Boundary and eigenvalue problems in mathematical physics / by Hans Sagan.

Por: Sagan, HansEditor: 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 description
Contenidos:
 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]
 
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Libros Libros Instituto de Matemática, CONICET-UNS
Libros ordenados por tema 35 Sa129 (Browse shelf) Available A-9361

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] --
--

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