Introduction to Partial Differential Equations

by Peter J. Olver

Springer-Verlag, Undergraduate Texts in Mathematics, New York, 2014

Click here for the book's local web site

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Δx = .1 Δt = .01 μ = 1.0	Δx = .1 Δt = .005 μ = .5

Δx = .01 Δt = .0001 μ = 1.0	Δx = .01 Δt = .00005 μ = .5

Δx = Δt = .0005 c = σ = .5	Δx = Δt = .0005 c = σ = -.5

Δx = Δt = .0005 c = σ = -1	Δx = Δt = .0005 c = σ = -1.5

c = 1.0 Δx = Δt = .01 σ = 1.0	c = 1.0 Δx = .01 Δt = .02 σ = 1.8

c = 1.0 Δx = .0111111 Δt = .01 σ = .9	c = 1.0 Δx = .0090909 Δt = .01 σ = 1.1

Multiply-valued solution	Equal area rule

Slow time scale	Faster time scale

Dirichlet boundary conditions

Neumann boundary conditions

Introduction to Partial Differential Equations

by Peter J. Olver

Springer-Verlag, Undergraduate Texts in Mathematics, New York, 2014

Click here for the book's local web site

Movies

Chapter 2

Figure 2.1. Stationary wave — page 16

Figure 2.3. Traveling wave with c > 0 — page 20

Figure 2.5. Decaying traveling wave — page 22

Figure 2.8. Solution to ut + ux / (x2 + 1) = 0 — page 27

Figure 2.10. Solution to ut + (x2 - 1) ux = 0 — page 29

Figure 2.11. Two solutions to ut + u ux = 0 — page 33

Figure 2.14. Rarefaction wave — page 35

Figure 2.15. Multiply-valued compression wave — page 36

Figure 2.17. Multiply-valued step wave — page 40

Figure 2.20. Rarefaction wave — page 43

Figure 2.21. Equal Area Rule for the triangular wave — page 44

Figure 2.22. Triangular-wave solution — page 45

Figure 2.24. Splitting of waves — page 53

Figure 2.25. Interaction of waves — page 54

Equation (2.84): Particles and waves — page 55

Figure 2.27. Error function solution to the wave equation — page 56

Example 2.19. Forcing and resonance — page 59

Chapter 4

Figure 4.1. A solution to the heat equation — page 127

Figure 4.2. Denoising a signal with the heat equation — page 128

Heat equation with periodic boundary conditions — pages 130-131

Figure 4.3. Plucked string solution of the wave equation — page 143

Figure 4.6. Solution to wave equation with fixed ends — page 148

Odd periodic extension of preceding solution

Chapter 5

Figure 5.2. Numerical solutions for the heat equation based on the explicit scheme — page 189

Figure 5.3. Numerical solutions for the heat equation based on the implicit scheme — page 191

Figure 5.4. Numerical Solutions for the heat equation based on the Crank-Nicolson scheme — page 192

Figure 5.5. Numerical solutions to the transport equation — page 196

Figure 5.8. Centered difference numerical solution to the transport equation — page 200

Figures 5.9 and 5.10. Numerically stable and unstable waves — page 204

Chapter 8

Figure 8.1. The fundamental solution to the one-dimensional heat equation — page 294

Figure 8.2. Error function solution to the heat equation — page 296

Figure 8.3. Effect of a concentrated heat source — page 299

Figure 8.4. Solution to the Black-Scholes equation — page 302

Figure 8.5. Traveling-wave solutions to Burgers' equation — page 317

Figure 8.6. Trignometric solution to Burgers' equation — page 319

Figure 8.7. Shock-wave solution to Burgers' equation — page 321

Figure 8.8. Triangular-wave solution to Burgers' equation — page 322

Figure 8.9. Gaussian solution to the dispersive wave equation — page 325

Figure 8.10. Fundamental solution to the dispersive wave equation — page 327

Figure 8.11. Periodic dispersion at irrational (with respect to π) times — page 328

Figure 8.12. Periodic dispersion at rational (with respect to π) times — page 329

Figure 8.13. Solitary wave/soliton — page 334

Figure 8.14. Interaction of two solitons — page 335

Interaction of three solitons — page 336

Chapter 11

Figure 11.2. Heat diffusion in a rectangle — page 448

Figure 11.7. Heat diffusion in a disk — page 478

Figure 11.8. Fundamental solution of the planar heat equation — page 483

Figure 11.9. Diffusion of a disk — page 484

Figure 11.10. Vibrations of a square — page 489

Figure 11.11. Vibrations of a disk — page 491

Chapter 12

Figure 12.10. Wave equation solution u(t,r) due to an initial velocity of the unit ball — page 557

Figure 12.11. Wave equation solution u(t,r) due to an initial displacement of the unit ball — page 559

Figure 12.12. Solution to the two-dimensional wave equation for a concentrated impulse — page 563

Figure 2.8. Solution to u_t + u_x / (x² + 1) = 0 — page 27

Figure 2.10. Solution to u_t + (x² - 1) u_x = 0 — page 29

Figure 2.11. Two solutions to u_t + u u_x = 0 — page 33