Carrier / Pearson Partial Differential Equations

Theory and Technique
1. Auflage 2014
ISBN: 978-1-4832-5916-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 332 Seiten, Web PDF

Partial Differential Equations: Theory and Technique provides formal definitions, notational conventions, and a systematic discussion of partial differential equations. The text emphasizes the acquisition of practical technique in the use of partial differential equations. The book contains discussions on classical second-order equations of diffusion, wave motion, first-order linear and quasi-linear equations, and potential theory. Certain chapters elaborate Green's functions, eigenvalue problems, practical approximation techniques, perturbations (regular and singular), difference equations, and numerical methods. Students of mathematics will find the book very useful.

Autoren/Hrsg.

Carrier, George F.

Pearson, Carl E.

Inhaltsverzeichnis

1;Front Cover;1
2;Partial Differential Equations: Theory and Technique;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;12
6;INTRODUCTION;14
6.1;I.1 DEFINITIONS AND EXAMPLES;14
7;CHAPTER 1. THE DIFFUSION EQUATION;20
7.1;1.1 DERIVATION;20
7.2;1.2 PROBLEMS;22
7.3;1.3 SIMPLE SOLUTIONS;23
7.4;1.4 PROBLEMS;25
7.5;1.5 SERIES SOLUTIONS;26
7.6;1.6 PROBLEMS;29
7.7;1.7 NONHOMOGENEOUS END CONDITIONS;30
7.8;1.8 PROBLEMS;31
7.9;1.9 THE MAXIMUM PRINCIPLE;32
7.10;1.10 PROBLEMS;33
8;CHAPTER 2. LAPLACE TRANSFORM METHODS;36
8.1;2.1 INTRODUCTORY EXAMPLE;37
8.2;2.2 PROBLEMS;38
8.3;2.3 A FINITE INTERVAL PROBLEM;40
8.4;2.4 PROBLEMS;42
8.5;2.5 DELTA FUNCTION;43
8.6;2.6 PROBLEMS;46
8.7;2.7 SUPPLEMENTARY PROBLEMS;47
9;CHAPTER 3. THE WAVE EQUATION;48
9.1;3.1 DERIVATION;48
9.2;3.2 PROBLEMS;49
9.3;3.3 AN INFINITE-INTERVAL PROBLEM;51
9.4;3.4 PROBLEMS;55
9.5;3.5 SERIES SOLUTIONS;58
9.6;3.6 PROBLEMS;59
9.7;3.7 A PROBLEM WITH RADIAL SYMMETRY;61
9.8;3.8 PROBLEMS;62
9.9;3.9 TRANSFORMS;63
9.10;3.10 PROBLEMS;64
9.11;3.11 UNIQUENESS;65
9.12;3.12 SUPPLEMENTARY PROBLEMS;66
10;CHAPTER 4. THE POTENTIAL EQUATION;68
10.1;4.1 LAPLACE'S AND POISSON 'S EQUATIONS;68
10.2;4.2 PROBLEMS;71
10.3;4.3 SIMPLE PROPERTIES OF HARMONIC FUNCTIONS;72
10.4;4.4 SOME SPECIAL SOLUTIONS—SERIES;75
10.5;4.5 PROBLEMS;77
10.6;4.6 DISCONTINUOUS BOUNDARY DATA;79
10.7;4.7 COMPLEX VARIABLES AND CON FORMAL MAPPING;81
10.8;4.8 PROBLEMS;85
11;CHAPTER 5. CLASSIFICATION OF SECOND-ORDER EQUATIONS;88
11.1;5.1 CAUCHY DATA ON y-AXIS;88
11.2;5.2 CAUCHY DATA ON ARBITRARY CURVE;90
11.3;5.3 PROBLEMS;91
11.4;5.4 CASE I: B2 – AC > 0;92
11.5;5.5 CASE II: B2 – AC = 0;94
11.6;5.6 CASE III: B2 – AC < 0;95
11.7;5.7 PROBLEMS;96
11.8;5.8 DISCONTINUITIES; SIGNAL PROPAGATION;99
11.9;5.9 PROBLEMS;100
11.10;5.10 SOME REMARKS;102
12;CHAPTER 6. FIRST-ORDER EQUATIONS;104
12.1;6.1 LINEAR EQUATION EXAMPLES;104
12.2;6.2 PROBLEMS;107
12.3;6.3 QUASI-LINEAR CASE;108
12.4;6.4 PROBLEMS;110
12.5;6.5 FURTHER PROPERTIES OF CHARACTERISTICS;112
12.6;6.6 PROBLEMS;114
12.7;6.7 MORE VARIABLES;114
13;CHAPTER 7. EXTENSIONS;116
