Articolo G.A. Partial Differential Equations and Boundary Value Problems with Maple V
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118 Chapter 2
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) sin
nπ ln(x)
ln(b)
√
2
√
ln(b)
(2.176)
Fourier coefficients
> F(n):=Int(f(x)*phi[n](x)*w(x),x=1..b);F(n):=‘F(n)’:
F(n) :=
b
1
f(x) sin
nπ ln(x)
ln(b)
√
2
√
ln(b)x
dx (2.177)
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) = x −1 over the
interval I ={x |1 <x<2} in terms of the preceding eigenfunctions. We assign the system
values
> a:=1;b:=2;f(x):=x−1;
a :=1
b :=2
f(x) :=x −1 (2.178)
SOLUTION: We evaluate the Fourier coefficients
> F(n):=eval(Int(f(x)*phi[n](x)*w(x),x=a..b));F(n):=simplify(value(%)):
F(n) :=
2
1
(x −1) sin
nπ ln(x)
ln(2)
√
2
√
ln(2)x
dx (2.179)
> F(n):=combine(%,trig):F(n):=subs(cos(n*Pi)=(−1)ˆn,sin(n*Pi)=0,F(n));
F(n) :=
−
√
ln(2)
√
2 n
2
π
2
(−1)
n
−ln(2)
5/2
√
2 +ln(2)
5/2
√
2(−1)
n
n
3
π
3
+nπ ln(2)
2
(2.180)
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 119
> Series:=eval(Sum(F(n)*phi[n](x),n=1..infinity));
Series :=
∞
n=1
−
√
ln(2)
√
2 n
2
π
2
(−1)
n
−ln(2)
5/2
√
2 +ln(2)
5/2
√
2(−1)
n
sin
nπ ln(x)
ln(2)
√
2
n
3
π
3
+nπ ln(2)
2
√
ln(2)
(2.181)
First few terms of expansion
> Series:=eval(sum(F(n)*phi[n](x),n=1..5)):
> plot({Series,f(x)},x=a..b,thickness=10);
x
1.21 1.4 1.6 1.8 2
0.0
0.2
0.4
0.6
0.8
1
Figure 2.13
The two curves in Figure 2.13 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 given boundary conditions at the left but fails to do so
at the right end point. The convergence is pointwise.
EXAMPLE 2.5.8: Consider the Cauchy-Euler operator with Neumann and 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 2 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
x
(1) = 0 and y(b) = 0
120 Chapter 2
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) (2.182)
General solution
> y(x):=C1*y1(x)+C2*y2(x);
y(x) := C1 +C2 ln(x) (2.183)
Substituting into the boundary conditions yields
> eval(subs(x=1,diff(y(x),x)))=0;
C2 = 0 (2.184)
> eval(subs(x=b,y(x)))=0;
C1 +C2 ln(b) = 0 (2.185)
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)) (2.186)
General solution
> y(x):=C1*y1(x)+C2*y2(x);
y(x) := C1 sin(μ ln(x)) +C2 cos(μ ln(x)) (2.187)
Substituting into the boundary conditions yields
> eval(subs(x=1,diff(y(x),x)))=0;
C1μ = 0 (2.188)
> eval(subs(x=b,y(x)))=0;
C1 sin(μ ln(b)) +C2 cos(μ ln(b)) = 0 (2.189)
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 121
The only nontrivial solutions to the preceding occur when C1 = 0, C2 is arbitrary, and μ
satisfies the following eigenvalue equation:
> cos(mu*ln(b))=0;
cos(μ ln(b)) = 0 (2.190)
Thus, μ takes on the values
> mu[n]:=(2*n−1)*Pi/(2*ln(b));
μ
n
:=
1
2
(2n −1)π
ln(b)
(2.191)
for n = 1, 2, 3,....
