Partial Differential Equations (21 pages)

Partial Differential Equations (21 pages)
Chapter 11
Partial Differential
Equations
A wide variety of partial differential equations occurs in technical computing. We
cannot begin to cover them all in this book. In this chapter, we limit ourselves to
three model problems for second-order partial differential equations in one or two
space dimensions.
11.1
Model Problems
All the problems we consider involve the Laplacian operator, which is
△=
∂2
∂x2
in one space dimension and
△=
∂2
∂2
+ 2
2
∂x
∂y
in two space dimensions. We let ⃗x denote the single variable x in one dimension
and the pair of variables (x, y) in two dimensions.
The first model problem is the Poisson equation. This elliptic equation does
not involve a time variable, and so describes the steady state, quiescent behavior of
a model variable:
△u = f (⃗x).
There are no initial conditions.
The second model problem is the heat equation. This parabolic equation occurs
in models involving diffusion and decay:
∂u
= △u − f (⃗x).
∂t
The initial condition is
u(⃗x, 0) = u0 (⃗x).
September 16, 2013
1
2
Chapter 11. Partial Differential Equations
The third model problem is the wave equation. This hyperbolic equation describes how a disturbance travels through matter. If the units are chosen so that
the wave propagation speed is equal to one, the amplitude of a wave satisfies
∂2u
= △u.
∂t2
Typical initial conditions specify the initial amplitude and take the initial velocity
to be zero:
∂u
(⃗x, 0) = 0.
u(⃗x, 0) = u0 (⃗x),
∂t
In one dimension, all the problems take place on a finite interval on the x-axis. In
more than one space dimension, geometry plays a vital role. In two dimensions, all
the problems take place in a bounded region Ω in the (x, y) plane. In all cases, f (⃗x)
and u0 (⃗x) are given functions of ⃗x. All the problems involve boundary conditions
where the value of u or some partial derivative of u is specified on the boundary of
Ω. Unless otherwise specified, we will take the boundary values to be zero.
11.2
Finite Difference Methods
Basic finite difference methods for approximating solutions to these problems use
a uniform mesh with spacing h. In one dimension, for the interval a ≤ x ≤ b, the
spacing is h = (b − a)/(m + 1) and the mesh points are
xi = a + ih, i = 0, . . . , m + 1.
The second derivative with respect to x is approximated by the 3-point centered
second difference:
△h u(x) =
u(x + h) − 2u(x) + u(x − h)
.
h2
In two dimensions, the mesh is the set of points
(xi , yj ) = (ih, jh)
that lie within the region Ω. Approximating the partial derivatives with centered
second differences gives the 5-point discrete Laplacian
△h u(x, y) =
u(x + h, y) − 2u(x, y) + u(x − h, y)
h2
u(x, y + h) − 2u(x, y) + u(x, y − h)
.
+
h2
Alternative notation uses P = (x, y) for a mesh point and N = (x, y + h), E =
(x + h, y), S = (x, y − h), and W = (x − h, y) for its four neighbors in the four
compass directions. The discrete Laplacian is
△h u(P ) =
u(N ) + u(W ) + u(E) + u(S) − 4u(P )
.
h2
11.2. Finite Difference Methods
3
The finite difference Poisson problem involves finding values of u so that
△h u(⃗x) = f (⃗x)
for each point ⃗x on the mesh.
If the source term f (⃗x) is zero, Poisson’s equation is called Laplace’s equation:
△h u(x) = 0.
In one dimension, Laplace’s equation has only trivial solutions. The value of u at a
mesh point x is the average of the values of u at its left and right neighbors, so u(x)
must be a linear function of x. Taking the boundary conditions into consideration
implies that u(x) is the linear function connecting the two boundary values. If the
boundary values are zero, then u(x) is identically zero. In more than one dimension,
solutions to Laplace’s equation are called harmonic functions and are not simply
linear functions of ⃗x.
The finite difference heat and wave equations also make use of first and second
differences in the t direction. Let δ denote the length of a time step. For the heat
equation, we use a difference scheme that corresponds to Euler’s method for ordinary
differential equations:
u(⃗x, t + δ) − u(⃗x, t)
= △h u(⃗x).
δ
Starting with the initial conditions u(⃗x, 0) = u0 (⃗x), we can step from any value of
t to t + δ with
u(⃗x, t + δ) = u(⃗x, t) + δ△h u(⃗x, t)
for all of the mesh points ⃗x in the region. The boundary conditions supply values on
the boundary or outside the region. This method is explicit because each new value
of u can be computed directly from values of u at the previous time step. More
complicated methods are implicit because they involve the solution of systems of
equations at each step.