13.1;7.1 MORE VARIABLES;116
13.2;7.2 PROBLEMS;120
13.3;7.3 SERIES AND TRANSFORMS;125
13.4;7.4 PROBLEMS;128
13.5;7.5 LEGENDRE FUNCTIONS;129
13.6;7.6 PROBLEMS;134
13.7;7.7 SPHERICAL HARMONICS;134
13.8;7.8 PROBLEMS;138
14;CHAPTER 8. PERTURBATIONS;140
14.1;8.1 A NONLINEAR PROBLEM;140
14.2;8.2 PROBLEMS;143
14.3;8.3 TWO EXAMPLES FROM FLUID MECHANICS;143
14.4;8.4 BOUNDARY PERTURBATIONS;147
14.5;8.5 PROBLEMS;148
15;CHAPTER 9. GREEN'S FUNCTIONS;152
15.1;9.1 SOME CONSEQUENCES OF THE DIVERGENCE THEOREM;152
15.2;9.2 THE LAPLACIAN OPERATOR;155
15.3;9.3 PROBLEMS;157
15.4;9.4 POTENTIALS OF VOLUME AND SURFACE DISTRIBUTIONS;160
15.5;9.5 PROBLEMS;163
15.6;9.6 MODIFIED LAPLACIAN;165
15.7;9.7 PROBLEMS;167
15.8;9.8 WAVE EQUATION;169
15.9;9.9 PROBLEMS;171
16;CHAPTER 10. VARIATIONAL METHODS;174
16.1;10.1 A MINIMIZATION PROBLEM;174
16.2;10.2 PROBLEMS;177
16.3;10.3 NATURAL BOUNDARY CONDITIONS;180
16.4;10.4 SUBSIDIARY CONDITIONS;181
16.5;10.5 PROBLEMS;185
16.6;10.6 APPROXIMATE METHODS;187
16.7;10.7 PROBLEMS;191
16.8;10.8 FINITE-ELEMENT METHOD;193
16.9;10.9 SUPPLEMENTARY PROBLEMS;196
17;CHAPTER 11. EIGENVALUE PROBLEMS;202
17.1;11.1 A PROTOTYPE PROBLEM;202
17.2;11.2 SOME EIGENVALUE PROPERTIES;203
17.3;11.3 PROBLEMS;206
17.4;11.4 PERTURBATIONS;210
17.5;11.5 APPROXIMATIONS;212
17.6;11.6 PROBLEMS;214
18;CHAPTER 12. MORE ON FIRST-ORDER EQUATIONS;216
18.1;12.1 ENVELOPES;216
18.2;12.2 CHARACTERISTIC STRIPS;218
18.3;12.3 COMPLETE INTEGRAL;222
18.4;12.4 PROBLEMS;225
18.5;12.5 LEGENDRE TRANSFORMATION;227
18.6;12.6 PROBLEMS;228
18.7;12.7 PROPAGATION OF A DISTURBANCE;230
18.8;12.8 COMPLETE INTEGRAL AND EIKONAL FUNCTION;233
18.9;12.9 HAMILTON-JACOBI EQUATION;235
18.10;12.10 PROBLEMS;237
19;CHAPTER 13. MORE ON CHARACTERISTICS;240
19.1;13.1 DISCONTINUITIES—A PRELIMINARY EXAMPLE;240
19.2;13.2 WEAK SOLUTIONS;243
19.3;13.3 BURGERS' EQUATION;246
19.4;13.4 PROBLEMS;249
19.5;13.5 A COMPRESSIBLE FLOW PROBLEM;252
19.6;13.6 A NUMERICAL APPROACH;255
19.7;13.7 PROBLEMS;259
19.8;13.8 MORE DEPENDENT VARIABLES;263
19.9;13.9 MORE INDEPENDENT VARIABLES;266
19.10;13.10 PROBLEMS;268
20;CHAPTER 14. FINITE-DIFFERENCE EQUATIONS AND NUMERICAL METHODS;270
20.1;14.1 ACCURACY AND STABILITY; A DIFFUSION EQUATION EXAMPLE;270
20.2;14.2 ERROR ANALYSIS;272
20.3;14.3 PROBLEMS;275
20.4;14.4 MORE DIMENSIONS, OR OTHER COMPLICATIONS;277
20.5;14.5 SERIES EXPANSIONS;280
20.6;14.6 PROBLEMS;282
20.7;14.7 WAVE EQUATION;283
20.8;14.8 A NONLINEAR EQUATION;285
20.9;14.9 PROBLEMS;286
20.10;14.10 BOUNDARY VALUE PROBLEMS;287
20.11;14.11 PROBLEMS;291
20.12;14.12 SERIES; FAST FOURIER TRANSFORM;291
20.13;14.13 PROBLEMS;294
21;CHAPTER 15. SINGULAR PERTURBATION METHODS;298
21.1;15.1 A BOUNDARY LAYER PROBLEM;298
21.2;15.2 A MORE GENERAL PROCEDURE;300
21.3;15.3 PROBLEMS;302
21.4;15.4 A TRANSITION SITUATION;305
21.5;15.5 PROBLEMS;308
21.6;15.6 ASYMPTOTIC ANALYSIS OF WAVE MOTION;310
21.7;15.7 BOUNDARY LAYER NEAR A CAUSTIC;313
21.8;15.8 PROBLEMS;316
21.9;15.9 MULTIPLE SCALING;316
21.10;15.10 PROBLEMS;319
22;REFERENCES;322
23;INDEX;326