Articolo G.A. Partial Differential Equations and Boundary Value Problems with Maple V
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298 Chapter 5
> B(n):=radsimp(subs(sin(sqrt(lambda[n])*a)=(1/sqrt(lambda[n]))*cos(sqrt(lambda[n])*a),
B(n)));
B(n) := −
2
√
2 cos
√
λ
n
(
λ
n
−1
)
2 −cos
√
λ
n
2
λ
2
n
sinh
√
λ
n
(5.65)
Generalized series terms
> u[n](x,y):=eval(X[n](x)*Y[n](y));
u
n
(x, y) := −
4 cos
√
λ
n
x
cos
√
λ
n
(
λ
n
−1
)
sinh
√
λ
n
(1 −y)
sin
√
λ
n
2
+1
2 −cos
√
λ
n
2
λ
2
n
sinh
√
λ
n
(5.66)
Series solution
> u(x,y):=Sum(u[n](x,y),n=1..infinity);
u(x, y) :=
∞
n=1
⎛
⎜
⎝
−
4 cos
√
λ
n
x
cos
√
λ
n
(
λ
n
−1
)
sinh
√
λ
n
(1 −y)
sin
√
λ
n
2
+1
2 −cos
√
λ
n
2
λ
2
n
sinh
√
λ
n
⎞
⎟
⎠
(5.67)
Evaluation of the eigenvalues from the roots of the eigenvalue equation gives
> tan(sqrt(lambda[n])*a)=(1/sqrt(lambda[n]));
tan
λ
n
=
1
√
λ
n
(5.68)
> plot({tan(v),1/v},v=0..20,y=0..3/2,thickness=10);
510
v
y
15 200
0
0.5
1.0
1.5
Figure 5.4
If we set v =
√
λa, then the eigenvalues are found from the intersection points of the curves
shown in Figure 5.4. An evaluation of the first few eigenvalues using the Maple fsolve
command yields
The Laplace Par tial Differential Equation 299
> lambda[1]:=(1/aˆ2)*(fsolve((tan(v)−1/v),v=0..1))ˆ2;
λ
1
:= 0.7401738844 (5.69)
> lambda[2]:=(1/aˆ2)*(fsolve((tan(v)−1/v),v=1..4))ˆ2;
λ
2
:= 11.73486183 (5.70)
> lambda[3]:=(1/aˆ2)*(fsolve((tan(v)−1/v),v=4..7))ˆ2;
λ
3
:= 41.43880785 (5.71)
First few terms of expansion
> u(x,y):=sum(u[n](x,y),n=1..3):
> plot3d(u(x,y),x=0..a,y=0..b,axes=framed,thickness=1);
0.8
0.6
0.4
0
0
0.2
0.4
y
0.6
11
0.8
0.6
0.4
x
0.2
0
0.8
0.2
Figure 5.5
The three-dimensional surface shown in Figure 5.5 depicts the steady-state temperature
distribution u(x, y) over the rectangular region. Note how the edges of the surface adhere to
the given boundary conditions. The temperature isotherms can be obtained from Maple by
clicking on the figure, choosing the special option “Render the plot using the polygon patch
and contour style” in the graphics bar, and then clicking the “redraw” button.
5.5 Laplace Equation in Cylindrical Coordinates
We now focus on the solution to the Laplace equation over the finite two-dimensional domain
D ={(r, θ) |0 <r<a,0 <θ<b} in cylindrical coordinates. We will be considering problems
in regions that have no extension in z; thus, the following laplacian is z independent.
In the cylindrical coordinate system, the Laplace equation reads
∂
∂r
u(r, θ) +r
∂
2
∂r
2
u(r, θ)
r
+
∂
2
∂θ
2
u(r, θ)
r
2
= 0
300 Chapter 5
We can rewrite this as
∂
∂r
u(r, θ) +r
∂
2
∂r
2
u(r, θ)
r
=−
∂
2
∂θ
2
u(r, θ)
r
2
We use the method of separation of variables to solve the preceding equation. We set
u(r, θ) = R(r)(θ)
Substituting this assumed solution into the partial differential equation, we get
d
dr
R(r)
(θ)+r
d
2
dr
2
R(r)
(θ)
r
=−
R(r)
d
2
dθ
2
(θ)
r
2
Dividing the preceding by the product solution yields
r
d
dr
R(r) +r
d
2
dr
2
R(r)
R(r)
=−
d
2
dθ
2
(θ)
(θ)
(5.72)
Setting each side of the preceding equal to a constant λ, we arrive at the two ordinary
differential equations
r
2
d
2
dr
2
R(r)
+r
d
dr
R(r)
−λR(r) = 0
and
d
2
dθ
2
(θ)+λ(θ) = 0
The first differential equation shown is of the Cauchy-Euler type and the second is of the
Euler type. To proceed further, we now consider the boundary conditions on the problem.