For the wave equation, we can use a centered second difference in t:
u(⃗x, t + δ) − 2u(⃗x, t) + u(⃗x, t − δ)
= △h u(⃗x, t).
δ2
This requires two “layers” of values of the solution, one at t − δ and one at t. In
our simple model problem, the initial condition
∂u
(⃗x, 0) = 0
∂t
allows us to start with both u(⃗x, 0) = u0 (⃗x) and u(⃗x, δ) = u0 (⃗x). We compute
subsequent layers with
u(⃗x, t + δ) = 2u(⃗x, t) − u(⃗x, t − δ) + δ 2 △h u(⃗x, t)
for all of the mesh points ⃗x in the region. The boundary conditions supply values
on the boundary or outside the region. Like our scheme for the heat equation, this
method for solving the wave equation is explicit.
4
Chapter 11. Partial Differential Equations
11.3
Matrix Representation
If a one-dimensional mesh function is represented as a vector, the one-dimensional
difference operator △h becomes the tridiagonal matrix


−2 1
 1 −2 1





1
−2
1
1 
.
..
..
..


2
h 
.
.
.


1 −2 1 
1 −2
This matrix is symmetric. (It is also negative definite.) Most importantly, even if
there are thousands of interior mesh points, there are at most three nonzero elements
in each row and column. Such matrices are the prime examples of sparse matrices.
When computing with sparse matrices, it is important to use data structures that
store only the locations and values of the nonzero elements.
With u represented as a vector and h2 △h as a matrix A, the Poisson problem
becomes
Au = b,
where b is a vector (the same size as u) containing the values of h2 f (x) at the
interior mesh points. The first and last components of b would also include any
nonzero boundary values.
In Matlab, the solution to the discrete Poisson problem is computed using
sparse backslash, which takes advantage of the sparsity in A:
u = A\b
The situation for meshes in two dimensions is more complicated. Let’s number
the interior mesh points in Ω from top to bottom and from left to right. For example,
the numbering of an L-shaped region would be
L =
0
0
0
0
0
0
0
0
0
0
0
0
1
2
3
4
0
0
0
0
0
0
0
5
6
7
8
0
0
0
0
0
0
0
9
10
11
12
0
0
0
0
0
0
0
13
14
15
16
0
0
0
0
0
0
0
17
18
19
20
0
0
0
0
0
0
0
21
22
23
24
25
26
27
28
29
0
0
30
31
32
33
34
35
36
37
38
0
0
39
40
41
42
43
44
45
46
47
0
0
48
49
50
51
52
53
54
55
56
0
0
0
0
0
0
0
0
0
0
0
0
The zeros are points on the boundary or outside the region. With this numbering,
the values of any function defined on the interior of the region can be reshaped into
a long column vector. In this example, the length of the vector is 56.
11.3. Matrix Representation
5
If a two-dimensional mesh function is represented as a vector, the finite difference Laplacian becomes a matrix. For example, at point number 43,
h2 △h u(43) = u(34) + u(42) + u(44) + u(52) − 4u(43).
If A is the corresponding matrix, then its 43rd row would have five nonzero elements:
a43,34 = a43,42 = a43,44 = a43,52 = 1, and a43,43 = −4.
A mesh point near the boundary has only two or three interior neighbors, so the
corresponding row of A has only three or four nonzero entries.
The complete matrix A has −4’s on its diagonal, four 1’s off the diagonal in
most of its rows, two or three 1’s off the diagonal in some of its rows, and zeros
elsewhere. For the example region above, A would be 56 by 56. Here is A if there
are only 16 interior points.
A =
-4 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
1 -4 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 -4 1 1 0 0 0 0 0 0 0 0 0 0 0
0 1 1 -4 0 1 0 0 0 0 0 0 0 0 0 0
0 0 1 0 -4 1 1 0 0 0 0 0 0 0 0 0
0 0 0 1 1 -4 0 1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 -4 1 0 0 0 1 0 0 0 0
0 0 0 0 0 1 1 -4 1 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1 -4 1 0 0 0 1 0 0
0 0 0 0 0 0 0 0 1 -4 1 0 0 0 1 0
0 0 0 0 0 0 0 0 0 1 -4 0 0 0 0 1
0 0 0 0 0 0 1 0 0 0 0 -4 1 0 0 0
0 0 0 0 0 0 0 1 0 0 0 1 -4 1 0 0
0 0 0 0 0 0 0 0 1 0 0 0 1 -4 1 0
0 0 0 0 0 0 0 0 0 1 0 0 0 1 -4 1
0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 -4
This matrix is symmetric, negative definite, and sparse. There are at most five
nonzero elements in each row and column.
