Articolo G.A. Partial Differential Equations and Boundary Value Problems with Maple V

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308 Chapter 5

The allowed eigenvalues and corresponding orthonormal eigenfunctions are obtained from

Example 2.5.1.

> lambda[n]:=(n*Pi/b)ˆ2;

:= 9n

(5.76)

for n = 1, 2, 3,....

Orthonormal eigenfunctions

> Theta[n](theta):=sqrt(2/b)*sin(n*Pi/b*theta);Theta[m](theta):=subs(n=m,Theta[n](theta)):



(θ) :=

√

6 sin(3nθ)

√

(5.77)

Statement of orthonormality with respect to the weight function w(θ) = 1

> w(theta):=1:Int(Theta[n](theta)*Theta[m](theta)*w(theta),theta=0..b)=delta(n,m);



6 sin(3nθ) sin(3mθ)

dθ = δ(n, m) (5.78)

Basis vectors for the r equation

> r1(r):=rˆ(sqrt(lambda));r2(r):=rˆ(−sqrt(lambda));

r1(r) := r

√

r2(r) := r

−

√

(5.79)

General solution to the r equation

> R[n](r):=C(n)*rˆ(sqrt(lambda[n]))+D(n)*rˆ(−sqrt(lambda[n]));

(r) := C(n)r

√

+D(n)r

−3

√

(5.80)

Substituting into the r-dependent boundary condition requiring a ﬁnite solution at r = 0 yields

> D(n):=0;

D(n) := 0 (5.81)

The r-dependent solution for the given eigenvalues reads

> R[n](r):=C(n)*rˆ(n*Pi/b);

(r) := C(n)r

(5.82)

The Laplace Par tial Differential Equation 309

Generalized series terms

> u[n](r,theta):=R[n](r)*Theta[n](theta);

(r, θ) :=

C(n)r

√

6 sin(3nθ)

√

(5.83)

Eigenfunction expansion

> u(r,theta):=Sum(u[n](r,theta),n=1..inﬁnity);

u(r, θ) :=

∞



n=1

C(n)r

√

6 sin(3nθ)

√

(5.84)

Evaluation of Fourier coefﬁcients from the speciﬁc remaining boundary condition

u(1,θ)= f(θ)

> f(theta):=theta*(Pi/3−theta);

f(θ) := θ



π −θ



(5.85)

Substituting this condition into the preceding solution yields

> f(theta)=subs(r=a,u(r,theta));



π −θ



∞



n=1

C(n)

√

6 sin(3nθ)

√

(5.86)

This equation is the Fourier series expansion of 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, and, taking

advantage of the previous statement of orthonormality, we get, for n = 1, 2, 3,...,

> C(n):=(1/(aˆ(n*Pi/b)))*Int(f(theta)*Theta[n](theta)*w(theta),theta=0..b);

C(n):=expand(value(%)):

C(n) :=





π −θ



√

6 sin(3nθ)

√

dθ (5.87)

> C(n):=simplify(subs({sin(n*Pi)=0,cos(n*Pi)=(−1)ˆn},C(n)));

C(n) := −

√

6(−1 +(−1)

)

√

πn

(5.88)

Generalized series terms

310 Chapter 5

> u[n](r,theta):=eval(R[n](r)*Theta[n](theta));

(r, θ) := −

(−1 +(−1)

sin(3nθ)

πn

(5.89)

Series solution

> u(r,theta):=Sum(u[n](r,theta),n=1..inﬁnity);

u(r, θ) :=

∞



n=1



−

(−1 +(−1)

sin(3nθ)

πn



(5.90)

First few terms of expansion

> u(r,theta):=sum(u[n](r,theta),n=1..5):

> cylinderplot([r,theta,u(r,theta)],r=0..a,theta=0..b,axes=framed,orientation=[−31,61],

thickness=1);

0.25

0.15

0.2

0.05

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.1

Figure 5.6

The three-dimensional surface shown in Figure 5.6 depicts the steady-state temperature

distribution u(r, θ) over the cylindrical 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.

EXAMPLE 5.7.2: 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 insulated, the solution is ﬁnite at r = 0, and the side r = 1 has a

temperature distribution f(θ) given as follows.

