Articolo G.A. Partial Differential Equations and Boundary Value Problems with Maple V
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108 Chapter 2
SOLUTION: We evaluate the Fourier coefficients
> F(n):=Int(eval(f(x)*phi[n](x))*w(x),x=a..b);F(n):=value(%):
F(n) :=
1
0
(1 −x) cos
√
λ
n
x
√
2
sin
√
λ
n
2
+1
dx (2.124)
> F(n):=radsimp(subs(tan(sqrt(lambda[n])*b)=h/sqrt(lambda[n]),F(n)));
F(n) :=
1 −cos
√
λ
n
√
2
λ
n
2 −cos
√
λ
n
2
(2.125)
> Series:=simplify(Sum(F(n)*phi[n](x),n=1..infinity));
Series :=
∞
n=1
2
−1 +cos
√
λ
n
cos
√
λ
n
x
λ
n
−2 +cos
√
λ
n
2
(2.126)
Evaluation of the eigenvalues from the roots of the eigenvalue equation yields
> tan(sqrt(lambda[n])*b)=h/sqrt(lambda[n]);
tan
λ
n
=
1
√
λ
n
(2.127)
> plot({tan(v),(h*b)/v},v=0..20,y=0..3/2,thickness=10);
v
0 5 10 15 20
y
0
0.5
1.0
1.5
Figure 2.9
If we set v =
√
λb, then the eigenvalues are found from the intersection points of the curves
shown in Figure 2.9. We evaluate a few of these eigenvalues from the Maple fsolve command
for the roots of the eigenvalue equation to be given as
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 109
> lambda[1]:=(1/b*(fsolve((tan(v)−(h*b)/v),v=0..1)))ˆ2;
λ
1
:= 0.7401738844 (2.128)
> lambda[2]:=(1/b*(fsolve((tan(v)−(h*b)/v),v=1..4)))ˆ2;
λ
2
:= 11.73486183 (2.129)
> lambda[3]:=(1/b*(fsolve((tan(v)−(h*b)/v),v=4..7)))ˆ2;
λ
3
:= 41.43880785 (2.130)
> lambda[4]:=(1/b*(fsolve((tan(v)−(h*b)/v),v=7..10)))ˆ2;
λ
4
:= 90.80821420 (2.131)
> lambda[5]:=(1/b*(fsolve((tan(v)−(h*b)/v),v=10..13)))ˆ2;
λ
5
:= 159.9032889 (2.132)
First five terms of expansion
> Series:=eval(sum(F(n)*phi[n](x),n=1..5)):
> plot({Series,f(x)},x=a..b,thickness=10);
x
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Figure 2.10
The two curves of Figure 2.10 depict the function f(x) and its Fourier series approximation in
terms of the orthonormal eigenfunctions for the particular operator and boundary conditions
given earlier. Note that f(x) does not satisfy either of the given boundary conditions at the left
or right end points. The convergence is pointwise.
EXAMPLE 2.5.6: Consider the Euler operator with Robin and Dirichlet conditions. We seek
the eigenvalues and corresponding orthonormal eigenfunctions for the Euler differential
equation [Sturm-Liouville type for p(x) = 1,q(x)= 0,w(x)= 1] over the interval
110 Chapter 2
I ={x |0 <x<b}. The boundary conditions are type 3 at the left and type 1 at the right end
points.