Matlab has two functions that involve the discrete Laplacian, del2 and
delsq. If u is a two-dimensional array representing a function u(x, y), then del2(u)
computes △h u, scaled by h2 /4, at interior points, and uses one-sided formulas at
points near the boundary. For example, the function u(x, y) = x2 + y 2 has △u = 4.
The statements
h = 1/20;
[x,y] = meshgrid(-1:h:1);
u = x.^2 + y.^2;
d = (4/h^2) * del2(u);
produce an array d, the same size as x and y, with all the elements equal to 4.
6
Chapter 11. Partial Differential Equations
If G is a two-dimensional array specifying the numbering of a mesh, then
A = -delsq(G) is the matrix representation of the operator h2 △h on that mesh.
The mesh numbering for several specific regions is generated by numgrid. For
example,
m = 5
L = numgrid(’L’,2*m+1)
generates the L-shaped mesh with 56 interior points shown above. And
m = 3
A = -delsq(numgrid(’L’,2*m+1))
generates the 16-by-16 matrix A shown above.
The function inregion can also generate mesh numberings. For example, the
coordinates of the vertices of the L-shaped domain are
xv = [0 0 1
yv = [0 -1 -1
1 -1 -1
1 1 0
0];
0];
The statement
[x,y] = meshgrid(-1:h:1);
generates a square grid of width h. The statement
[in,on] = inregion(x,y,xv,yv);
generates arrays of zeros and ones that mark the points that are contained in the
domain, including the boundary, as well as those that are strictly on the boundary.
The statements
p = find(in-on);
n = length(p);
L = zeros(size(x));
L(p) = 1:n;
number the n interior points from top to bottom and left to right. The statement
A = -delsq(L);
generates the n-by-n sparse matrix representation of the discrete Laplacian on the
mesh.
With u represented as a vector with n components, the Poisson problem becomes
Au = b,
where b is a vector (the same size as u) containing the values of h2 f (x, y) at the
interior mesh points. The components of b that correspond to mesh points with
neighbors on the boundary or outside the region also include any nonzero boundary
values.
As in one dimension, the solution to the discrete Poisson problem is computed
using sparse backslash.
u = A\b
11.4. Numerical Stability
11.4
7
Numerical Stability
The time-dependent heat and wave equations generate a sequence of vectors, u(k) ,
where the k denotes the kth time step. For the heat equation, the recurrence is
u(k+1) = u(k) + σAu(k) ,
where
σ=
δ
.
h2
This can be written
u(k+1) = M u(k) ,
where
M = I + σA.
In one dimension, the iteration matrix M has 1 − 2σ on the diagonal and one
or two σ’s off the diagonal in each row. In two dimensions, M has 1 − 4σ on the
diagonal and two, three, or four σ’s off the diagonal in each row. Most of the row
sums in M are equal to 1; a few are less than 1. Each element of u(k+1) is a linear
combination of elements of u(k) with coefficients that add up to 1 or less. Now here
is the key observation. If the elements of M are nonnegative, then the recurrence is
stable. In fact, it is dissipative. Any error or noise in u(k) is not magnified in u(k+1) .
But if the diagonal elements of M are negative, then the recurrence can be unstable.
Error and noise, including roundoff error and noise in the initial conditions, can be
magnified with each time step. Requiring 1 − 2σ or 1 − 4σ to be positive leads to
a very important stability condition for this explicit method for the heat equation.
In one dimension,
1
σ≤ .
2
And, in two dimensions,
1
σ≤ .
4
If this condition is satisfied, the iteration matrix has positive diagonal elements and
the method is stable.
Analysis of the wave equation is a little more complicated because it involves
three levels, u(k+1) , u(k) , and u(k−1) . The recurrence is
u(k+1) = 2u(k) − u(k−1) + σAu(k) ,
where
δ2
.
h2
The diagonal elements of the iteration matrix are now 2 − 2σ, or 2 − 4σ. In
one dimension, the stability condition is
σ=
σ ≤ 1.
