The Heat Equation

1;Front Cover;1 2;The Heat Equation;4 3;Copyright Page ;5 4;Contents;8 5;Preface;12 6;Symbols and Notation;14 7;Chapter I. Introduction;18 7.1;1. Introduction;18 7.2;2. The Physical Model;18 7.3;3. The Heat Equation;20 7.4;4. Generalities;22 7.5;5. Basic Solutions of the Heat Equation;25 7.6;6. Methods of Generating Solutions;27 7.7;7. Definitions and Notations;31 8;Chapter II. Boundary-Value Problems;34 8.1;I. Introduction;34 8.2;2. Uniqueness;35 8.3;3. The Maximum Principle;37 8.4;4. A Criterion for Temperature Functions;40 8.5;5. Solution of Problem 1 in a Special Case;41 8.6;6. Uniqueness for the Infinite Rod;43 9;Chapter III. Further Developments;47 9.1;1. Introduction;47 9.2;2. The Source Solution;47 9.3;3. The Addition Formula for k(x, t);49 9.4;4. The Homogeneity of k(x, t );51 9.5;5. An Integral Representation of k(x, t );52 9.6;6. A Further Addition Formula for k(x, t);54 9.7;7. Laplace Transform of k(x, ts);55 9.8;8. Laplace Transform of h(x, t);56 9.9;9. Operational Calculus;58 9.10;10. Three Classes of Functions;61 9.11;11. Examples of Class II;63 9.12;12. Relation among the Classes;66 9.13;13. Series Expansions of Functions in Class I;67 9.14;14. Series Expansions of Functions in Class II;69 9.15;15. Series Expansions of Functions in Class III;70 9.16;16. A Temperature Function Which Is Not Entire in the Space Variable;75 10;Chapter IV. Integral Transforms;77 10.1;1. Poisson Transforms;77 10.2;2. Convergence;79 10.3;3. Poisson Transform in H;81 10.4;4. Analyticity;81 10.5;5. Inversion of the PoissonLebesgue Transform;82 10.6;6. Inversion of the PoissonStieltjes Transform;85 10.7;7. The h-Transform;87 10.8;8. h-Transform in H;91 10.9;9. Analyticity;92 10.10;10. Inversion of the h-Lebesgue Transform;95 10.11;11. The k-Transform;97 10.12;12. A Basic Integral Representation;99 10.13;13. Analytic Character of Every Temperature Function;101 11;Chapter V. Theta-Functions;103 11.1;1. Introduction;103 11.2;2. Analyticity;105 11.3;3..-Functions in H;106 11.4;4.
Alternate Expansions;107 11.5;5. Two Positive Kernels;109 11.6;6. A .-Transform;111 11.7;7. A f-Transform;114 11.8;8. Fouriers Ring;117 11.9;9. A Solution of the First Boundary-Value Problem;118 11.10;10. Uniqueness;119 12;Chapter VI. Greens Function;124 12.1;1. Greens Function for a Rectangle;124 12.2;2. An Integral Representation;126 12.3;3. Problem I Again;127 12.4;4. A Property of G(x, t ; ., n);129 12.5;5. Greens Function for an Arbitrary Rectangle;130 12.6;6. Series of Temperature Functions;131 12.7;7. The Reflection Principle;132 12.8;8. Isolated Singularities;133 13;Chapter VII. Bounded Temperature Functions;139 13.1;1. The Infinite Rod;139 13.2;2. TheSemi-Infinite Rod;141 13.3;3. Semi-Infinite Rod, Continued;143 13.4;4. Semi-Infinite Rod, General Case;144 13.5;5. The Finite Rod;147 14;Chapter VIII. Positive Temperature Functions;149 14.1;1. The Infinite Rod;149 14.2;2. Uniqueness, Positive Temperatures on an Infinite Rod;150 14.3;3. Stieltjes Integral Representation, Infinite Rod;153 14.4;4. Uniqueness, Semi-Infinite Rod;154 14.5;5. Representation, Semi-Infinite Rod;157 14.6;6. The Finite Rod;164 14.7;7. Examples;168 14.8;8. Further Classes of Temperature Functions;170 15;Chapter IX. The Huygens Property;172 15.1;1. Introduction;172 15.2;2. Blackmans Example;176 15.3;3. Conditionally Convergent Poisson Integrals;178 15.4;5. Heat Polynomials and Associated Functions;182 16;Chapter X. Series Expansions of Temperature Functions;186 16.1;1. Introduction;186 16.2;2. Asymptotic Estimates;188 16.3;3. A Generating Function;192 16.4;4. Region of Convergence;194 16.5;5. Strip of Convergence;197 16.6;6. Representation by Series of Heat Polynomials;200 16.7;7. The Growth of an Entire Function;202 16.8;8. Expansions in Series of Associated Functions;203 16.9;9. A Further Criterion;205 16.10;10. Examples;208 17;Chapter XI. Analogies;212 17.1;1. Introduction;212 17.2;2. The Appell Transformation;214 17.3;3. Heat Polynomials;214 17.4;4. Associated Functions;215 17.5;5. The
Huygens Property;215 17.6;6. The Operators ecD and eCD2;216 17.7;7. Biorthogonality;216 17.8;8. Generating Functions;217 17.9;9. Polynomial Expansions;217 17.10;10. Associated Function Expansions;217 17.11;11. Criteria for Polynomial Expansions;218 17.12;12. Criteria for Expansions in Series of Associated Functions;218 18;Chapter XII. Higher Dimensions;221 18.1;1. Introduction;221 18.2;2. The Heat Equation or Solids;222 18.3;3. Notations and Definitions;224 18.4;4. Generating Functions;226 18.5;5. Expansions in Series of Polynomials;227 18.6;6. An Example;231 19;Chapter XIII. Homogeneous Temperature Functions;233 19.1;1. Introduction;233 19.2;2. The Totality of Homogeneous Temperature Functions;235 19.3;3. Recurrence Relations;239 19.4;4. Continued Fraction Developments;240 19.5;5. Decomposition of the Basic Functions;243 19.6;6. Summary;244 19.7;7. Series of Polynomials;244 19.8;8. First Kind, Negative Degree;247 19.9;9. Second Kind, Positive Degree;248 19.10;10. Second Kind, Negative Degree;249 19.11;11. Examples;249 20;Chapter XIV. Miscellaneous Topics;252 20.1;1. Positive Temperature Functions;252 20.2;2.Positive Definite Functions;254 20.3;3. Positive Temperature Functions, Concluded;256 20.4;4. A Statistical Problem;258 20.5;5. Examples;261 20.6;6. Statistical Problem Concluded;263 20.7;7. Alternate Inversion of the h-Transform;266 20.8;8. Time-Variable Singularities;270 21;Bibliography;276 22;Index;280