Euler differential equation
d
2
dx
2
y(x) +λy(x) = 0
Boundary conditions (h>0)
y
x
(0) −hy(0) = 0 and y(b) = 0
SOLUTION: We consider two possibilities for values of λ. We first consider λ = 0. For this
case, the system basis vectors are
> restart:y1(x):=1;y2(x):=x;
y1(x) := 1
(2.133)
y2(x) := x
General solution
> y(x):=C1*y1(x)+C2*y2(x);
y(x) := C1 +C2x (2.134)
Substituting into the boundary conditions yields
> eval(subs(x=0,diff(y(x),x)−h*y(x)))=0;
C2 −hC1 = 0 (2.135)
> eval(subs(x=b,y(x)))=0;
C1 +C2b = 0 (2.136)
The only solution to the above is the trivial solution. We next consider λ>0. We set λ = μ
2
,
and, for this case, the system basis vectors are
> y1(x):=sin(mu*x);y2(x):=cos(mu*x);
y1(x) := sin(μx)
(2.137)
y2(x) := cos(μx)
General solution
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 111
> y(x):=C1*y1(x)+C2*y2(x);
y(x) := C1 sin(μx) +C2 cos(μx) (2.138)
Substituting into the boundary conditions yields
> eval(subs(x=0,diff(y(x),x)−h*y(x)))=0;
C1μ −hC2 = 0 (2.139)
> eval(subs(x=b,y(x)))=0;
C1 sin(μb) +C2 cos(μb) = 0 (2.140)
The only nontrivial solutions to the preceding occur when both C1 and C2 are arbitrary and μ
satisfies the following eigenvalue equation:
> h*sin(mu*b)+mu*cos(mu*b)=0;
h sin(μb) +μ cos(μb) = 0 (2.141)
We identify these roots as μ
n
for n = 1, 2, 3,....
Allowed eigenvalues are λ
n
= μ
2
n
> lambda[n]=mu[n]ˆ2;
λ
n
= μ
2
n
(2.142)
Nonnormalized eigenfunctions are
> phi[n](x):=h*sin(sqrt(lambda[n])*x)+sqrt(lambda[n])*cos(sqrt(lambda[n])*x);
φ
n
(x) := h sin
λ
n
x
+
λ
n
cos
λ
n
x
(2.143)
Normalization
Evaluating the norm from the inner product of the eigenfunctions with respect to the weight
function w(x) = 1 over the interval yields
> w(x):=1:unprotect(norm):norm:=sqrt(Int(phi[n](x)ˆ2*w(x),x=0..b));norm:=value(%):
norm :=
b
0
h sin
λ
n
x
+
λ
n
cos
λ
n
x
2
dx (2.144)
112 Chapter 2
Substitution of the eigenvalue equation simplifies the preceding equation:
> norm:=radsimp(subs(sin(sqrt(lambda[n])*b)=−sqrt(lambda[n])/
h*cos(sqrt(lambda[n])*b),norm));
norm :=
1
2
√
2
−
−2h
2
+h
2
cos
√
λ
n
b
2
−h
3
b +λ
n
cos
√
λ
n
b
2
−λ
n
bh
h
h
(2.145)
Orthonormal eigenfunctions
> phi[n](x):=phi[n](x)/norm;phi[m](x):=subs(n=m,phi[n](x)):
φ
n
(x) :=
h sin
√
λ
n
x
+
√
λ
n
cos
√
λ
n
x
√
2h
−
−2h
2
+h
2
cos
√
λ
n
b
2
−h
3
b +λ
n
cos
√
λ
n
b
2
−λ
n
bh
h
(2.146)
Statement of orthonormality
> Int(phi[n](x)*phi[m](x)*w(x),x=0..b)=delta(n,m);;
b
0
2
h sin
λ
n
x
+
λ
n
cos
λ
n
x
h
2
h sin
λ
m
x
+
λ
m
cos
λ
m
x
−
−2h
2
+h
2
cos
λ
n
b
2
−h
3
b +λ
n
cos
λ
n
b
2
−λ
n
bh
h
−
−2h
2
+h
2
cos
λ
m
b
2
−h
3
b +λ
m
cos
λ
m
b
2
−λ
m
bh
h
dx = δ(n, m)
(2.147)
Generalized Fourier series expansion
> f(x):=Sum(F(n)*phi[n](x),n=1..infinity);f(x):=‘f(x)’:
f(x) :=
∞
n=1
F(n)
h sin
√
λ
n
x
+
√
λ
n
cos
√
λ
n
x
√
2h
−
−2h
2
+h
2
cos
√
λ
n
b
2
−h
3
b +λ
n
cos
√
λ
n
b
2
−λ
n
bh
h
(2.148)
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 113
Fourier coefficients
> F(n):=Int(f(x)*phi[n](x)*w(x),x=0..b);F(n):=‘F(n)’:
F(n) :=
b
0
f(x)
h sin
√
λ
n
x
+
√
λ
n
cos
√
λ
n
x
√
2h
−
−2h
2
+h
2
cos
√
λ
n
b
2
−h
3
b +λ
n
cos
√
λ
n
b
2
−λ
n
bh
h
dx (2.149)
This is the generalized series expansion of f(x) in terms of the “complete” set of eigenfunctions
for the particular Sturm-Liouville operator and boundary conditions over the interval.