Generally, one seeks a solution to this problem over the finite two-dimensional domain
D ={(r, θ) |0 <r<a,0 <θ<b} subject to the regular homogeneous boundary conditions on
one of the variables, r or θ. We seek out that variable whose boundary conditions are
homogeneous.
For example, consider the case where the boundary conditions on u(r, θ) are homogeneous
with respect to the variable θ; that is,
κ
1
u(r, 0) +κ
2
u
θ
(r, 0) = 0
and
κ
3
u(r, b) +κ
4
u
θ
(r, b) = 0
The earlier ordinary differential equation on θ combined with the preceding equivalent
homogeneous boundary conditions constitute a Sturm-Liouville problem in the variable θ.
We already have solutions for this type of problem.
The Laplace Par tial Differential Equation 301
5.6 Sturm-Liouville Problem for the Laplace Equation in
Cylindrical Coordinates
The differential equation in the spatial variable θ in the preceding section must be solved
subject to the equivalent homogeneous spatial boundary conditions over the finite interval
I ={θ |0 <θ<b}. The Sturm-Liouville problem for the θ-dependent equation consists of the
ordinary differential equation
d
2
dθ
2
(θ)+λ(θ) = 0
along with the corresponding equivalent homogeneous boundary conditions on the θ variable,
which read as
κ
1
(0) +κ
2
θ
(0) = 0
and
κ
3
(b)+κ
4
θ
(b) = 0
We see that the problem in the variable θ is a regular Sturm-Liouville eigenvalue problem
where the allowed values of λ are called the system “eigenvalues” and the corresponding
solutions are called the “eigenfunctions.” Note that the preceding ordinary differential equation
is of the Euler type with the weight function w(θ) = 1.
As before, we can show that for regular Sturm-Liouville problems over finite domains, an
infinite number of eigenvalues exist that can be indexed by the positive integer n. The indexed
eigenvalues and corresponding eigenfunctions are given, respectively, as
λ
n
,
n
(θ)
for n = 0, 1, 2, 3,....
What we stated in Section 2.1 about regular Sturm-Liouville eigenvalue problems holds here;
that is, the eigenfunctions form a “complete” set with respect to any piecewise smooth function
over the finite interval I ={θ |0 <θ<b}.
The remaining r-dependent differential equation is a Cauchy-Euler-type differential equation
and, from Section 1.5, the solution to the corresponding r-dependent differential equation in
terms of the allowed eigenvalues is
R
n
(r) = C(n)r
√
λ
n
+D(n)r
−
√
λ
n
Thus, for each allowed value of the index n, we have a solution
u
n
(r, θ) =
C(n)r
√
λ
n
+D(n)r
−
√
λ
n
n
(θ) (5.73)
for n = 0, 1, 2, 3,....
302 Chapter 5
These functions form a set of basis vectors for the solution space of the partial differential
equation, and using the same concepts of the superposition principle as in Chapters 3 and 4, the
general solution to the homogeneous partial differential equation can now be written as an
infinite sum over all basis vectors as follows:
u(r, θ) =
∞
n=0
C(n)r
√
λ
n
+D(n)r
−
√
λ
n
n
(θ)
The arbitrary constants C(n) and D(n) are evaluated by taking advantage of the orthonormality
of the θ-dependent eigenfunctions over the finite interval I ={θ |0 <θ<b} with respect to the
weight function w(θ) = 1. The statement of orthonormality reads
b
0
n
(θ)
m
(θ) dθ = δ(n, m)
for n, m = 0, 1, 2, 3,....
We now consider the alternative possibility. If the homogeneous boundary conditions on
u(r, θ) are with respect to the variable r instead of θ, then we rewrite the two earlier ordinary
differential equations as
r
2
d
2
dr
2
R(r)
+r
d
dr
R(r)
+λR(r) = 0
and
d
2
dθ
2
(θ)−λ(θ) = 0
We consider the solution to this problem over the interval I ={r |1 <r<a}; we are avoiding
the origin because the differential equation in r is not normal (it is singular) at r = 0. Let the
equivalent homogeneous boundary conditions with respect to the variable r read as
κ
1
u(1,θ)+κ
2
u
r
(1,θ)= 0
and
κ
3
u(a, θ) +κ
4
u
r
(a, θ) = 0
The preceding ordinary differential equation in r combined with the preceding homogeneous
boundary conditions again constitutes a Sturm-Liouville problem, for which we already
have solutions from Chapter 2.