SOLUTION: The homogeneous Laplace equation is

∂

∂r

u(r, θ) +r



∂

∂r

u(r, θ)



∂

∂θ

u(r, θ)

= 0

The Laplace Par tial Differential Equation 311

The boundary conditions are

u(0,θ)

< ∞ and u(1,θ)= θ



−θ



and

(r, 0) = 0 and u





= 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

dθ

(θ)+λ(θ) = 0

and



R(r)





R(r)



−λR(r) = 0

Homogeneous boundary conditions on the θ equation are type 2 at θ = 0 and type 2 at

θ = π/3



(0) = 0 and 





= 0

Assignment of system parameters

> restart:with(plots):a:=1:b:=Pi/3:

Allowed eigenvalues and corresponding orthonormal eigenfunctions for θ equation are

obtained from Example 2.5.3. For n = 0,

> lambda[0]:=0;

:= 0 (5.91)

Orthonormal eigenfunctions

> Theta[0](theta):=1/sqrt(b);



(θ) :=

√

(5.92)

For n = 1, 2, 3,...,

> lambda[n]:=(n*Pi/b)ˆ2;

:= 9n

(5.93)

Orthonormal eigenfunctions

312 Chapter 5

> Theta[n](theta):=sqrt(2/b)*cos(n*Pi/b*theta);Theta[m](theta):=subs(n=m,Theta[n](theta)):



(θ) :=

√

6 cos(3nθ)

√

(5.94)

Statement of orthonormality with respect to the weight function w(θ) = 1

> w(theta):=1:Int(Theta[n](theta)*Theta[m](theta)*w(theta),theta=0..b)=delta(n,m);



6 cos(3nθ) cos(3mθ)

dθ = δ(n, m) (5.95)

Basis vectors for the r equation for λ>0

> r1(r):=rˆ(sqrt(lambda));r2(r):=rˆ(−sqrt(lambda));

r1(r) := r

√

r2(r) := r

−

√

(5.96)

General solution to the r equation for λ>0is

> R[n](r):=A(n)*rˆ(sqrt(lambda))+B(n)*rˆ(−sqrt(lambda));

(r) := A(n)r

√

+B(n)r

−

√

(5.97)

Substituting into the r-dependent boundary condition requiring a ﬁnite solution at r = 0 yields

> B(n):=0;

B(n) := 0 (5.98)

The r-dependent solution for n = 1, 2, 3,..., for the given eigenvalues, reads

> R[n](r):=A(n)*rˆ(n*Pi/b);

(r) := A(n)r

(5.99)

Basis vectors for the r equation for λ = 0

> r1(r):=1;r2(r):=ln(r);

r1(r) := 1

r2(r) := ln(r) (5.100)

General solution to the r equation for λ = 0

The Laplace Par tial Differential Equation 313

> R[0](r):=A(0)+B(0)*ln(r);

(r) := A(0) +B(0) ln(r) (5.101)

Substituting into the r-dependent boundary condition at r = 0 yields

> B(0):=0;

B(0) := 0 (5.102)

The r-dependent solution for n = 0 reads

> R[0](r):=A(0);

(r) := A(0) (5.103)

Generalized series terms

> u[0](r,theta):=R[0](r)*Theta[0](theta);u[n](r,theta):=R[n](r)*Theta[n](theta);

(r, θ) :=

A(0)

√

(r, θ) :=

A(n)r

√

6 cos(3nθ)

√

(5.104)

Eigenfunction expansion

> u(r,theta):=u[0](r,theta)+Sum(u[n](r,theta),n=1..inﬁnity);

u(r, θ) :=

A(0)

√

∞



n=1

A(n)r

√

6 cos(3nθ)

√

(5.105)

Evaluation of Fourier coefﬁcients from the speciﬁc remaining boundary condition

u(1,θ)= f(θ)

> f(theta):=theta*(Pi/3−theta);

f(θ) := θ



π −θ



(5.106)

Substituting this condition into the preceding solution yields

> f(theta)=subs(r=a,u(r,theta));



π −θ



A(0)

√

∞



n=1

A(n)

√

6 cos(3nθ)

√

(5.107)

This equation is the Fourier series expansion of f(θ) in terms of the orthonormal

Sturm-Liouville eigenfunctions found earlier. To evaluate the Fourier cofﬁcients, we take the

314 Chapter 5

inner product of both sides with respect to the orthonormal eigenfunctions, and, taking

advantage of the preceding orthonormality statement, we get, for n = 1, 2, 3,...,

> A(n):=eval((1/(aˆ(n*Pi/b)))*Int(f(theta)*Theta[n](theta)*w(theta),theta=0..b));

A(n):=expand(value(%)):

A(n) :=





π −θ



√

6 cos(3nθ)

√

dθ (5.108)

> A(n):=simplify(subs({sin(n*Pi)=0,cos(n*Pi)=(−1)ˆn},A(n)));

A(n) := −

√

6(1 +(−1)

)

(5.109)

For n = 0,

> A(0):=Int(f(theta)*Theta[0](theta)*w(theta),theta=0..b);

A(0) :=





π −θ



√

dθ (5.110)

> A(0):=value(%);

A(0) :=

162

√

3π

5/2

(5.111)

Generalized series terms

> u[0](r,theta):=eval(R[0](r)*Theta[0](theta));u[n](r,theta):=eval(R[n](r)*Theta[n](theta));

(r, θ) :=

(r, θ) := −

(1 +(−1)

cos(3nθ)

(5.112)