8
Chapter 11. Partial Differential Equations
And, in two dimensions,
1
.
2
These stability conditions are known as the CFL conditions, after Courant, Friedrichs
and Lewy, who wrote a paper in 1928 that used finite difference methods to prove
existence of solutions to the partial differential equations of mathematical physics.
Stability conditions are restrictions on the size of the time step, δ. Any attempt to
speed up the computation by taking larger time steps is likely to be disastrous. For
the heat equation, the stability condition is particularly severe—the time step must
be smaller than the square of the space mesh width. More sophisticated methods,
often involving some implicit equation solving at each step, have less restrictive or
unlimited stability conditions.
The M-file pdegui illustrates the concepts discussed in this chapter by offering the choice among several domains and among the model partial differential
equations. For Poisson’s equation, pdegui uses sparse backslash to solve
σ≤
△h u = 1
in the chosen domain. For the heat and wave equations, the stability parameter
σ can be varied. If the critical value, 0.25 for the heat equation and 0.50 for the
wave equation, is exceeded by even a small amount, the instability rapidly becomes
apparent.
You will find much more powerful capabilities in the Matlab Partial Differential Equation Toolbox.
11.5
The L-Shaped Membrane
Separating out periodic time behavior in the wave equation leads to solutions of the
form
√
u(⃗x, t) = cos ( λ t) v(⃗x).
The functions v(⃗x) depend upon λ. They satisfy
△v + λv = 0
and are zero on the boundary. The quantities λ that lead to nonzero solutions
are the eigenvalues, and the corresponding functions v(⃗x) are the eigenfunctions or
modes. They are determined by the physical properties and the geometry of each
particular situation. The square roots of the eigenvalues are resonant frequencies. A
periodic external driving force at one of these frequencies generates an unboundedly
strong response in the medium.
Any solution of the wave equation can be expressed as a linear combination of
these eigenfunctions. The coefficients in the linear combination are obtained from
the initial conditions.
In one dimension, the eigenvalues and eigenfunctions are easily determined.
The simplest example is a violin string, held fixed at the ends of the interval of
length π. The eigenfunctions are
vk (x) = sin (kx).
11.5. The L-Shaped Membrane
9
The eigenvalues are determined by the boundary condition, vk (π) = 0. Hence, k
must be an integer and
λk = k 2 .
If the initial condition, u0 (x), is expanded in a Fourier sine series,
∑
u0 (x) =
ak sin (kx),
k
then the solution to the wave equation is
∑
u(x, t) =
ak cos (kt) sin (kx)
=
k
∑
ak cos (
√
λk t) vk (x).
k
In two dimensions, an L-shaped region formed from three unit squares is interesting for several reasons. It is one of the simplest geometries for which solutions
to the wave equation cannot be expressed analytically, so numerical computation
is necessary. Furthermore, the 270◦ nonconvex corner causes a singularity in the
solution. Mathematically, the gradient of the first eigenfunction is unbounded near
the corner. Physically, a membrane stretched over such a region would rip at the
corner. This singularity limits the accuracy of finite difference methods with uniform grids. The MathWorks has adopted a surface plot of the first eigenfunction of
the L-shaped region as the company logo. The computation of this eigenfunction
involves several of the numerical techniques we have described in this book.
Simple model problems involving waves on an L-shaped region include an
L-shaped membrane, or L-shaped tambourine, and a beach towel blowing in the
wind, constrained by a picnic basket on one fourth of the towel. A more practical
example involves ridged microwave waveguides. One such device, shown in Figure
11.1, is a waveguide-to-coax adapter. The active region is the channel with the
H-shaped cross section visible at the end of the adapter. The ridges increase the
bandwidth of the guide at the expense of higher attenuation and lower powerhandling capability. Symmetry of the H about the dotted lines shown in the contour
plot of the electric field implies that only one quarter of the domain needs to be
considered and that the resulting geometry is our L-shaped region. The boundary
conditions are different than our membrane problem, but the differential equation
and the solution techniques are the same.
