Elements of Partial Differential Equations Pavel Drábek / Gabriela Holubová ISBN:...

Elements of Partial Differential EquationsPavel Drábek / Gabriela Holubová ISBN: 978-3-11-031665-0

© 2014 Walter de Gruyter GmbH, Berlin/Boston

Abbildungsübersicht / List of Figures

Tabellenübersicht / List of Tables

Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0


2

Figure 1.1. An isolated tube with cross-section A; the quantities considered change only in the direction of the x-axis.



3

Figure 1.2. Traveling wave.



4

Figure 1.3. String segment.



5

Figure 1.4. Vibrating membrane over the subdomain Ω.



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Figure 3.1. Characteristic lines.



7

Figure 3.2. Solution from Example 3.1.



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9

Figure 3.4. Solution of ut +3ux = 0, u(x, 0) = sin x.



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Figure 3.5. Transformation of the coordinate system.



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Figure 3.6. Vector field v = (1, y).



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Figure 3.7. Characteristics of the equation ux + 2xy2uy = 0.



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Figure 3.9. Solution of the problem ux + uy = 0, u(cos s, sin s) = s, s [0, π/2].∈



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Figure 3.13. Solution of the problem ut + 2ux = −3u, u(x, 0) = 1/(1 + x2).



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Figure 4.1. Standing waves – a solution of the initial value problem (4.9) with c = 4.



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Figure 4.2. Solution of Example 4.4 on particular time levels.



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Figure 4.3. Graph of solution from Example 4.4 for c = 2, b = 1, a = 2.



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Figure 4.4. Solution of Example 4.5 on particular time levels.



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Figure 4.5. Graph of solution from Example 4.5 for c = 2.3, a = 1.3.



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Figure 4.6. Domain of influence of the point (x0, 0) at time t ≥ 0.



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Figure 4.7. Domain of influence of the interval (−R, R) at time t ≥ 0.



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Figure 4.8. Domain of dependence (characteristic triangle) of the point (x, t).



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Figure 4.9. Characteristic triangle of the point (x0, t0).



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Figure 4.10. Solution of problem (4.19).



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Figure 5.1. Temperature profile on several time levels for a step initial temperature.



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Figure 5.2. Fundamental solution of the diffusion equation (here, with the choice k = 0.5).



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Figure 5.3. Solution of Example 5.4 with k = 2.



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Figure 6.1. Polar coordinates r and θ.



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Figure 7.1. The odd extension.



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Figure 7.2. Solution of problem (7.5) with k = 1.



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Figure 7.3. Reflection method for the wave equation.



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Figure 7.4. Solution of problem (7.10) for c = 2.



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Figure 7.5. Solution of problem (7.11) for c = 4.



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Figure 7.6. Reflection method for the wave equation on finite interval.



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Figure 7.7. Graphic illustration of the solution of problem (7.26).



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Figure 7.8. Graphic illustration of the solution of Example 7.8.



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Figure 7.9. Graphic illustration of the solution of problem (7.38).



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Figure 7.10. Schematic illustration of problem (7.39).



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Figure 7.12. Graphic illustration of the solution u(x, t) of problem (7.39) for h = 1, k = 1.



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Figure 8.1. The rectangle R and boundary conditions of (8.1).



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Figure 8.2. Decomposition of the nonhomogeneous boundary value problem for the Laplace equation on a rectangle.



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Figure 9.1. A string falling due to the gravitation.



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Figure 10.1. Domain of influence of the point (x0, 0) and domain of dependence of the point (x, t).



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Figure 10.2. Trapezoid of characteristic triangle.



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Figure 10.4. Covering of Ω by circular neighborhoods.



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Figure 10.5. Point (x, y) and its “neighbors”.



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Figure 11.1. Spherical coordinates.



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Figure 11.2. Half-space and reflection method.



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Figure 11.3. Verification of the property (ii) of Green’s function.



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Figure 11.4. Ball Ω and spherical inversion.



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Figure 11.5. Congruent triangles and the proportionality of ρ and ρ .∗



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Figure 12.1. Graphic illustration of the solution of the initial boundary value problem (12.11) with constant initial condition on

time levels t = 0, 0.01, 0.04, 0.09.



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Figure 12.2. Graphic illustration of the solution of the initial boundary value problem (12.15) with initial condition (12.16) on

time levels t = 0, 0.01, 0.04, 0.09.



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Figure 12.3. Space-time cylinder Ω X [0, T].



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Figure 13.1. “Hammer blow” in one, two and three dimensions.



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Figure 13.3. Solid cone frustum F.



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Figure 13.4. Initial condition (13.23) with the choice a = 2, b = 3.



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Figure 13.5. Graphic illustration of the solution of the initial boundary value problem (13.22) for the data c = 3, a = 2, b = 3,

on time levels t = 0, 0.2, 0.4, 0.8.



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Figure 13.6. The initial displacement (13.29).



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time levels t = 0, 0.4, 0.8, 1.2.



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time levels t = 0, 0.4, 0.8, 1.2.



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Figure 13.9. Graphic illustration of the solution of the initial boundary value problem (13.39) (symmetric vibrations in a unit

ball) with initial condition (13.42) – dependence on r and t.



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Figure 13.10. Graphic illustration of the solution of the initial boundary value problem (13.27) (symmetric vibrations in a unit

disc) with initial condition (13.42) – dependence on r and t.



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Figure 13.11. Radially symmetric solutions of the Dirichlet

problem for the wave equation in a disc (2D) and in a ball

(3D) with the initial condition ψ(r) = 1 for 0 ≤ r ≤ 1 and zero

otherwise.



100

Figure B.1. Bessel functions of the first kind for n = 0, 1, 2.



101

Figure B.2. Bessel functions of the second kind for n = 0, 1, 2.

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Elements of Partial Differential Equations Pavel Drábek / Gabriela Holubová ISBN:...

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