Series solution

> u(r,theta):=u[0](r,theta)+Sum(u[n](r,theta),n=1..inﬁnity);

u(r, θ) :=

∞



n=1



−

(1 +(−1)

cos(3nθ)



(5.113)

First few terms of expansion

> u(r,theta):=u[0](r,theta)+sum(u[n](r,theta),n=1..5):

The Laplace Par tial Differential Equation 315

> cylinderplot([r,theta,u(r,theta)],r=0..a,theta=0..b,axes=framed,orientation=[−31,61],

thickness=1);

0.25

0.2

0.15

0.1

0.05

0.8

0.6

0.4

0.2

0.4

0.6

0.8

Figure 5.7

The three-dimensional surface shown in Figure 5.7 depicts the steady-state temperature

distribution u(r, θ) over the cylindrical 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.

EXAMPLE 5.7.3: We seek the electrostatic potential in a cylindrical region over the domain

D ={(r, θ) |1 <r<2, 0 <θ<π/2}. The sides r = 1,r= 2, and θ = π/2 are held at ﬁxed

potential zero. The side θ = 0 has an electrostatic potential distribution f(r) given as follows.

SOLUTION: The homogeneous Laplace equation is

∂

∂r

u(r, θ) +r



∂

∂r

u(r, θ)



∂

∂θ

u(r, θ)

= 0

The boundary conditions are

u(1,θ)= 0 and u(2,θ)= 0

and

u(r, 0) = (r −1)(2 −r) and u





= 0

Here we see that the boundary conditions are homogeneous with respect to the r variable.

Ordinary differential equations from the method of separation of variables are

dθ

(θ)−λ(θ) = 0

and



R(r)





R(r)



+λR(r) = 0

316 Chapter 5

Homogeneous boundary conditions on the r equation are type 1 at r = 1 and type 1 at r = 2

R(1) = 0 and R(2) = 0

Assignment of system parameters

> restart:with(plots):a:=2:b:=Pi/2:

Allowed eigenvalues and corresponding orthonormal eigenfunctions for the r equation are

obtained from Example 2.5.7.

> lambda[n]:=(n*Pi/(ln(a)))ˆ2;

ln(2)

(5.114)

for n = 1, 2, 3,....

Orthonormal eigenfunctions

> R[n](r):=sqrt(2/ln(a))*sin(n*Pi*ln(r)/(ln(a)));R[n](s):=subs(r=s,R[n](r)):R[m](r):=

subs(n=m,R[n](r)):

(r) :=

√

2 sin



nπ ln(r)

ln(2)



√

ln(2)

(5.115)

Statement of orthonormality with respect to the weight function w(r) =

> w(r):=1/r:Int(R[n](r)*R[m](r)/r,r=1..a)=delta(n,m);



2 sin



nπ ln(r)

ln(2)



sin



mπ ln(r)

ln(2)



ln(2)r

dr = δ(n, m) (5.116)

Basis vectors for the θ equation

> theta1(theta):=sinh(sqrt(lambda)*(b−theta));theta2(theta):=cosh(sqrt(lambda)*(b−theta));

θ1(θ) := sinh



√



π −θ



θ2(θ) := cosh



√



π −θ



(5.117)

General solution to the θ equation

The Laplace Par tial Differential Equation 317

> Theta[n](theta):=A(n)*cosh(n*Pi/(ln(a))*(b−theta))+B(n)*sinh(n*Pi/(ln(a))*(b−theta));



(θ) := A(n) cosh

⎛

⎝

nπ



π −θ



ln(2)

⎞

⎠

+B(n) sinh

⎛

⎝

nπ



π −θ



ln(2)

⎞

⎠

(5.118)

Substituting into the θ-dependent boundary condition yields

> eval(subs(theta=b,Theta[n](theta)))=0;

A(n) = 0 (5.119)

The θ-dependent solution

> Theta[n](theta):=B(n)*sinh(n*Pi*(b−theta)/ln(a));



(θ) := B(n) sinh

⎛

⎝

nπ



π −θ



ln(2)

⎞

⎠

(5.120)

Generalized series terms

> u[n](r,theta):=R[n](r)*Theta[n](theta);

(r, θ) :=

√

2 sin



nπ ln(r)

ln(2)



B(n) sinh



nπ



π−θ



ln(2)



√

ln(2)

(5.121)

Eigenfunction expansion

> u(r,theta):=Sum(u[n](r,theta),n=1..inﬁnity);

u(r, θ) :=

∞



n=1

√

2 sin



nπ ln(r)

ln(2)



B(n) sinh



nπ



π−θ



ln(2)



√

ln(2)

(5.122)

Evaluation of Fourier coefﬁcients from the speciﬁc remaining boundary condition

u(r, 0) = f(r)

> f(r):=(r−1)*(2−r);

f(r) := (r −1)(2 −r) (5.123)