Eigenvalues and eigenfunctions of the L-shaped domain can be computed by
finite difference methods. The Matlab statements
m = 200
h = 1/m
A = delsq(numgrid(’L’,2*m+1))/h^2
set up the 5-point finite difference approximation to the Laplacian on an 200-by-200
mesh in each of the three squares that make up the domain. The resulting sparse
matrix A has order 119201 and 594409 nonzero entries. The statement
10
Chapter 11. Partial Differential Equations
Figure 11.1. A double-ridge microwave-to-coax adapter and its H-shaped
region. Photo courtesy Advanced Technical Materials, Inc. [1].
lambda = eigs(A,6,0)
uses Arnoldi’s method from the Matlab implementation of ARPACK to compute
the first six eigenvalues. It takes less than 2 min on a 1.4 GHz Pentium laptop to
produce
lambda =
9.64147
15.19694
19.73880
29.52033
31.91583
41.47510
The exact values are
9.63972
15.19725
19.73921
29.52148
31.91264
41.47451
You can see that even with this fine mesh and large matrix calculation, the computed
eigenvalues are accurate to only three or four significant digits. If you try to get
more accuracy by using a finer mesh and hence a larger matrix, the computation
requires so much memory that the total execution time is excessive.
For the L-shaped domain and similar problems, a technique using analytic
solutions to the underlying differential equation is much more efficient and accurate than finite difference methods. The technique involves polar coordinates and
11.5. The L-Shaped Membrane
11
fractional-order Bessel functions. With parameters α and λ, the functions
√
v(r, θ) = Jα ( λ r) sin (α θ)
are exact solutions to the polar coordinate version of the eigenfunction equation
∂ 2 v 1 ∂v
1 ∂2v
+
+
+ λv = 0.
∂r2
r ∂r
r2 ∂θ2
For any value of λ, the functions v(r, θ) satisfy the boundary conditions
v(r, 0) = 0 and v(r, π/α) = 0
√
on the two straight edges of a circular
sector with angle π/α. If λ is chosen to be a
√
zero of the Bessel function, Jα ( λ) = 0, then v(r, θ) is also zero on the circle, r = 1.
Figure 11.2 shows a few of the eigenfunctions of the circular sector with angle 3π/2.
The eigenfunctions have been chosen to illustrate symmetry about 3π/4 and π/2.
8.9494
14.3559
20.7146
33.4927
44.0711
55.6455
Figure 11.2. Eigenfunctions of the three-quarter disc.
We approximate the eigenfunctions of the L-shaped domain and other regions
with corners by linear combinations of the circular sector solutions:
∑
√
v(r, θ) =
cj Jαj ( λ r) sin (αj θ).
j
The angle of the reentrant 270◦ corner in the L-shaped region is 3π/2, or π/(2/3),
so the values of α are integer multiples of 2/3:
αj =
2j
.
3
These functions v(r, θ) are exact solutions to the eigenfunction differential equation.
There is no finite difference mesh involved. The functions also satisfy the boundary
12
Chapter 11. Partial Differential Equations
conditions on the two edges that meet at the reentrant corner. All that remains is
to pick the parameter λ and the coefficients cj so that the boundary conditions on
the remaining edges are satisfied.
A least squares approach involving the SVD is used to determine λ and the
cj . Pick m points, (ri , θi ), on the remaining edges of the boundary. Let n be the
number of fundamental solutions to be used. Generate an m-by-n matrix A with
elements that depend upon λ:
√
Ai,j (λ) = Jαj ( λ ri ) sin (αj θi ), i = 1, . . . , m, j = 1, . . . , n.
Then, for any vector c, the vector Ac is the vector of boundary values, v(ri , θi ).
We want to make ||Ac|| small without taking ||c|| small. The SVD provides the
solution.
Let σn (A(λ)) denote the smallest singular value of the matrix A(λ). and let
λk denote a value of λ that produces a local minimum of the smallest singular value:
λk = kth minimizer(σn (A(λ))).
Each λk approximates an eigenvalue of the region. The corresponding right singular
vector provides the coefficients for the linear combination c = V(:,n).
9.6397
15.1973
19.7392
31.9126
44.9485
49.3480
Figure 11.3. Eigenfunctions of the L-shaped region.
It is worthwhile to take advantage of symmetries. It turns out that the eigenfunctions fall into three symmetry classes:
• symmetric about the center line at θ = 3π/4, so v(r, θ) = v(r, 3π/2 − θ);
• antisymmetric about the center line at θ = 3π/4, so v(r, θ) = −v(r, 3π/2 − θ);
• eigenfunction of the square, so v(r, π/2) = 0 and v(r, π) = 0.
