function into a Fourier Series, [102] 6. Formal Rules for the Evaluation of Fourier Coefficients, [103] §2. THE THEORY OF FOURIER SERIES, [105] 1. Illustrative Examples, [105] 2. Convergence of Fourier Series, [111] 3. The Convergence Proof, [113] 4. Bessel’s Inequality, [118] 5. Absolute and Uniform Convergence of Fourier Series, [120] CONTENTS 6. Completeness, [126] 7. Integration and Differentiation of Fourier Series, [129] §3. APPLICATIONS, [132] 1. The Plucked String, [132] 2. The Validity of the Solution of the String Equation for a Displacement Function with a Sectionally Continuous Derivative, [136] 3. The General Solution of the String Problem D’Alembert’s Method, [140] Chapter V SELF-ADJOINT BOUNDARY VALUE PROBLEMS [145] §1. SELF-ADJOINT DIFFERENTIAL EQUATIONS OF THE SECOND ORDER, [145] 1. Self-adjoint Differential Equations, [145] 2. Examples, [150] 3. The -Diagram, [152] 4. The Zeros of the Solutions of a Self-adjoint Differential Equation, [154] §2. THE SELF-ADJOINT BOUNDARY VALUE PROBLEM OF STURM AND LIOUVILLE, [163] 1. Eigenvalues and Eigenfunctions, [163] 2. The Sturm-Liouville Problem in the x, -Diagram, [167] 3. Number and Distribution of the Eigenvalues of the Sturm-Liouville Boundary Value Problem, [168] 4. The Orthogonality of Eigenfunctions, [175] 5. Fourier Expansions in Terms of Eigenfunctions, [177] 6. Completeness and the Expansion Problem, [178] Chapter VI LEGENDRE POLYNOMIALS AND BESSEL FUNCTIONS [182] 1. THE LEGENDRE EQUATION AND LEGENDRE POLYNOMIALS, [182] 1. Senes Solution of a Linear Differential Equation of the Second Order with Analytic Coefficients, [182] 2. Senes Solutions the Legendre Equation, [188] 3. Polynomial Solutions of the Legendre Equation, [193] 4. Generating Function—Legendre Polynomials, [194] 5. Orthogonality and Normalization, [198] 6. Legendre Functions of the Second Kind, [201] 7. Legendre Polynomials as Potentials of Multipoles with Respect to an Action Point at Unit Distance, [202] §2. THE BESSEL EQUATION AND BESSEL FUNCTIONS, [204] 1. Introductory Remarks, [204] 2. Solution of a Linear Differential Equation of the Second Order at a Regular Singular Point, [209] 3. Bessel Functions of Integral Order, [214] 4. Bessel Functions of Nonintegral Order and of Negative Order, [216] 5. The Linear Independence of Two Solutions of Bessel’s Equation of Nonintegral Order, [220] 6. Bessel Functions of the Second Kind, [222] 7. Circular Membrane with Axially Symmetric Initial Displacement, [227] 8. The Normalization of the Bessel Functions, [230] 9. Integral Representation of Bessel Functions of Integral Order, [232] 10. The Almost-Periodic Behavior of the Functions [235] Chapter VII CHARACTERIZATION OF EIGENVALUES BY A VARIATIONAL PRINCIPLE [239] §1. INTRODUCTION AND GENERAL EXPOSITION, [239] 1. The Sturm-Liouville Equation as Euler-Lagrange Equation of a Certain Isoperimetric Problem, [239] 2. Numerical Examples, [241] 3. Minimizing Sequences, [244] 4. The Existence of a Minimum, [248] §2. MINIMUM PROPERTIES OF THE EIGENVALUES OF A SELF-ADJOINT BOUNDARY VALUE PROBLEM, [253] 1. The Solutions of a Certain Isoperimetric Problem as the Eigenfunctions of a Self-adjoint Boundary Value Problem, [253] 2. The Eigenfunctions of the Sturm-Liouville Boundary Value Problem as Solutions of the Associated Variational Problem, [260] §3. THE METHOD OF RAYLEIGH AND RITZ, [269] 1. Theorems Regarding the Secular Equation, [269] 2. The Method of Rayleigh and Ritz, [273] 3. The Eigenvalues of Ritz’s Problem as Upper Bounds for the Eigenvalues of the Original Problem, [279] 4. Successive Approximations to the Eigenvalues from Above, [281] 5. Numerical Examples, [284] Chapter VIII SPHERICAL HARMONICS [288] §1. ASSOCIATED