1 Introduction

1

Page References

2

How to Use this Book

3

Plotting Functions

2 Graphical Facilities

5

Downloading the Free FlexPDE Program

5

The FlexPDE Editor

6

Plotting Functions of One Variable

6

Running the Problem Descriptor File

8

Plotting Functions of Two Variables

9

Purging FlexPDE Files

14

Help and Manual

15

3 "Curly" Velocity Fields

16

Liquid, Rotating as a Disk

16

Non-Constant w

18

More General Velocity Fields

23

4 Fields of Gravitation

25

Earth and Moon as Point Masses

25

Planets of Finite Size

27

Divergence of g

32

Electricity

5 Fields around Electric Charges

34

Field around Two Positive Point Charges

34

Positive and Negative Point Charges

36

The Dipole Field

38

Field around Charged Wires

39

Dipole of Charged Wires

41

The Gauss Integral

42

6 Laplace and Poisson Equations

45

Known Values on Boundaries

45

Values and Derivatives on Boundaries

47

Multiplying through the PDE

48

Derivative Boundary Conditions Only

49

Solution Over a Quarter-Circle

50

The Poisson Equation

51

7 Electrostatic Fields in (x,y) Space

53

Metal Rod in a Metal Box

53

Metal Bar in a Tube

55

Metal Rod across a Parallel Field

56

8 Electrical Conduction in (x, y) Space

60

Conduction in a Rectangular Plate

61

Conduction in a Trapezoidal Plate

62

Checking the Solution

64

Radial Conduction in a Foil

65

Constricted Rectangular Plate

66

Plate Made of Two Different Metals

68

Plate with an Elliptic Insert

70

9 Dielectrics in (x, y) Space

73

Coaxial Cable

73

Capacitance

75

Parallel Plate Capacitor

75

Exploiting Symmetry

80

Glass Rod across a Parallel Field

81

Surface Charge of Polarization

84

Magnetism

10 Magnetostatics in (x, y) Space

87

Magnetic Field around a Wire

88

Field around Two Wires

91

Boundary Conditions for Magnetic Vectors

93

Model of a Permanent Magnet

95

Heat Transport

11 Heat Conduction in (x, y) Space

98

Two Hot-Water Tubes

99

Uniformly Heated, Semi-Circular Rod

101

Cooling by Forced Convection

103

Conduction in Anisotropic Wood

105

12 Non-Linear Heat Transfer

108

Temperature-Dependent Conductivity

108

Steel Foil Emitting Infrared Radiation

111

13 Transient in One Dimension

114

Iron Bar with a Temperature Step

114

Bar with a Temperature Step Halfway

117

Iron Bar Soldered to a Copper Bar

118

Ramp Function at a Boundary

120

Internally Heated Steel Bar with a Loss

122

14 Time-Sinusoidal Problems

125

Oscillating Temperature in a Steel Block

126

Animation of an Oscillating Temperature

130

Sinusoidal Volume Heating of a Steel Foil

131

Problems in 3D Space

15 Fields in (x,y,z)

134

Two Point Charges of Different Signs

134

Non-Linear Set of Point Charges

137

Coin in a Metal Box

138

Electrical Conduction in a Cone

143

Glass Block in a Parallel Electric Field

145

Steady Heat Conduction

148

Temperature Transients

151

Electromagnetic Waves

16 Electromagnetic Waves in (x,y)

154

Constant Permeability

154

Complex Formalism

156

Plane Wave in a Conducting Material

156

Interpretation of Complex Amplitudes

161

Animation of a Plane Wave

161

Non-Uniform Conductivity

163

Plane Wave in Two Different Media

164

Scanning for Resonance Frequencies

166

The Magnetic Wave Field

167

Running Wave

170

17 Resonators in (x,y)

174

Loss-Free Rectangular Cavity

174

Resonance Model

178

Eigenvalues and Eigenstates

180

Cavity of Circular Cross-Section

182

Degenerate Eigenstates

184

Wave Mechanics

18 Elementary Wave Mechanics in 1D

189

The Schrödinger Wave Equations

189

Particle Transmission

190

Particle Scattering

193

Bound Particle

197

19 Harmonic Oscillators in 1D and 2D

203

Oscillator in One Dimension

203

Expectation Values

205

Harmonic Oscillator in Two Dimensions

208

Degenerate Eigenstates

210

Fluid Dynamics

20 Irrotational Flow of Liquids in (x,y)

214

Flow through a Constricted Channel

215

Cylindrical Obstacle across a Straight Channel

219

Obstacle Close to a Wall

222

Drag and Lift on an Inclined Plate

223

21 Viscous Flow in Channels

228

Boundary Conditions

231

Steady Flow at Small Speeds (Re<<1)

232

Flow Due to a Moving Wall

232

Pressure-Driven Flow through a Channel

235

Viscous Flow through a Constricted Channel

237

Comparison with Irrotational Flow

242

Tangential Input Velocity

243

Channel with a Lateral Cavity

244

Uniform Velocity of Injection

246

22 Viscous Flow past an Obstacle

249

Viscous Flow past a Circular Cylinder

249

Viscous Force on a Solid Surface

252

Forces on a Circular Cylinder

253

Force Equilibrium

255

Drag and Lift on an Inclined Plate

256

23 Seeping through Porous Materials

259

Simple Percolation in (x,y) Space

259

Percolation in (x,y) by Navier-Stokes PDE

261

24 Viscous Flow at Re>>1 in (x,y)

266

Viscous Flow in a Channel

268

Viscous Flow past a Circular Cylinder

270

Viscous Boundary Layer

272

Viscous Flow past an Inclined Plate

275

25 Viscous Flow in Three Dimensions

280

Extension of the N-S Formalism to 3D

280

Flow through a Rectangular Duct

281

Flow through a Box with Two Orifices

285

Viscous Flow around a Cubical Obstacle

288

Viscous Flow by Gravity through a Funnel

291

Seeping through a Concrete Plate with a Pillar

294

Appendix

Principles of Finite Element Analysis

298

Interpolation and Differentiationin 2D

298

Interpolating to Obtain a Solution

300

Solving for Node Values

301

Natural Boundary Conditions

304

Conclusion

305

References

306

Vocabulary

307