Similarity Solutions of Nonlinear Partial Differential Equations (Research Notes in Mathematics Series)

Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Dedication......Page 7
Preface......Page 9
Contents......Page 11
1 INTRODUCTION......Page 13
2.1 First-order ordinary differential equations......Page 17
2.2 Lie's formula for the integrating factor......Page 18
2.3 Example......Page 19
2.4 Analysis of the direction field......Page 20
2.5 Separatrices......Page 22
2.6 Separation of variables......Page 23
2.7 Second-order ordinary differential equations......Page 24
2.8 Example: Emden-Fowler equations......Page 26
3.1 Birkhoff's idea......Page 29
3.3 Concentration polarization......Page 33
4 NONLINEAR DIFFUSION......Page 37
4. 1 Boltzmann's roblem: C_t=(CC_z)_z......Page 0
4.2 Clamped flux......Page 39
4.3 The associated group......Page 43
4.4 Clamped concentration (temperature)......Page 45
4 .5 C_t = (C^nC_z )_z......Page 46
4 .7 CC_t = C_zz......Page 47
4.9 Transient heat transfer in superfluid helium......Page 53
4.10 Clamped flux......Page 54
4.11 Clamped temperature......Page 57
4.13 Isothermal percolation of a turbulent liquid into a porous half-space......Page 59
4.14 Other groups......Page 62
4.15 Resemblance to dimensional analysis......Page 64
5.1 Prandtl-Blasius problem of a flat plate......Page 67
5.2 Blasius's differential equation......Page 69
5.3 The associated differential equation......Page 71
5.4 Flat plate with uniform suction or injection......Page 75
5.5 Thermal boundary layers......Page 77
5.6 Free convection boundary layer......Page 79
6.1 Introduction......Page 81
6.2 von Karman-Duwez-Taylor problems......Page 82
6.3 Elastic (Hookean) wire......Page 83
6.4 Non-Hookean wire......Page 86
6.5 Characteristics and Riemann invariants......Page 89
6.6 Shock conditions......Page 93
6.7 "Superelastic" wire......Page 94
6.8 Transverse waves......Page 99
6.9 Elastic (Hookean) wire - transverse waves......Page 104
6.10 Long waves in a channel......Page 107
6.11 Travelling waves......Page 109
7.1 Approximate solutions: diffusion in cylindrical geometry......Page 113
7.2 Diffusion in cylindrical geometry (cont'd): clamped temperature......Page 117
7.3 The method of local similarity......Page 119
7.4 Eigenvalue problems......Page 123
PROBLEMS......Page 125
SOLUTIONS TO PROBLEMS......Page 128
REFERENCES......Page 133
INDEX......Page 135
Back Cover......Page 137