LEGENDRE FUNCTIONS, SPHERICAL HARMONICS, AND LAGUERRE POLYNOMIALS, [288] 1. The Equation of Wave Propagation, [288] 2. Schroedinger’s Wave Equation, [292] 3. The Hydrogen Atom, [296] 4. Legendre’s Associated Equation, [299] 5. Spherical Harmonics, [300] 6. Laguerre Polynomials and Associated Functions, [302] 7. Solution of Schroedinger’s Equation for the Hydrogen Atom, [305] §2. SPHERICAL HARMONICS AND POISSON’S INTEGRAL, [308] 1. Stationary Temperature Distribution Generated by a Spherical Stove, [308] 2. Fourier Expansion in Terms of Associated Legendre Functions, [311] 3. Poisson’s Integral Representation of the Solution of the Potential Equation, [314] 4. Expansion in Terms of Laplace Coefficients, [316] Chapter IX THE NONHOMOGENEOUS BOUNDARY VALUE PROBLEM [322] §1. THE INFLUENCE FUNCTION (GREEN’S FUNCTION), [322] 1. The Nonhomogeneous String Equation, [322] 2. Determination of Green’s Function, [323] 3. Examples, [326] 4. A General Discussion of the Nonhomogeneous and the Associated Homogeneous Boundary Value Problem, [329] 5. The Existence of Green’s Function, [331] §2. THE GENERALIZED GREEN FUNCTION, [335] 1. Construction of Green’s Function in the Case Where the Associated Homogeneous Boundary Value Problem Has a Nontrivial Solution, [335] 2. Examples, [340] §3. FURTHER ASPECTS OF GREEN’S FUNCTION, [344] 1. Green’s Function and the Sturm-Liouville Boundary Value Problem, [344] 2. Green’s Function for Partial Differential Equations, [348] APPENDIX [353] KEY FOR REFERENCE SYMBOLS USED IN THE APPENDIX, [353] I. VECTOR ANALYSIS, [354] A. Vector Operations, [354] B. Differential Operations, [355] C. Integral Theorems, [358] II. CONVERGENCE, [359] A. Definitions, [359] B. Convergence Tests, [360] C. Theorems, [360] D. Power Series, [361] III. ORDINARY DIFFERENTIAL EQUATIONS, [362] A. Theorem on Implicit Functions, [362] B. Systems of Ordinary Differential Equations of the First Order, [363] answers and hints to even numbered PROBLEMS [365] INDEX [375]
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Reprint. Originally published: New York : Wiley, 1961.
Includes bibliographies and index.
CONTENTS --
Chapter I --
HAMILTON’S PRINCIPLE AND THE THEORY OF THE FIRST VARIATION [1] --
§1. VARIATIONAL PROBLEMS IN ONE INDEPENDENT VARIABLE, [1] --
1. Newton’s Equations of Motion, [1] --
2. The Euler-Lagrange Equation, [7] --
§2. VARIATIONAL PROBLEMS IN TWO AND MORE INDEPENDENT VARIABLES, [15] --
1. Vibrations of a Stretched String, [15] --
2. The Euler-Lagrange Equation for the Two-Dimensional Problem, [16] --
§3. THE ISOPERIMETRIC PROBLEM, [22] --
1. The Problem of Dido, [22] --
2. The Euler-Lagrange Equation for the Isoperimetric Problem in One Independent Variable, [23] --
§4. NATURAL BOUNDARY CONDITIONS, [28] --
1. A Problem of Zermelo in Modified Form, [28] --
2. Natural Boundary Conditions for the One-Dimensional Problem, [29] --
3. Natural Boundary Condition for the Two-Dimensional Problem, [30] --
REPRESENTATION OF SOME PHYSICAL PHENOMENA BY PARTIAL DIFFERENTIAL EQUATIONS --
§1. THE VIBRATING STRING, [34] --
1. Vibrations of a Stretched String—Vectorial Approach, [34] --
2. An Attempt to Solve a Specific String Problem, [37] --
3. Boundary and Initial Conditions in Differential Equations, [42] --
4. Boundary and Initial Value Problem for the String, [45] --
5. The Uniqueness of the Solution of the String Equation, [46] --
§2. THE VIBRATING MEMBRANE, [49] --
1. Vibrations of a Stretched Membrane, [49] --
2. The Vibrations of a Membrane as a Variational Problem, [53] --
3. The Boundary and Initial Value Problem for the Membrane, [55] --
4. An Attempt to Solve a Specific Membrane Problem, [58] --
5. The Uniqueness of the Solution of the Membrane Equation, [62] --
