Heat Equation (eBook)

Front Cover 1
The Heat Equation 4
Copyright Page 5
Contents 8
Preface 12
Symbols and Notation 14
Chapter I. Introduction 18
1. Introduction 18
2. The Physical Model 18
3. The Heat Equation 20
4. Generalities 22
5. Basic Solutions of the Heat Equation 25
6. Methods of Generating Solutions 27
7. Definitions and Notations 31
Chapter II. Boundary-Value Problems 34
I. Introduction 34
2. Uniqueness 35
3. The Maximum Principle 37
4. A Criterion for Temperature Functions 40
5. Solution of Problem 1 in a Special Case 41
6. Uniqueness for the Infinite Rod 43
Chapter III. Further Developments 47
1. Introduction 47
2. The Source Solution 47
3. The Addition Formula for k(x, t) 49
4. The Homogeneity of k(x, t ) 51
5. An Integral Representation of k(x, t ) 52
6. A Further Addition Formula for k(x, t) 54
7. Laplace Transform of k(x, ts) 55
8. Laplace Transform of h(x, t) 56
9. Operational Calculus 58
10. Three Classes of Functions 61
11. Examples of Class II 63
12. Relation among the Classes 66
13. Series Expansions of Functions in Class I 67
14. Series Expansions of Functions in Class II 69
15. Series Expansions of Functions in Class III 70
16. A Temperature Function Which Is Not Entire in the Space Variable 75
Chapter IV. Integral Transforms 77
1. Poisson Transforms 77
2. Convergence 79
3. Poisson Transform in H 81
4. Analyticity 81
5. Inversion of the Poisson–Lebesgue Transform 82
6. Inversion of the Poisson–Stieltjes Transform 85
7. The h-Transform 87
8. h-Transform in H 91
9. Analyticity 92
10. Inversion of the h-Lebesgue Transform 95
11. The k-Transform 97
12. A Basic Integral Representation 99
13. Analytic Character of Every Temperature Function 101
Chapter V. Theta-Functions 103
1. Introduction 103
2. Analyticity 105
3..-Functions in H 106
4. Alternate Expansions 107
5. Two Positive Kernels 109
6. A .-Transform 111
7. A f-Transform 114
8. Fourier’s Ring 117
9. A Solution of the First Boundary-Value Problem 118
10. Uniqueness 119
Chapter VI. Green’s Function 124
1. Green’s Function for a Rectangle 124
2. An Integral Representation 126
3. Problem I Again 127
4. A Property of G(x, t ., n)
5. Green’s Function for an Arbitrary Rectangle 130
6. Series of Temperature Functions 131
7. The Reflection Principle 132
8. Isolated Singularities 133
Chapter VII. Bounded Temperature Functions 139
1. The Infinite Rod 139
2. TheSemi-Infinite Rod 141
3. Semi-Infinite Rod, Continued 143
4. Semi-Infinite Rod, General Case 144
5. The Finite Rod 147
Chapter VIII. Positive Temperature Functions 149
1. The Infinite Rod 149
2. Uniqueness, Positive Temperatures on an Infinite Rod 150
3. Stieltjes Integral Representation, Infinite Rod 153
4. Uniqueness, Semi-Infinite Rod 154
5. Representation, Semi-Infinite Rod 157
6. The Finite Rod 164
7. Examples 168
8. Further Classes of Temperature Functions 170
Chapter IX. The Huygens Property 172
1. Introduction 172
2. Blackman’s Example 176
3. Conditionally Convergent Poisson Integrals 178
5. Heat Polynomials and Associated Functions 182
Chapter X. Series Expansions of Temperature Functions 186
1. Introduction 186
2. Asymptotic Estimates 188
3. A Generating Function 192
4. Region of Convergence 194
5. Strip of Convergence 197
6. Representation by Series of Heat Polynomials 200
7. The Growth of an Entire Function 202
8. Expansions in Series of Associated Functions 203
9. A Further Criterion 205
10. Examples 208
Chapter XI. Analogies 212
1. Introduction 212
2. The Appell Transformation 214
3. Heat Polynomials 214
4. Associated Functions 215
5. The Huygens Property 215
6. The Operators ecD and eCD2 216
7. Biorthogonality 216
8. Generating Functions 217
9. Polynomial Expansions 217
10. Associated Function Expansions 217
11. Criteria for Polynomial Expansions 218
12. Criteria for Expansions in Series of Associated Functions 218
Chapter XII. Higher Dimensions 221
1. Introduction 221
2. The Heat Equation €or Solids 222
3. Notations and Definitions 224
4. Generating Functions 226
5. Expansions in Series of Polynomials 227
6. An Example 231
Chapter XIII. Homogeneous Temperature Functions 233
1. Introduction 233
2. The Totality of Homogeneous Temperature Functions 235
3. Recurrence Relations 239
4. Continued Fraction Developments 240
5. Decomposition of the Basic Functions 243
6. Summary 244
7. Series of Polynomials 244
8. First Kind, Negative Degree 247
9. Second Kind, Positive Degree 248
10. Second Kind, Negative Degree 249
11. Examples 249
Chapter XIV. Miscellaneous Topics 252
1. Positive Temperature Functions 252
2.Positive Definite Functions 254
3. Positive Temperature Functions, Concluded 256
4. A Statistical Problem 258
5. Examples 261
6. Statistical Problem Concluded 263
7. Alternate Inversion of the h-Transform 266
8. Time-Variable Singularities 270
Bibliography 276
Index 280