These symmetries allow us to restrict the values of αj used in each expansion:
Exercises
13
• αj =
2j
3 ,
j odd and not a multiple of 3;
• αj =
2j
3 ,
j even and not a multiple of 3;
• αj =
2j
3 ,
j a multiple of 3.
The M-file membranetx in the NCM directory computes eigenvalues and eigenfunctions of the L-membrane using these symmetries and a search for local minima of
σn (A(λ)). The M-file membrane, distributed with Matlab in the demos directory,
uses an older version of the algorithm based on the QR decomposition instead of the
SVD. Figure 11.3 shows six eigenfunctions of the L-shaped region, with two from
each of the three symmetry classes. They can be compared with the eigenfunctions
√
of the sector shown in Figure 11.2. By taking the radius of the sector to be 2/ π,
both regions have the same area and the eigenvalues are comparable.
The demo M-file logo makes a surf plot of the first eigenfunction, then adds
lighting and shading to create The MathWorks logo. After being so careful to satisfy
the boundary conditions, the logo uses only the first two terms in the circular sector
expansion. This artistic license gives the edge of the logo a more interesting curved
shape.
Exercises
11.1. Let n be an integer and generate n-by-n matrices A, D, and I with the
statements
e
I
D
A
=
=
=
=
ones(n,1);
spdiags(e,0,n,n);
spdiags([-e e],[0 1],n,n);
spdiags([e -2*e e],[-1 0 1],n,n);
(a) For an appropriate value of h, the matrix (1/h2 )A approximates △h for
the interval 0 ≤ x ≤ 1. Is that value of h equal to 1/(n−1), 1/n, or 1/(n+1)?
(b) What does (1/h)D approximate?
(c) What are DT D and DDT ?
(d) What is A2 ?
(e) What is kron(A,I)+kron(I,A)?
(f) Describe the output produced by plot(inv(full(-A))).
11.2. (a) Use finite differences to compute a numerical approximation to the solution u(x) to the one-dimensional Poisson problem
d2 u
= exp (−x2 )
dx2
on the interval −1 ≤ x ≤ 1. The boundary conditions are u(−1) = 0 and
u(1) = 0. Plot your solution.
(b) If you have access to dsolve in the Symbolic Toolbox, or if you are very
good at calculus, find the analytic solution of the same problem and compare
it with your numerical approximation.
14
Chapter 11. Partial Differential Equations
11.3. Reproduce the contour plot in Figure 11.1 of the first eigenfunction of the
H-shaped ridge waveguide formed from four L-shaped regions.
11.4. Let h(x) be the function defined by the M-file humps(x). Solve four different
problems involving h(x) on the interval 0 ≤ x ≤ 1.
(a) One-dimensional Poisson problem with humps as the source term:
d2 u
= −h(x),
dx2
with boundary conditions
u(0) = 0, u(1) = 0.
Make plots, similar to Figure 11.4, of h(x) and u(x). Compare diff(u,2)
with humps(x).
100
5
80
4
3
60
2
40
1
20
0
0
0
0.5
1
−1
0
0.5
1
Figure 11.4. h(x) and u(x).
(b) One-dimensional heat equation with humps as the source term:
∂u
∂2u
=
+ h(x),
∂t
∂x2
with initial value
u(0, x) = 0
and boundary conditions
u(0, t) = 0, u(1, t) = 0.
Create an animated plot of the solution as a function of time. What is the
limit as t → ∞ of u(x, t)?
(c) One-dimensional heat equation with humps as the initial value:
∂2u
∂u
=
,
∂t
∂x2
with initial value
u(x, 0) = h(x)
Exercises
15
and boundary conditions
u(0, t) = h(0), u(1, t) = h(1).
Create an animated plot of the solution as a function of time. What is the
limit as t → ∞ of u(x, t)?
(d) One-dimensional wave equation with humps as the initial value:
∂2u
∂2u
=
,
∂t2
∂x2
with initial values
u(x, 0) = h(x),
∂u
(x, 0) = 0,
∂t
and boundary conditions
u(0, t) = h(0), u(1, t) = h(1).
Create an animated plot of the solution as a function of time. For what values
of t does u(x, t) return to its initial value h(x)?
11.5. Let p(x, y) be the function defined by the M-file peaks(x,y). Solve four
different problems involving p(x, y) on the square −3 ≤ x ≤ 3, −3 ≤ y ≤ 3.
2
2
0
0
−2
−2
−2
0
2
−2
0
2
Figure 11.5. p(x, y) and u(x, y).