§3.iTHE EQUATION OF HEAT CONDUCTION AND THE POTENTIAL EQUATION, [66] --
1. Heat Conduction without Convection, 66 2. The Potential Equation, [69] --
3. The Initial and Boundary Value Problem for Heat Conduction, [72] --
4. Stationary Temperature Distribution Generated by a Spherical Stove, [74] --
5. The Uniqueness of the Solution of the Potential Equation, [77] --
6 The Uniqueness of the Solution of the Heat Equation, [79] --
[34] --
THEOREMS RELATED TO PARTIAL DIFFERENTIAL EQUATIONS AND THEIR SOLUTIONS [82] --
§1. GENERAL REMARKS ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS, [82] --
1. The Problem of Minimal Surfaces, [82] --
2. The Problem of Cauchy-Kowalewski, [85] --
§2. INFINITE SERIES AS SOLUTIONS OF LINEAR HOMOGENEOUS PARTIAL DIFFERENTIAL EQUATIONS WITH HOMOGENEOUS BOUNDARY OR INITIAL CONDITIONS, [88] --
1. Bernoulli’s Separation Method, [88] --
2. Solution of the Homogeneous Boundary Value Problem by Infinite Series, [91] --
Chapter IV --
FOURIER SERIES [95] --
§1. THE FORMAL APPROACH, [95] --
1. Introductory Remarks, [95] --
2. Approximation of a Given Function by a Trigonometric Polynomial, [96] --
3. The Trigonometric Functions as an Orthogonal System, [97] --
4. Approximations by Trigonometric Polynomials, Continued, [101] --
5. Formal Expansion of a Function into a Fourier Series, [102] --
6. Formal Rules for the Evaluation of Fourier Coefficients, [103] --
§2. THE THEORY OF FOURIER SERIES, [105] --
1. Illustrative Examples, [105] --
2. Convergence of Fourier Series, [111] --
3. The Convergence Proof, [113] --
4. Bessel’s Inequality, [118] --
5. Absolute and Uniform Convergence of Fourier Series, [120] --
CONTENTS --
6. Completeness, [126] --
7. Integration and Differentiation of Fourier Series, [129] --
§3. APPLICATIONS, [132] --
1. The Plucked String, [132] --
2. The Validity of the Solution of the String Equation for a Displacement Function with a Sectionally Continuous Derivative, [136] --
3. The General Solution of the String Problem D’Alembert’s Method, [140] --
Chapter V --
SELF-ADJOINT BOUNDARY VALUE PROBLEMS [145] --
§1. SELF-ADJOINT DIFFERENTIAL EQUATIONS OF THE SECOND ORDER, [145] --
1. Self-adjoint Differential Equations, [145] --
2. Examples, [150] --
3. The -Diagram, [152] --
4. The Zeros of the Solutions of a Self-adjoint Differential Equation, [154] --
§2. THE SELF-ADJOINT BOUNDARY VALUE PROBLEM OF STURM AND LIOUVILLE, [163] --
1. Eigenvalues and Eigenfunctions, [163] --
2. The Sturm-Liouville Problem in the x, -Diagram, [167] --
3. Number and Distribution of the Eigenvalues of the Sturm-Liouville Boundary Value Problem, [168] --
4. The Orthogonality of Eigenfunctions, [175] --
5. Fourier Expansions in Terms of Eigenfunctions, [177] --
6. Completeness and the Expansion Problem, [178] --
Chapter VI --
LEGENDRE POLYNOMIALS AND BESSEL FUNCTIONS [182] --
1. THE LEGENDRE EQUATION AND LEGENDRE POLYNOMIALS, [182] --
1. Senes Solution of a Linear Differential Equation of the Second Order with Analytic Coefficients, [182] --
2. Senes Solutions the Legendre Equation, [188] --
3. Polynomial Solutions of the Legendre Equation, [193] --
4. Generating Function—Legendre Polynomials, [194] --
5. Orthogonality and Normalization, [198] --
6. Legendre Functions of the Second Kind, [201] --
7. Legendre Polynomials as Potentials of Multipoles with Respect to an Action Point at Unit Distance, [202] --
§2. THE BESSEL EQUATION AND BESSEL FUNCTIONS, [204] --
1. Introductory Remarks, [204] --
2. Solution of a Linear Differential Equation of the Second Order at a Regular Singular Point, [209] --
3. Bessel Functions of Integral Order, [214] --
4. Bessel Functions of Nonintegral Order and of Negative Order, [216] --
