Introduction to Partial Differential Equations

by Peter J. Olver

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

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Chapter 2

Figure 2.1.   Stationary wave  —  page 16

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

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Figure 2.5.   Decaying traveling wave  —  page 22

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

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

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

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Figure 2.14.   Rarefaction wave  —  page 35

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Figure 2.15.   Multiply-valued compression wave  —  page 36

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Figure 2.17.   Multiply-valued step wave  —  page 40

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Figure 2.20.   Rarefaction wave  —  page 43

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

Multiply-valued solution	Equal area rule

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Figure 2.22.   Triangular-wave solution  —  page 45

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Figure 2.24.   Splitting of waves  —  page 53

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Figure 2.25.   Interaction of waves  —  page 54

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Equation (2.84): Particles and waves  —  page 55

The wave solution u(t,x) = cos t sin x = (sin(x-t) + sin(x+t))/2

Constitutent traveling waves (particles)

Particles and Waves

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

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Example 2.19.   Forcing and resonance  —  page 59

Periodic solution

Quasiperiodic solution

Resonant solution

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Chapter 4

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

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

Slow time scale	Faster time scale

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Heat equation with periodic boundary conditions  —  pages 130-131

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

Dirichlet boundary conditions
Neumann boundary conditions

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

Odd periodic extension of preceding solution

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Chapter 5

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

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

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

Δx = .1         Δt = .01         μ = 1.0	Δx = .01         Δt = .01         μ = 100.

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

Δx = .1         Δt = .01         μ = 1.0	Δx = .01         Δt = .01         μ = 100.

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

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

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

Δx = Δt = .0005         c = σ = .5

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

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

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Chapter 8

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

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

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

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

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

γ = .25

γ = .1

γ = .025

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

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

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

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

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

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

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

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Figure 8.13.   Solitary wave/soliton  —  page 334

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Figure 8.14.   Interaction of two solitons  —  page 335

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Interaction of three solitons  —  page 336

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Chapter 11

Figure 11.2.   Heat diffusion in a rectangle  —  page 448

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Figure 11.7.   Heat diffusion in a disk  —  page 478

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

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Figure 11.9.   Diffusion of a disk  —  page 484

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Figure 11.10.   Vibrations of a square  —  page 489

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Figure 11.11.   Vibrations of a disk  —  page 491

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Chapter 12

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

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

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

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