DEMONSTRATION: Develop the generalized series expansion for f(x) = 1 −x over the
interval I ={x |0 <x<1} in terms of the preceding eigenfunctions for h = 1. We assign the
system values
> a:=0;b:=1;h:=1;f(x):=1−x;
a := 0
b := 1
h := 1
f(x) := 1 −x (2.150)
SOLUTION: We evaluate the Fourier coefficients
> F(n):=Int(eval(f(x)*phi[n](x))*w(x),x=a..b);F(n):=value(%):
F(n) :=
1
0
(1 −x)
sin
√
λ
n
x
+
√
λ
n
cos
√
λ
n
x
√
2
3 −cos
√
λ
n
2
−λ
n
cos
√
λ
n
2
+λ
n
dx (2.151)
> F(n):=radsimp(subs(sin(sqrt(lambda[n])*b)=−sqrt(lambda[n])/
h*cos(sqrt(lambda[n])*b),F(n)));
F(n) :=
2
√
2
√
λ
n
3 −cos
√
λ
n
2
−λ
n
cos
√
λ
n
2
+λ
n
(2.152)
> Series:=simplify(Sum(F(n)*phi[n](x),n=1..infinity));
Series :=
∞
n=1
⎛
⎝
−
4
sin
√
λ
n
x
+
√
λ
n
cos
√
λ
n
x
√
λ
n
−3 +cos
√
λ
n
2
+λ
n
cos
√
λ
n
2
−λ
n
⎞
⎠
(2.153)
114 Chapter 2
Evaluation of the eigenvalues from the roots of the eigenvalue equation yields
> tan(sqrt(lambda[n])*b)=−sqrt(lambda[n])/h;
tan
λ
n
=−
λ
n
(2.154)
> plot({tan(v),−v/(h*b)},v=0..20,y=−20..0,thickness=10);
v
5101520
y
220
215
210
25
0
Figure 2.11
If we set v =
√
λb, then the eigenvalues are found from the intersection points of the curves
shown in Figure 2.11. We evaluate a few of these eigenvalues from the Maple fsolve command
for the roots of the eigenvalue equation to be given as
> lambda[1]:=(1/b*(fsolve((tan(v)+v/(b*h)),v=1..3)))ˆ2;
λ
1
:= 4.115858365 (2.155)
> lambda[2]:=(1/b*(fsolve((tan(v)+v/(b*h)),v=3..6)))ˆ2;
λ
2
:= 24.13934203 (2.156)
> lambda[3]:=(1/b*(fsolve((tan(v)+v/(b*h)),v=6..9)))ˆ2;
λ
3
:= 63.65910654 (2.157)
> lambda[4]:=(1/b*(fsolve((tan(v)+v/(b*h)),v=10..12)))ˆ2;
λ
4
:= 122.8891618 (2.158)
> lambda[5]:=(1/b*(fsolve((tan(v)+v/(b*h)),v=13..15)))ˆ2;
λ
5
:= 201.8512584 (2.159)
First five terms of expansion
> Series:=eval(sum(F(n)*phi[n](x),n=1..5)):
> plot({Series,f(x)},x=a..b,thickness=10);
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 115
x
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Figure 2.12
The two curves shown in Figure 2.12 depict the function f(x) and its Fourier series
approximation in terms of the orthonormal eigenfunctions for the particular operator and
boundary conditions given earlier. Note that f(x) satisfies the boundary condition at the right
but not at the left end point. The convergence is pointwise.