The Laplace Par tial Differential Equation 303
The differential equation in the preceding spatial variable r must be solved subject to the
homogeneous spatial boundary conditions over the finite interval. The Sturm-Liouville
problem for the r-dependent equation consists of the ordinary differential equation
r
2
d
2
dr
2
R(r)
+r
d
dr
R(r)
+λR(r) = 0 (5.74)
along with the equivalent corresponding regular homogeneous boundary conditions on r,
which read
κ
1
R(1) +κ
2
R
r
(1) = 0
and
κ
3
R(a) +κ
4
R
r
(a) = 0
We see the preceding problem in the spatial variable r over the interval I ={r |1 <r<a} to be
a regular Sturm-Liouville eigenvalue problem where the allowed values of λ are called the
system “eigenvalues” and the corresponding solutions are called the “eigenfunctions.” Note
that the preceding ordinary differential equation is of the Cauchy-Euler type, and, from
Section 2.5, the weight function is w(r) = 1/r.
As before, we can show that for regular Sturm-Liouville problems over finite domains, an
infinite number of eigenvalues exist that can be indexed by a positive integer n. The indexed
eigenvalues and corresponding eigenfunctions are given, respectively, as
λ
n
,R
n
(r)
for n = 0, 1, 2, 3,....
What we stated in Section 2.1 about regular Sturm-Liouville eigenvalue problems holds here;
that is, the eigenfunctions form a “complete” set with respect to any piecewise smooth function
over the finite r interval I ={r |1 <r<a}.
From Section 1.4 on second-order linear differential equations, the solution to the remaining
θ-dependent differential equation in terms of the allowed eigenvalues is
n
(θ) = A(n) cosh
λ
n
θ
+B(n) sinh
λ
n
θ
Thus, for each allowed value of the index n, we have a solution
u
n
(r, θ) = R
n
(r)
A(n) cosh
λ
n
θ
+B(n) sinh
λ
n
θ
for n = 0, 1, 2, 3,....
304 Chapter 5
Using the concept of the superposition principle as in Chapters 3 and 4, the general solution to
the homogeneous partial differential equation can again be written as the infinite sum of the
given basis vectors, in terms of the eigenfunctions with respect to the variable r, as follows:
u(r, θ) =
∞
n=0
R
n
(r)
A(n) cosh
λ
n
θ
+B(n) sinh
λ
n
θ
Again, the arbitrary constants A(n) and B(n) can be evaluated by taking advantage of the
orthonormality of the r-dependent eigenfunctions, over the finite interval I ={r |1 <r<a},
with respect to the weight function w(r) = 1/r. The statement of orthonormality reads
a
1
R
n
(r)R
m
(r)
r
dr = δ(n, m)
for n, m = 0, 1, 2, 3,....
The original statement of the boundary conditions on the problem determines whether the
solution is written in terms of the eigenfunctions with respect to r or θ. Remember that the
decision comes from our seeking that particular variable that is experiencing the homogeneous
boundary conditions.
DEMONSTRATION: We seek the steady-state temperature distribution in a thin cylindrical
plate over the domain D ={(r, θ) |0 <r<1, 0 <θ<π/3} whose lateral surface is insulated.
The sides θ = 0 and θ = π/3 are fixed at zero temperature, and the side r = 1 has a temperature
distribution f(θ) given as follows. The solution is required to be finite at the origin.
