PARTIAL DIFFERENTIAL EQUATIONS - UC Santa Barbara

PARTIAL DIFFERENTIAL EQUATIONS

Math 124A ? Fall 2010

?Viktor Grigoryan

grigoryan@math.ucsb.edu Department of Mathematics University of California, Santa Barbara

These lecture notes arose from the course "Partial Differential Equations" ? Math 124A taught by the author in the Department of Mathematics at UCSB in the fall quarters of 2009 and 2010. The selection of topics and the order in which they are introduced is based on [Str]. Most of the problems appearing in this text are also borrowed from Strauss. A list of other references that were consulted while teaching this course appears in the bibliography at the end. These notes are copylefted, and may be freely used for noncommercial educational purposes. I will appreciate any and all feedback directed to the e-mail address listed above. The most up-to date version of these notes can be downloaded from the URL given below.

Version 0.1 - December 2010

Contents

1 Introduction

1

1.1 Classification of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Problem Set 1

4

2 First-order linear equations

5

2.1 The method of characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 General constant coefficient equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Variable coefficient equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Method of characteristics revisited

11

3.1 Transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Quasilinear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Rarefaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Problem Set 2

15

4 Vibrations and heat flow

16

4.1 Vibrating string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 Vibrating drumhead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 Heat flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.4 Stationary waves and heat distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.6 Examples of physical boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Classification of second order linear PDEs

21

5.1 Hyperbolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 Parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.3 Elliptic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Problem Set 3

26

6 Wave equation: solution

27

6.1 Initial value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.2 The Box wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.3 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7 The energy method

34

7.1 Energy for the wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.2 Energy for the heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Problem Set 4

38

i

8 Heat equation: properties

39

8.1 The maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

8.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9 Heat equation: solution

43

9.1 Invariance properties of the heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9.2 Solving a particular IVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9.3 Solving the general IVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Problem Set 5

47

10 Heat equation: interpretation of the solution

48

10.1 Dirac delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

10.2 Interpretation of the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

11 Comparison of wave and heat equations

53

11.1 Comparison of wave to heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

11.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Problem Set 6

57

12 Heat conduction on the half-line

58

12.1 Neumann boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

12.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

13 Waves on the half-line

62

13.1 Neumann boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

13.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

14 Waves on the finite interval

66

14.1 The parallelogram rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

14.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Problem Set 7

70

15 Heat with a source

71

15.1 Source on the half-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

15.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

16 Waves with a source

75

16.1 Source on the half-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

16.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

17 Waves with a source: the operator method

79

17.1 The operator method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

17.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Problem Set 8

83

ii

18 Separation of variables: Dirichlet conditions

84

18.1 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

18.2 Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

18.3 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

18.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

19 Separation of variables: Neumann conditions

88

19.1 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

19.2 Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

19.3 Mixed boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

19.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Problem Set 9

91

Bibliography

92

iii

1 Introduction

Recall that an ordinary differential equation (ODE) contains an independent variable x and a dependent

variable u, which is the unknown in the equation. The defining property of an ODE is that derivatives

of the unknown function u

=

du dx

enter the equation.

Thus, an equation that relates the independent

variable x, the dependent variable u and derivatives of u is called an ordinary differential equation. Some

examples of ODEs are:

u (x) = u u + 2xu = ex u + x(u )2 + sin u = ln x

In general, and ODE can be written as F (x, u, u , u , . . . ) = 0.

In contrast to ODEs, a partial differential equation (PDE) contains partial derivatives of the depen-

dent variable, which is an unknown function in more than one variable x, y, . . . . Denoting the partial

derivative

of

u x

=

ux,

and

u y

=

uy ,

we

can

write

the

general

first

order

PDE

for

u(x, y)

as

F (x, y, u(x, y), ux(x, y), uy(x, y)) = F (x, y, u, ux, uy) = 0.

(1.1)

Although one can study PDEs with as many independent variables as one wishes, we will be primarily concerned with PDEs in two independent variables. A solution to the PDE (1.1) is a function u(x, y) which satisfies (1.1) for all values of the variables x and y. Some examples of PDEs (of physical significance) are:

ux + uy = 0 transport equation ut + uux = 0 inviscid Burger's equation uxx + uyy = 0 Laplace's equation utt - uxx = 0 wave equation ut - uxx = 0 heat equation ut + uux + uxxx = 0 KdV equation iut - uxx = 0 Shro?dinger's equation

(1.2) (1.3) (1.4) (1.5) (1.6) (1.7) (1.8)

It is generally nontrivial to find the solution of a PDE, but once the solution is found, it is easy to verify whether the function is indeed a solution. For example to see that u(t, x) = et-x solves the wave

equation (1.5), simply substitute this function into the equation:

(et-x)tt - (et-x)xx = et-x - et-x = 0.

1.1 Classification of PDEs

There are a number of properties by which PDEs can be separated into families of similar equations. The two main properties are order and linearity. Order. The order of a partial differential equation is the order of the highest derivative entering the equation. In examples above (1.2), (1.3) are of first order; (1.4), (1.5), (1.6) and (1.8) are of second order; (1.7) is of third order. Linearity. Linearity means that all instances of the unknown and its derivatives enter the equation linearly. To define this property, rewrite the equation as

Lu = 0,

(1.9)

where

L

is

an

operator,

which

assigns

u

a

new

function

Lu.

For

example

L

=

2 x2

+ 1,

then

Lu

=

uxx + u.

The operator L is called linear if

L(u + v) = Lu + Lv, and L(cu) = cLu

(1.10)

1

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