Partial Differential Equations: An Introduction, 2nd Edition
CONTENTS
(The sta
ed sections form the basic part of the book.)
Chapter 1/Where PDEs Come From
1.1* What is a Partial Differential Equation? 1
1.2* First-Order Linear Equations 6
1.3* Flows, Vi
ations, and Diffusions 10
1.4* Initial and Boundary Conditions 20
1.5 Well-Posed Problems 25
1.6 Types of Second-Order Equations 28
Chapter 2/Waves and Diffusions
2.1* The Wave Equation 33
2.2* Causality and Energy 39
2.3* The Diffusion Equation 42
2.4* Diffusion on the Whole Line 46
2.5* Comparison of Waves and Diffusions 54
Chapter 3/Reflections and Sources
3.1 Diffusion on the Half-Line 57
3.2 Reflections of Waves 61
3.3 Diffusion with a Source 67
3.4 Waves with a Source 71
3.5 Diffusion Revisited 80
Chapter 4/Boundary Problems
4.1* Separation of Variables, The Dirichlet Condition 84
4.2* The Neumann Condition 89
4.3* The Robin Condition 92
viii
CONTENTS ix
Chapter 5/Fourier Series
5.1* The Coefficients 104
5.2* Even, Odd, Periodic, and Complex Functions 113
5.3* Orthogonality and General Fourier Series 118
5.4* Completeness 124
5.5 Completeness and the Gi
s Phenomenon 136
5.6 Inhomogeneous Boundary Conditions 147
Chapter 6/Harmonic Functions
6.1* Laplace’s Equation 152
6.2* Rectangles and Cubes 161
6.3* Poisson’s Formula 165
6.4 Circles, Wedges, and Annuli 172
(The next four chapters may be studied in any order.)
Chapter 7/Green’s Identities and Green’s Functions
7.1 Green’s First Identity 178
7.2 Green’s Second Identity 185
7.3 Green’s Functions 188
7.4 Half-Space and Sphere 191
Chapter 8/Computation of Solutions
8.1 Opportunities and Dangers 199
8.2 Approximations of Diffusions 203
8.3 Approximations of Waves 211
8.4 Approximations of Laplace’s Equation 218
8.5 Finite Element Method 222
Chapter 9/Waves in Space
9.1 Energy and Causality 228
9.2 The Wave Equation in Space-Time 234
9.3 Rays, Singularities, and Sources 242
9.4 The Diffusion and Schrödinger Equations 248
9.5 The Hydrogen Atom 254
Chapter 10/Boundaries in the Plane and in Space
10.1 Fourier’s Method, Revisited 258
10.2 Vi
ations of a Drumhead 264
10.3 Solid Vi
ations in a Ball 270
10.4 Nodes 278
10.5 Bessel Functions 282
x CONTENTS
10.6 Legendre Functions 289
10.7 Angular Momentum in Quantum Mechanics 294
Chapter 11/General Eigenvalue Problems
11.1 The Eigenvalues Are Minima of the Potential Energy 299
11.2 Computation of Eigenvalues 304
11.3 Completeness 310
11.4 Symmetric Differential Operators 314
11.5 Completeness and Separation of Variables 318
11.6 Asymptotics of the Eigenvalues 322
Chapter 12/Distributions and Transforms
12.1 Distributions 331
12.2 Green’s Functions, Revisited 338
12.3 Fourier Transforms 343
12.4 Source Functions 349
12.5 Laplace Transform Techniques 353
Chapter 13/PDE Problems from Physics
13.1 Electromagnetism 358
13.2 Fluids and Acoustics 361
13.3 Scattering 366
13.4 Continuous Spectrum 370
13.5 Equations of Elementary Particles 373
Chapter 14/Nonlinear PDEs
14.1 Shock Waves 380
14.2 Solitons 390
14.3 Calculus of Variations 397
14.4 Bifurcation Theory 401
14.5 Water Waves 406
Appendix
A.1 Continuous and Differentiable Functions 414
A.2 Infinite Series of Functions 418
A.3 Differentiation and Integration 420
A.4 Differential Equations 423
A.5 The Gamma Function 425
References 427
Answers and Hints to Selected Exercises 431
Index 446
1
WHERE PDEs
COME FROM
After thinking about the meaning of a partial differential equation, we will
flex our mathematical muscles by solving a few of them. Then we will see
how naturally they arise in the physical sciences. The physics will motivate
the formulation of boundary conditions and initial conditions.
1.1 WHAT IS A PARTIAL DIFFERENTIAL EQUATION?
The key defining property of a partial differential equation (PDE) is that there
is more than one independent variable x, y, XXXXXXXXXXThere is a dependent variable
that is an unknown function of these variables u(x, y, XXXXXXXXXXWe will often
denote its derivatives by subscripts; thus ∂u/∂x = ux , and so on. A PDE is an
identity that relates the independent variables, the dependent variable u, and
the partial derivatives of u. It can be written as
F(x, y, u(x, y), ux (x, y), uy(x, y)) = F(x, y, u, ux , uy) = 0. (1)
This is the most general PDE in two independent variables of first order. The
order of an equation is the highest derivative that appears. The most general
second-order PDE in two independent variables is
F(x, y, u, ux , uy, uxx , uxy, uyy) = 0. (2)
A solution of a PDE is a function u(x, y, XXXXXXXXXXthat satisfies the equation
identically, at least in some region of the x, y, . . . variables.
