Introduction to Partial Differential Equations

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Chapter 2

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

 Multiply-valued solution Equal area rule

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

Example 2.19. Forcing and resonance   — page 59

 Periodic solution Quasiperiodic solution Resonant solution

Chapter 4

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

 Slow time scale Faster time scale

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

 Dirichlet boundary conditions Neumann boundary conditions

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

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.

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.

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

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

 Δx = Δt = .0005         c = σ = .5

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

Chapter 8

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

 γ = .25

 γ = .1

 γ = .025