(a) Two-dimensional Poisson problem with peaks as the source term:
∂2u ∂2u
+ 2 = p(x, y),
∂x2
∂y
with boundary conditions
u(x, y) = 0 if |x| = 3 or |y| = 3.
Make contour plots, similar to Figure 11.5, of p(x, y) and u(x, y).
16
Chapter 11. Partial Differential Equations
(b) Two-dimensional heat equation with peaks as the source term:
∂u
∂2u ∂2u
=
+ 2 − p(x, y),
∂t
∂x2
∂y
with initial value
u(x, y, 0) = 0
and boundary conditions
u(x, y, t) = 0 if |x| = 3 or |y| = 3.
Create an animated contour plot of the solution as a function of time. What
is the limit as t → ∞ of u(x, t)?
(c) Two-dimensional heat equation with peaks as the initial value:
∂u
∂2u
=
,
∂t
∂x2
with initial value
u(x, y, 0) = p(x, y)
and boundary conditions
u(x, y, t) = p(x, y) if |x| = 3 or |y| = 3.
Create an animated contour plot of the solution as a function of time. What
is the limit as t → ∞ of u(x, t)?
(d) Two-dimensional wave equation with peaks as the initial value:
∂2u
∂2u
=
,
∂t2
∂x2
with initial values
u(x, y, 0) = p(x, y),
∂u
(x, y, 0) = 0
∂t
and boundary conditions
u(x, y, t) = p(x, y) if |x| = 3 or |y| = 3.
Create an animated contour plot of the solution as a function of time. Does
the limit as t → ∞ of u(x, t) exist?
11.6. The method of lines is a convenient technique for solving time-dependent
partial differential equations. Replace all the spatial derivatives with finite
differences, but leave the time derivatives intact. Then use a stiff ordinary
differential equation solver on the resulting system. In effect, this is an implicit time-stepping finite difference algorithm with the time step determined
Exercises
17
automatically and adaptively by the ODE solver. For our model heat and
wave equations, the ODE systems are simply
u̇ = (1/h2 )Au
and
ü = (1/h2 )Au.
The matrix (1/h2 )A represents △h , and u is the vector-valued function of t
formed from all the elements u(xi , t) or u(xi , yj , t) at the mesh points.
(a) The Matlab function pdepe implements the method of lines in a general
setting. Investigate its use for our one- and two-dimensional model heat
equations.
(b) If you have access to the Partial Differential Equation Toolbox, investigate
its use for our two-dimensional model heat and wave equations.
(c) Implement your own method of lines solutions for our model equations.
11.7. Answer the following questions about pdegui.
(a) How does the number of points n in the grid depend upon the grid size
h for the various regions?
(b) How does the time step for the heat equation and for the wave equation
depend upon the grid size h?
(c) Why are the contour plots of the solution to the Poisson problem and the
eigenvalue problem with index = 1 similar?
(d) How do the contour plots produced by pdegui of the eigenfunctions of
the L-shaped domain compare with those produced by
contourf(membranetx(index))
(e) Why are the regions Drum1 and Drum2 interesting? Search the Web for
“isospectral” and “Can you hear the shape of a drum?” You should find
many articles and papers, including ones by Gordon, Webb, and Wolpert [3]
and Driscoll [2].
11.8. Add the outline of your hand that you obtained in exercise 3.4 as another
region to pdegui. Figure 11.6 shows one of the eigenfunctions of my hand.
11.9. The electrostatic capacity of a region Ω is the quantity
∫ ∫
u(x, y)dxdy,
Ω
where u(x, y) is the solution to the Poisson problem
△u = −1 in Ω
and u(x, y) = 0 on the boundary of Ω.
(a) What is the capacity of the unit square?
(b) What is the capacity of the L-shaped domain?
(c) What is the capacity of your hand?
18
Chapter 11. Partial Differential Equations
Figure 11.6. An eigenfunction of a hand.
11.10. The statements
load penny
P = flipud(P)
contour(P,1:12:255)
colormap(copper)
axis square
access a file in the Matlab demos directory and produce Figure 11.7. The
data were obtained in 1984 at what was then the National Bureau of Standards by an instrument that makes precise measurements of the depth of a
mold used to mint the U.S. one cent coin.
Figure 11.7. The depth of a mold used to mint the U.S. one cent coin.
