The Mollification Method and the Numerical Solution of Ill-Posed Problems

✍ Scribed by Diego A. Murio

• Publisher: Wiley-Interscience
• Year: 1993
• Tongue: English
• Leaves: 270
• Edition: 1
• Category: Library

✦ Synopsis

Uses a strong computational and truly interdisciplinary treatment to introduce applied inverse theory. The author created the Mollification Method as a means of dealing with ill-posed problems. Although the presentation focuses on problems with origins in mechanical engineering, many of the ideas and techniques can be easily applied to a broad range of situations.

✦ Table of Contents

Preface......Page 8
Acknowledgments......Page 12
1.1. Description of the Problem......Page 17
1.2. Stabilized Problem......Page 20
1.3. Differentiation as an Inverse Problem......Page 25
1.4. Parameter Selection......Page 27
1.5. Numerical Procedure......Page 28
1.6. Numerical Results......Page 29
1.7. Exercises......Page 33
1.8. References and Comments......Page 35
2.1. Description of the Problem......Page 38
2.2. Stabilized Problems......Page 43
2.3. Numerical Implementations......Page 54
2.4. Numerical Results and Comparisons......Page 65
2.5. Exercises......Page 71
2.6. References and Comments......Page 73
3.1. One-Dimensional IHCP in a Semi-infinite Body......Page 76
3.2. Stabilized Problems......Page 78
3.3. One-Dimensional IHCP with Finite Slab Symmetry......Page 88
3.4. Finite-Difference Approximations......Page 93
3.5. Integral Equation Approximations......Page 101
3.6. Numerical Results......Page 104
3.7. Exercises......Page 117
3.8. References and Comments......Page 119
4.1. Two-Dimensional IHCP in a Semi-infinite Slab......Page 123
4.2. Stabilized Problem......Page 125
4.3. Numerical Procedure and Error Analysis......Page 129
4.4. Numerical Results......Page 133
4.5. Exercises......Page 145
4.6. References and Comments......Page 146
5.1. Identification of Boundary Source Functions......Page 147
5.2. Numerical Procedure......Page 151
5.3. IHCP with Phase Changes......Page 157
5.4. Description of the Problems......Page 159
5.5. Numerical Procedure......Page 162
5.7. Semi-infinite Body......Page 171
5.8. Finite Slab Symmetry......Page 173
5.9. Stabilized Problems......Page 175
5.10. Numerical Results......Page 177
5.11. Exercises......Page 181
5.12. References and Comments......Page 182
6.1. Numerical Identification of Forcing Terms......Page 185
6.2. Stabilized Problem......Page 186
6.3. Numerical Results......Page 189
6.4. Identification of the Transmissivity Coefficient in the One-Dimensional Elliptic Equation......Page 191
6.5. Stability Analysis......Page 193
6.6. Numerical Method......Page 196
6.7. Numerical Results......Page 201
6.8. Identification of the Transmissivity Coefficient in the One-Dimensional Parabolic Equation......Page 205
6.9. Stability Analysis......Page 206
6.10. Numerical Method......Page 210
6.11. Numerical Results......Page 215
6.12. Exercises......Page 219
6.13. References and Comments......Page 221
A.l. L" Spaces......Page 224
A.2. The Hilbert Space L2......Page 227
A.3. Approximation of Functions in Æ......Page 230
A.4. Mollifiers......Page 233
A.5. Fourier Transform......Page 236
A.6. Discrete Functions......Page 240
A.7. References and Comments......Page 246
Appendix B. References to the Literature on the IHCP......Page 248
Index......Page 265