SOLUTION: The homogeneous Laplace equation is
∂
∂r
u(r, θ) +r
∂
2
∂r
2
u(r, θ)
r
+
∂
2
∂θ
2
u(r, θ)
r
2
= 0
The boundary conditions are
|
u(0,θ)
|
< ∞ and u(1,θ)= f(θ)
and
u(r, 0) = 0 and u
r,
π
3
= 0
We seek that variable whose boundary conditions are homogeneous. Since the boundary
conditions are homogeneous with respect to the θ variable, we write the ordinary differential
equations from the method of separation of variables as follows:
d
2
dθ
2
(θ)+λ(θ) = 0
The Laplace Par tial Differential Equation 305
and
r
2
d
2
dr
2
R(r)
+r
d
dr
R(r)
−λR(r) = 0
We first consider the θ-dependent differential equation, since the boundary conditions on this
equation are homogeneous. The boundary conditions on the θ equation are type 1 at θ = 0 and
type 1 at θ = π/3:
(0) = 0 and
π
3
= 0
This same boundary value problem was considered in Example 2.5.1 in Chapter 2. The allowed
eigenvalues are
λ
n
= 9n
2
for n = 1, 2, 3,..., and the corresponding orthonormal eigenfunctions are
n
(θ) =
6
π
sin(3nθ)
The statement of orthonormality with respect to the weight function w(θ) = 1 over the interval
I ={θ |0 <θ<π/3} reads
π
3
0
6 sin(3nθ) sin(3mθ)
π
dθ = δ(n, m)
Focusing on the r-dependent equation, we see that we have a Cauchy-Euler-type differential
equation and a set of basis vectors for this equation for λ
n
= 9n
2
is
r1 = r
3n
and
r2 = r
−3n
Thus, the general solution to the r equation reads
R
n
(r) = C(n)r
3n
+D(n)r
−3n
Since the r-dependent solution must be finite at r = 0, we must set the coefficient D(n) = 0,
and the final solution to the r-dependent equation reads
R
n
(r) = C(n)r
3n
306 Chapter 5
From the superposition of the products of the θ- and r-dependent solutions, the final
eigenfunction expansion for the solution reads
u(r, θ) =
∞
n=1
C(n)r
3n
6
π
sin(3nθ)
The Fourier coefficients C(n) are to be determined from the specific remaining boundary
condition u(1,θ)= f(θ). We consider the special case for f(θ) = θ
π
3
−θ
. Substitution of this
condition into the preceding solution yields
θ
π
3
−θ
=
∞
n=1
C(n)
6
π
sin(3nθ) (5.75)
This equation is the Fourier series expansion of the boundary function f(θ) in terms of the
orthonormal Sturm-Liouville eigenfunctions found earlier. To evaluate the Fourier cofficients,
we take the inner product of both sides with respect to the orthonormal eigenfunctions over the
interval I ={θ |0 <θ<π/3} with respect to the weight function w(θ) = 1. Doing so yields
π
3
0
θ
π
3
−θ
6
π
sin(3mθ) dθ =
∞
n=1
C(n)
⎛
⎜
⎝
π
3
0
6
π
sin(3nθ)
6
π
sin(3mθ) dθ
⎞
⎟
⎠
Taking advantage of the preceding statement of orthonormality, the Fourier coefficients are
C(n) =
π
3
0
θ
π
3
−θ
6
π
sin(3nθ)dθ
Evaluation of this integral yields
C(n) =
24
π
(1 −(−1)
n
)
(3n)
3
for n = 1, 2, 3,....Thus, the final series solution to our problem reads
u(r, θ) =
∞
n=1
4(1 −(−1)
n
)r
3n
sin(3nθ)
9n
3
π
The details for the development of the solution and the graphics are given later in one of the
Maple worksheet examples.
The Laplace Par tial Differential Equation 307
5.7 Example Laplace Problems in the Cylindrical
Coordinate System
We now consider Laplace equation examples in the cylindrical coordinate system. The
examples here have no extension along the z-axis; thus, the Laplacian here is z-independent.
Note that all the r-dependent differential equations are of the Cauchy-Euler type and the
θ-dependent equations are of the Euler type.
EXAMPLE 5.7.1: We seek the steady-state temperature distribution in a thin cylindrical plate
over the domain D ={(r, θ) |0 <r<1, 0 <θ<π/3} whose lateral surface is insulated. The
sides θ = 0 and θ = π/3 are fixed at temperature zero, and the side r = 1 has a temperature
distribution f(θ) given as follows. The solution is required to be finite at the origin.
SOLUTION: The homogeneous Laplace equation is
∂
∂r
u(r, θ) +r
∂
2
∂r
2
u(r, θ)
r
+
∂
2
∂θ
2
u(r, θ)
r
2
= 0
The boundary conditions are
|
u(0,θ)
|
< ∞ and u(1,θ)= θ
π
3
−θ
and
u(r, 0) = 0 and u
r,
π
3
= 0
Here we see that the boundary conditions are homogeneous with respect to the θ variable.
Ordinary differential equations from the method of separation of variables are
d
2
dθ
2
(θ)+λ(θ) = 0
and
r
2
d
2
dr
2
R(r)
+r
d
dr
R(r)
−λR(r) = 0
Homogeneous boundary conditions on the θ equation are type 1 at θ = 0 and type 1 at θ = π/3
(0) = 0 and
π
3
= 0
Assignment of system parameters
> restart:with(plots):a:=1:b:=Pi/3: