1. Elements of Partial Differential Equations Pavel Drábek / Gabriela Holubová ISBN: 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Abbildungsübersicht / List of Figures Tabellenübersicht / List of Tables

2. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 2 Figure 1.1. An isolated tube with cross-section A; the quantities considered change only in the direction of the x-axis.

3. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 3 Figure 1.2. Traveling wave.

4. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 4 Figure 1.3. String segment.

5. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 5 Figure 1.4. Vibrating membrane over the subdomain Ω.

6. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 6 Figure 3.1. Characteristic lines.

7. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 7 Figure 3.2. Solution from Example 3.1.

8. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 8

9. 9 Figure 3.4. Solution of u t +3u x = 0, u(x, 0) = sin x.

10. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 10 Figure 3.5. Transformation of the coordinate system.

11. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 11 Figure 3.6. Vector field v = (1, y).

12. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 12 Figure 3.7. Characteristics of the equation u x + 2xy 2 u y = 0.

13. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 13

14. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 14

15. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 15 Figure 3.9. Solution of the problem u x + u y = 0, u(cos s, sin s) = s, s ∈ [0, π/2].

16. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 16

17. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 17

18. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 18

19. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 19 Figure 3.13. Solution of the problem u t + 2u x = −3u, u(x, 0) = 1/(1 + x 2 ).

20. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 20

21. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 21 Figure 4.1. Standing waves – a solution of the initial value problem (4.9) with c = 4.

22. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 22 Figure 4.2. Solution of Example 4.4 on particular time levels.

23. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 23 Figure 4.3. Graph of solution from Example 4.4 for c = 2, b = 1, a = 2.

24. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 24 Figure 4.4. Solution of Example 4.5 on particular time levels.

25. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 25 Figure 4.5. Graph of solution from Example 4.5 for c = 2.3, a = 1.3.

26. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 26 Figure 4.6. Domain of influence of the point (x 0, 0) at time t ≥ 0.

27. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 27 Figure 4.7. Domain of influence of the interval (−R, R) at time t ≥ 0.

28. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 28 Figure 4.8. Domain of dependence (characteristic triangle) of the point (x, t).

29. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 29

30. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 30 Figure 4.9. Characteristic triangle of the point (x 0, t 0 ).

31. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 31 Figure 4.10. Solution of problem (4.19).

32. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 32 Figure 5.1. Temperature profile on several time levels for a step initial temperature.

33. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 33 Figure 5.2. Fundamental solution of the diffusion equation (here, with the choice k = 0.5).

34. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 34

35. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 35 Figure 5.3. Solution of Example 5.4 with k = 2.

36. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 36

37. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 37 Figure 6.1. Polar coordinates r and θ.

38. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 38 Figure 7.1. The odd extension.

39. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 39 Figure 7.2. Solution of problem (7.5) with k = 1.

40. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 40 Figure 7.3. Reflection method for the wave equation.

41. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 41 Figure 7.4. Solution of problem (7.10) for c = 2.

42. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 42 Figure 7.5. Solution of problem (7.11) for c = 4.

43. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 43

44. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 44 Figure 7.6. Reflection method for the wave equation on finite interval.

45. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 45 Figure 7.7. Graphic illustration of the solution of problem (7.26).

46. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 46 Figure 7.8. Graphic illustration of the solution of Example 7.8.

47. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 47 Figure 7.9. Graphic illustration of the solution of problem (7.38).

48. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 48 Figure 7.10. Schematic illustration of problem (7.39).

49. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 49

50. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 50

51. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 51 Figure 7.12. Graphic illustration of the solution u(x, t) of problem (7.39) for h = 1, k = 1.

52. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 52 Figure 8.1. The rectangle R and boundary conditions of (8.1).

53. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 53 Figure 8.2. Decomposition of the nonhomogeneous boundary value problem for the Laplace equation on a rectangle.

54. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 54

55. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 55

56. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 56 Figure 9.1. A string falling due to the gravitation.

57. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 57

58. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 58 Figure 10.1. Domain of influence of the point (x 0, 0) and domain of dependence of the point (x, t).

59. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 59 Figure 10.2. Trapezoid of characteristic triangle.

60. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 60

61. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 61

62. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 62

63. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 63

64. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 64

65. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 65

66. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 66

67. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 67

68. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 68 Figure 10.4. Covering of Ω by circular neighborhoods.

69. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 69

70. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 70 Figure 10.5. Point (x, y) and its “neighbors”.

71. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 71

72. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 72 Figure 11.1. Spherical coordinates.

73. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 73

74. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 74

75. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 75

76. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 76

77. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 77

78. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 78

79. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 79 Figure 11.2. Half-space and reflection method.

80. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 80 Figure 11.3. Verification of the property (ii) of Green’s function.

81. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 81 Figure 11.4. Ball Ω and spherical inversion.

82. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 82 Figure 11.5. Congruent triangles and the proportionality of ρ and ρ ∗.

83. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 83

84. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 84 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.

85. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 85 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.

86. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 86 Figure 12.3. Space-time cylinder Ω X [0, T].

87. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 87

88. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 88 Figure 13.1. “Hammer blow” in one, two and three dimensions.

89. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 89

90. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 90

91. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 91 Figure 13.3. Solid cone frustum F.

92. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 92 Figure 13.4. Initial condition (13.23) with the choice a = 2, b = 3.

93. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 93 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.

94. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 94 Figure 13.6. The initial displacement (13.29).

95. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 95 Figure 13.7. Graphic illustration of the solution of the initial boundary value problem (13.27) with initial condition (13.29) on time levels t = 0, 0.4, 0.8, 1.2.

96. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 96 Figure 13.8. Graphic illustration of the solution of the initial boundary value problem (13.30) with initial condition (13.33) on time levels t = 0, 0.4, 0.8, 1.2.

97. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 97 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.

98. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 98 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.

99. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 99 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. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 100 Figure B.1. Bessel functions of the first kind for n = 0, 1, 2.

101. Elements of Partial Differential Equations, Pavel Drábek / Gabriela Holubová ISBN 978-3-11-031665-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 101 Figure B.2. Bessel functions of the second kind for n = 0, 1, 2.