When solving an ordinary differential equation (ODE), one sometimes
everses the roles of the independent and the dependent variables—for in-
stance, for the separable ODE
du
dx
= u3. For PDEs, the distinction between
the independent variables and the dependent variable (the unknown) is always
maintained.
1
2 CHAPTER 1 WHERE PDEs COME FROM
Some examples of PDEs (all of which occur in physical theory) are:
1. ux + uy = 0 (transport)
2. ux + yuy = 0 (transport)
3. ux + uuy = 0 (shock wave)
4. uxx + uyy = 0 (Laplace’s equation)
5. utt − uxx + u3 = 0 (wave with interaction)
6. ut + uux + uxxx = 0 (dispersive wave)
7. utt + uxxxx = 0 (vi
ating bar)
8. ut − iuxx = 0 (i =
√−1) (quantum mechanics)
Each of these has two independent variables, written either as x and y o
as x and t. Examples 1 to 3 have order one; 4, 5, and 8 have order two; 6 has
order three; and 7 has order four. Examples 3, 5, and 6 are distinguished from
the others in that they are not “linear.” We shall now explain this concept.
Linearity means the following. Write the equation in the form lu = 0,
wherel is an operator. That is, if v is any function,lv is a new function. Fo
instance, l = ∂/∂x is the operator that takes v into its partial derivative vx .
In Example 2, the operator l is l = ∂/∂x + y∂/∂y. (lu = ux + yuy.) The
definition we want for linearity is
l(u + v) = lu + lv l(cu) = clu (3)
for any functions u, v and any constant c. Whenever (3) holds (for all choices
of u, v, and c), l is called linear operator. The equation
lu = 0 (4)
is called linear if l is a linear operator. Equation (4) is called a homogeneous
linear equation. The equation
lu = g, (5)
where g �= 0 is a given function of the independent variables, is called an
inhomogeneous linear equation. For instance, the equation
(cos xy2)ux − y2uy = tan(x2 + y2) (6)
is an inhomogeneous linear equation.
As you can easily verify, five of the eight equations above are linea
as well as homogeneous. Example 5, on the other hand, is not linear because
although (u + v)xx = uxx + vxx and (u + v)t t = utt + vt t satisfy property (3),
the cubic term does not:
(u + v)3 = u3 + 3u2v + 3uv2 + v3 �= u3 + v3.
1.1 WHAT IS A PARTIAL DIFFERENTIAL EQUATION? 3
The advantage of linearity for the equation lu = 0 is that if u and v are
oth solutions, so is (u + v). If u1, . . . , un are all solutions, so is any linea
combination
c1u1(x) + · · · + cnun(x) =
n∑
j=1
c j uj (x) (cj = constants).
(This is sometimes called the superposition principle.) Another consequence
of linearity is that if you add a homogeneous solution [a solution of (4)] to an
inhomogeneous solution [a solution of (5)], you get an inhomogeneous solu-
tion. (Why?) The mathematical structure that deals with linear combinations
and linear operators is the vector space. Exercises 5–10 are review problems
on vector spaces.
We’ll study, almost exclusively, linear systems with constant coefficients.
Recall that for ODEs you get linear combinations. The coefficients are the
a
itrary constants. For an ODE of order m, you get m a
itrary constants.
Let’s look at some PDEs.
Example 1.
Find all u(x, y) satisfying the equation uxx = 0. Well, we can integrate
once to get ux = constant. But that’s not really right since there’s anothe
variable y. What we really get is ux(x, y) = f (y), where f (y) is a
itrary.
Do it again to get u(x, y) = f (y)x + g(y). This is the solution formula.
Note that there are two a
itrary functions in the solution. We see this
as well in the next two examples. �
Example 2.
Solve the PDE uxx + u = 0. Again, it’s really an ODE with an extra
variable y. We know how to solve the ODE, so the solution is
u = f (y) cos x + g(y) sin x,
where again f (y) and g(y) are two a
itrary functions of y. You can easily
check this formula by differentiating twice to verify that uxx = −u. �
Example 3.
Solve the PDE uxy = 0. This isn’t too hard either. First let’s integrate in
x, regarding y as fixed. So we get
uy(x, y) = f (y).
Next let’s integrate in y regarding x as fixed. We get the solution
u(x, y) = F(y) + G(x),
where F ′ = f. �
4 CHAPTER 1 WHERE PDEs COME FROM
Moral A PDE has a
itrary functions in its solution. In these examples the
a
itrary functions are functions of one variable that combine to produce a
function u(x, y) of two variables which is only partly a
itrary.
A function of two variables contains immensely more information than
a function of only one variable. Geometrically, it is obvious that a surface
{u = f (x, y)}, the graph of a function of two variables, is a much more com-
plicated object than a curve {u = f (x)}, the graph of a function of one variable.
To illustrate this, we can ask how a computer would record a function
u = f (x). Suppose that we choose 100 points to describe it using equally spaced
values of x : x1, x2, x3, . . . , x100. We could write them down in a column, and
next to each xj we could write the co
esponding value uj = f (xj ). Now how
about a