Solution Manual for Introduction to Partial Differential Equations by Olver – 2014 | All 12 Chapters Covered

Covers All 12 Chapters




SOLUTIONS MANUAL

,Selected Solutions Manual
for Instructors
for

Introduction to Partial
Differential Equations
by
Peter J. Olver

Undergraduate Texts in Mathematics
Springer, 2014

ISBN 978–3–319–02098–3


c 2020 Peter J. Olver

,Table of Contents


Chapter 1. What Are Partial Differential Equations? . . . . . . . . . . 1

Chapter 2. Linear and Nonlinear Waves . . . . . . . . . . . . . . . . 3

Chapter 3. Fourier Series . . . . . . . . . . . . . . . . . . . . . 14

Chapter 4. Separation of Variables . . . . . . . . . . . . . . . . . 29

Chapter 5. Finite Differences . . . . . . . . . . . . . . . . . . . 43

Chapter 6. Generalized Functions and Green’s Functions . . . . . . . 60

Chapter 7. Fourier Transforms . . . . . . . . . . . . . . . . . . . 71

Chapter 8. Linear and Nonlinear Evolution Equations . . . . . . . . 77

Chapter 9. A General Framework for
Linear Partial Differential Equations . . . . . . 87

Chapter 10. Finite Elements and Weak Solutions . . . . . . . . . . . 99

Chapter 11. Dynamics of Planar Media . . . . . . . . . . . . . . . 110

Chapter 12. Partial Differential Equations in Space . . . . . . . . . . 126




c 2020 Peter J. Olver

, Selected Solutions to
Chapter 1: What Are Partial Differential Equations?

Note: Solutions marked with a ⋆ do not appear in the Student Solutions Manual.



1.1. (a) Ordinary differential equation, equilibrium, order = 1;
(c) partial differential equation, dynamic, order = 2;
(e) partial differential equation, equilibrium, order = 2;
⋆ (g) partial differential equation, equilibrium, order = 2;
⋆ (i) partial differential equation, dynamic, order = 3;
⋆ (k) partial differential equation, dynamic, order = 4.
∂2u ∂2u
1.2. (a) (i) + = 0, (ii) uxx + uyy = 0;
∂x2 ∂y2
∂u ∂2u ∂2u
⋆ (c) (i)
∂t
=
∂x2
+
∂y2
, (ii) ut = uxx + uyy .

1.4. (a) independent variables: x, y; dependent variables: u, v; order = 1;
⋆ (b) independent variables: x, y; dependent variables: u, v; order = 2;
⋆ (d) independent variables: t, x, y; dependent variables: u, v, p; order = 1.



∂2u ∂2u
1.5. (a) + = ex cos y − ex cos y = 0; defined and C∞ on all of R 2 .
∂x2 ∂y2
∂2u ∂2u
⋆ (c) + = 6 x − 6 x = 0; defined and C∞ on all of R 2 .
∂x2 ∂y2
∂2u ∂2u 2 y2 − 2 x2 2 x2 − 2 y2
⋆ (d) + = + = 0; defined and C∞ on R 2 \ {0}.
∂x2 ∂y2 (x2 + y2 )2 (x2 + y2 )2
h i
1.7. u = log c (x − a)2 + c (y − b)2 , for a, b, c arbitrary constants.

⋆ 1.8. (a) c0 + c1 x + c2 y + c3 z + c4 (x2 − y2 ) + c5 (x2 − z 2 ) + c6 x y + c7 x z + c8 y z,
where c0 , . . . , c8 are arbitrary constants.
∂2u ∂2u ∂2u ∂2u
1.10. (a)
∂t2
− 4
∂x2
= 8 − 8 = 0; ⋆ (c) ∂t2
− 4
∂x2
= −4 sin 2 t cos x + 4 sin 2 t cos x = 0.

1.11. (a) c0 + c1 t + c2 x + c3 (t2 + x2 ) + c4 t x, where c0 , . . . , c4 are arbitrary constants.
b b
⋆ 1.13. u = a +
r
=a+ q , where a, b are arbitrary constants.
x2 + y2 + z 2

c 2020 Peter J. Olver