EXAMPLE 2.5.7: Consider the Cauchy-Euler operator with Dirichlet conditions. We seek the
eigenvalues and corresponding orthonormal eigenfunctions for the Cauchy-Euler differential
equation [Sturm-Liouville type for p(x) = x, q(x) = 0, w(x) =
1
x
] over the interval
I ={x |1 <x<b}. The boundary conditions are type 1 at the left and type 1 at the right end
points.
Cauchy-Euler differential equation
x
2
d
2
dx
2
y(x)
+x
d
dx
y(x)
+λy(x) = 0
Boundary conditions
y(1) = 0 and y(b) = 0
SOLUTION: We consider two possibilities for values of λ. We first consider λ = 0. For this
case, the system basis vectors are
> restart:y1(x):=1;y2(x):=ln(x);
y1(x) := 1
y2(x) := ln(x) (2.160)
General solution
> y(x):=C1*y1(x)+C2*y2(x);
y(x) := C1 +C2 ln(x) (2.161)
116 Chapter 2
Substituting into the boundary conditions yields
> eval(subs(x=1,y(x)))=0;
C1 = 0 (2.162)
> eval(subs(x=b,y(x)))=0;
C1 +C2 ln(b) = 0 (2.163)
The only solution to the preceding is the trivial solution. We next consider λ>0. We set
λ = μ
2
, and, for this case, the system basis vectors are
> y1(x):=sin(mu*ln(x));y2(x):=cos(mu*ln(x));
y1(x) := sin(μ ln(x))
y2(x) := cos(μ ln(x)) (2.164)
General solution
> y(x):=C1*y1(x)+C2*y2(x);
y(x) := C1 sin
(
μ ln(x)
)
+C2 cos
(
μ ln(x)
)
(2.165)
Substituting into the boundary conditions yields
> eval(subs(x=1,y(x)))=0;
C2 = 0 (2.166)
> eval(subs(x=b,y(x)))=0;
C1 sin(μ ln(b)) +C2 cos(μ ln(b)) = 0 (2.167)
The only nontrivial solutions to the preceding occur when C2 = 0, C1 is arbitrary, and μ
satisfies the following eigenvalue equation:
> sin(mu*ln(b))=0;
sin(μ ln(b)) = 0 (2.168)
Thus, the values for μ are
> mu[n]:=n*Pi/ln(b);
μ
n
:=
nπ
ln(b)
(2.169)
for n = 1, 2, 3,....
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 117
Allowed eigenvalues are λ
n
= μ
2
n
> lambda[n]:=(n*Pi/ln(b))ˆ2;
λ
n
:=
n
2
π
2
ln(b)
2
(2.170)
Nonnormalized eigenfunctions are
> phi[n](x):=(sin(mu[n]*ln(x)));
φ
n
(x) := sin
nπ ln(x)
ln(b)
(2.171)
Normalization
Evaluating the norm from the inner product of the eigenfunctions with respect to the weight
function w(x) =
1
x
over the interval yields
> w(x):=1/x:unprotect(norm):norm:=sqrt(Int(phi[n](x)ˆ2*w(x),x=1..b));norm:=value(%):
norm :=
b
1
sin
nπ ln(x)
ln(b)
2
x
dx (2.172)
Substitution of the eigenvalue equation simplifies the norm
> norm:=radsimp(subs({sin(n*Pi)=0,cos(n*Pi)=(−1)ˆn},norm));
norm :=
1
2
√
2
ln(b) (2.173)
Orthonormal eigenfunctions
> phi[n](x):=phi[n](x)/norm;phi[m](x):=subs(n=m,phi[n](x)):
φ
n
(x) :=
sin
nπ ln(x)
ln(b)
√
2
√
ln(b)
(2.174)
Statement of orthonormality
> Int(phi[n](x)*phi[m](x)*w(x),x=1..b)=delta(n,m);;
b
1
2 sin
nπ ln(x)
ln(b)
sin
mπ ln(x)
ln(b)
ln(b)x
dx = δ(n, m) (2.175)