The NCM function pennymelt uses this penny data as the initial condition,
u(x, y, 0), for the heat equation and produces an animated, evolving plot
of the solution, u(x, y, t). You can choose either a lighted, surface plot or
Exercises
19
a contour plot. You can choose the time step δ with uicontrols or with
pennymelt(delta). You can also choose a time-stepping algorithm know as
the ADI or alternating direction implicit method. Each time step involves
two half-steps, one implicit in the x direction and the other implicit in the y
direction.
−σu(k+1/2) (N ) + (1 + 2σ)u(k+1/2) (P ) − σu(k+1/2) (S)
= σu(k) (E) + (1 − 2σ)u(k) (P ) + σu(k) (W ),
−σu
(E) + (1 + 2σ)u(k+1) (P ) − σu(k+1) (W )
= σu(k+1/2) (N ) + (1 − 2σ)u(k+1/2) (P ) + σu(k+1/2) (S).
(k+1)
Solving these implicit equations on an m-by-n grid requires the solution of m
tridiagonal systems of order n for the first half-step and then n tridiagonal
systems of order m for the second half-step.
Answer these questions about pennymelt.
(a) What is the limiting behavior of u(x, y, t) as t → ∞?
(b) For what values of δ is the explicit time-stepping algorithm stable?
(c) Demonstrate that the ADI method is stable for any value of δ.
11.11. Let p(x, y) be the function defined on a 128-by-128 square by the penny data
described in the previous exercise.
(a) Make a contour plot of p(x, y) and make a lighted surface plot using the
section of code in pennymelt.m.
(b) Solve the discrete Poisson problem
△h u = p
with u(x, y) = 0 outside the square, and plot the solution u(x, y).
(c) Use del2 to compute
f = △h u,
and compare f (x, y) with p(x, y).
11.12. Modify pennymelt.m to solve the wave equation instead of the heat equation.
11.13. Modify waves.m to use nine eigenfunctions instead of four.
11.14. The eigenvalues and eigenfunctions of the unit square are
λm,n = (m2 + n2 )π 2 ,
um,n = sin mx sin ny.
If the λm,n are indexed with one subscript and listed in increasing order, we
have
λk = (2, 5, 5, 8, 10, 10, 13, 13, 17, 17, 18, 20, 20, . . .)π 2 .
We see that λ1 , λ4 , and λ11 are simple eigenvalues, but that most of the
eigenvalues are double.
(a) What is the smallest triple eigenvalue of the unit square and what is its
index? In other words, what is the smallest integer that can be written as
the sum of two squares in three different ways?
(b) What is the smallest quadruple eigenvalue of the unit square?
20
Chapter 11. Partial Differential Equations
11.15. By reflecting the eigenfunctions of the unit square twice, we obtain some of
the eigenfunctions of the L-shaped domain. The indexing is different because
the L also has eigenfunctions that are not derived from the square. For example, λ3 of the L is 2π 2 because it is equal to λ1 of the square. And λ8 = λ9
of the L is a double eigenvalue, 5π 2 , corresponding to λ2 = λ3 of the square.
(a) Roughly what fraction of the eigenvalues of the L-shaped region are also
eigenvalues of the square?
(b) What is the smallest triple eigenvalue of the L-shaped region and what
is its index?
(c) What is the smallest quadruple eigenvalue of the L-shaped region?
(d) Neither membranetx nor pdegui uses the sin mx sin ny representation of
eigenfunctions of the square. This is OK because these eigenfunctions are not
unique and can have other representations. How do membranetx and pdegui
compute eigenfunctions? How do they get a set of linearly independent eigenfunctions for eigenvalues with multiplicity greater than one?
11.16. Enter the commands
ncmlogo
cameratoolbar
Or, just enter the command ncmlogo and then select Camera Toolbar
from the View tab on the figure window. Experiment with the various icons
available on the new toolbar. What do they all do?
11.17. Make your own copy of ncmlogo and modify it to create a logo for your own
book or company.
Bibliography
[1] Advanced Technical Materials, Inc.
http://www.atmmicrowave.com
[2] T. A. Driscoll, Eigenmodes of isospectral drums, SIAM Review, 39 (1997),
pp. 1–17.
http://www.math.udel.edu/~driscoll/pubs/drums.pdf
[3] C. Gordon, D. Webb, and S. Wolpert, Isospectral plane domains and
surfaces via Riemannian orbifolds, Inventiones Mathematicae, 110 (1992), pp.
1–22.
21
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