5. The Linear Independence of Two Solutions of Bessel’s Equation of Nonintegral Order, [220] --
6. Bessel Functions of the Second Kind, [222] --
7. Circular Membrane with Axially Symmetric Initial Displacement, [227] --
8. The Normalization of the Bessel Functions, [230] --
9. Integral Representation of Bessel Functions of Integral Order, [232] --
10. The Almost-Periodic Behavior of the Functions [235] --
Chapter VII --
CHARACTERIZATION OF EIGENVALUES BY A VARIATIONAL PRINCIPLE [239] --
§1. INTRODUCTION AND GENERAL EXPOSITION, [239] --
1. The Sturm-Liouville Equation as Euler-Lagrange Equation of a Certain Isoperimetric Problem, [239] --
2. Numerical Examples, [241] --
3. Minimizing Sequences, [244] --
4. The Existence of a Minimum, [248] --
§2. MINIMUM PROPERTIES OF THE EIGENVALUES OF A SELF-ADJOINT BOUNDARY VALUE PROBLEM, [253] --
1. The Solutions of a Certain Isoperimetric Problem as the Eigenfunctions of a Self-adjoint Boundary Value Problem, [253] --
2. The Eigenfunctions of the Sturm-Liouville Boundary Value Problem as Solutions of the Associated Variational Problem, [260] --
§3. THE METHOD OF RAYLEIGH AND RITZ, [269] --
1. Theorems Regarding the Secular Equation, [269] --
2. The Method of Rayleigh and Ritz, [273] --
3. The Eigenvalues of Ritz’s Problem as Upper Bounds for the Eigenvalues of the Original Problem, [279] --
4. Successive Approximations to the Eigenvalues from Above, [281] --
5. Numerical Examples, [284] --
Chapter VIII --
SPHERICAL HARMONICS [288] --
§1. ASSOCIATED LEGENDRE FUNCTIONS, SPHERICAL HARMONICS, AND LAGUERRE POLYNOMIALS, [288] --
1. The Equation of Wave Propagation, [288] --
2. Schroedinger’s Wave Equation, [292] --
3. The Hydrogen Atom, [296] --
4. Legendre’s Associated Equation, [299] --
5. Spherical Harmonics, [300] --
6. Laguerre Polynomials and Associated Functions, [302] --
7. Solution of Schroedinger’s Equation for the Hydrogen Atom, [305] --
§2. SPHERICAL HARMONICS AND POISSON’S INTEGRAL, [308] --
1. Stationary Temperature Distribution Generated by a Spherical Stove, [308] --
2. Fourier Expansion in Terms of Associated Legendre Functions, [311] --
3. Poisson’s Integral Representation of the Solution of the Potential Equation, [314] --
4. Expansion in Terms of Laplace Coefficients, [316] --
Chapter IX --
THE NONHOMOGENEOUS BOUNDARY VALUE PROBLEM [322] --
§1. THE INFLUENCE FUNCTION (GREEN’S FUNCTION), [322] --
1. The Nonhomogeneous String Equation, [322] --
2. Determination of Green’s Function, [323] --
3. Examples, [326] --
4. A General Discussion of the Nonhomogeneous and the Associated Homogeneous Boundary Value Problem, [329] --
5. The Existence of Green’s Function, [331] --
§2. THE GENERALIZED GREEN FUNCTION, [335] --
1. Construction of Green’s Function in the Case Where the Associated Homogeneous Boundary Value Problem Has a Nontrivial Solution, [335] --
2. Examples, [340] --
§3. FURTHER ASPECTS OF GREEN’S FUNCTION, [344] --
1. Green’s Function and the Sturm-Liouville Boundary Value Problem, [344] --
2. Green’s Function for Partial Differential Equations, [348] --
APPENDIX [353] --
KEY FOR REFERENCE SYMBOLS USED IN THE APPENDIX, [353] --
I. VECTOR ANALYSIS, [354] --
A. Vector Operations, [354] --
B. Differential Operations, [355] --
C. Integral Theorems, [358] --
II. CONVERGENCE, [359] --
A. Definitions, [359] --
B. Convergence Tests, [360] --
C. Theorems, [360] --
D. Power Series, [361] --
III. ORDINARY DIFFERENTIAL EQUATIONS, [362] --
A. Theorem on Implicit Functions, [362] --
B. Systems of Ordinary Differential Equations of the First Order, [363] --
answers and hints to even numbered PROBLEMS [365] --
INDEX [375] --
--
MR, REVIEW #
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