Allowed eigenvalues are λ
n
= μ
2
n
> lambda[n]:=mu[n]ˆ2;
λ
n
:=
1
4
(2n −1)
2
π
2
ln(b)
2
(2.192)
Nonnormalized eigenfunctions are
> phi[n](x):=cos(mu[n]*ln(x));
φ
n
(x) := cos
1
2
(2n −1)π ln(x)
ln(b)
(2.193)
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:=expand(value(%)):
norm :=
b
1
cos
1
2
(2n−1)π ln(x)
ln(b)
2
x
dx (2.194)
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) (2.195)
122 Chapter 2
Orthonormal eigenfunctions
> phi[n](x):=phi[n](x)/norm;phi[m](x):=subs(n=m,phi[n](x)):
φ
n
(x) :=
cos
1
2
(2n−1)π ln(x)
ln(b)
√
2
√
ln(b)
(2.196)
Statement of orthonormality
> Int(phi[n](x)*phi[m](x)*w(x),x=1..b)=delta(n,m);
b
1
2 cos
1
2
(2n−1)π ln(x)
ln(b)
cos
1
2
(2m−1)π ln(x)
ln(b)
ln(b)x
dx = δ(n, m) (2.197)
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) cos
1
2
(2n−1)π ln(x)
ln(b)
√
2
√
ln(b)
(2.198)
Fourier coefficients
> F(n):=Int(f(x)*phi[n](x)*w(x),x=1..b);F(n):=‘F(n)’:
F(n) :=
b
1
f(x) cos
1
2
(2n−1)π ln(x)
ln(b)
√
2
√
ln(b)x
dx (2.199)
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) = 2x −x
2
over the
interval I ={x |1 <x<2} in terms of the preceding eigenfunctions. We assign the system
values
> a:=1;b:=2;f(x):=2*x−xˆ2;
a := 1
b := 2
f(x) := 2x −x
2
(2.200)
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 123
SOLUTION: We evaluate the Fourier coefficients
> F(n):=eval(Int(f(x)*phi[n](x)*w(x),x=a..b));F(n):=simplify(value(%)):
F(n) :=
2
1
(2x −x
2
) cos
1
2
(2n−1)π ln(x)
ln(2)
√
2
√
ln(2)x
dx (2.201)
> F(n):=combine(%,trig):F(n):=factor(simplify(subs(cos(n*Pi)=(−1)ˆn,sin(n*Pi)=0,F(n))));
F(n) := −
96 ln(2)
5/2
√
2
2πn(−1)
n
−π(−1)
n
+ln(2)
16 ln(2)
2
−4π
2
n +π
2
+4π
2
n
2
4π
2
n
2
−4π
2
n +4ln(2)
2
+π
2
(2.202)
> Series:=eval(Sum(F(n)*phi[n](x),n=1..infinity));
Series :=
∞
n=1
⎛
⎝
−
192 ln(2)
2
2πn(−1)
n
−π(−1)
n
+ln(2)
cos
1
2
(2n−1)π ln(x)
ln(2)
16 ln(2)
2
−4π
2
n +π
2
+4π
2
n
2
4π
2
n
2
−4π
2
n +4ln(2)
2
+π
2
⎞
⎠
(2.203)
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
1.21 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
Figure 2.14
The two curves in Figure 2.14 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 given boundary conditions at both the left and the
right end points. The convergence is uniform.
124 Chapter 2
EXAMPLE 2.5.9: Consider the Cauchy-Euler operator with Dirichlet and Robin 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 3 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 (h>0)
y(1) = 0 and y
x
(b) +hy(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) (2.204)
General solution
> y(x):=C1*y1(x)+C2*y2(x);
y(x) := C1 +C2 ln(x) (2.205)
Substituting into the boundary conditions yields
> eval(subs(x=1,y(x)))=0;
C1 = 0 (2.206)
> eval(subs(C1=0,x=b,diff(y(x),x)+h*y(x)))=0;
C2
b
+hC2 ln(b) = 0 (2.207)
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)) (2.208)
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 125
General solution
> y(x):=C1*y1(x)+C2*y2(x);
y(x) := C1 sin(μ ln(x)) +C2 cos(μ ln(x)) (2.209)
Substituting into the boundary conditions yields
> eval(subs(x=1,y(x)))=0;
C2 = 0 (2.210)
> eval(subs(C2=0,x=b,diff(y(x),x)+h*y(x)))=0;
C1 cos(μ ln(b))μ
b
+hC1 sin(μ ln(b)) = 0 (2.211)
The only nontrivial solutions to the preceding are that C2 = 0, C1 is arbitrary, and μ must
satisfy the following eigenvalue equation:
> h*sin(mu*ln(b))+mu/b*cos(mu*ln(b))=0;
h sin(μ ln(b)) +
μ cos(μ ln(b))
b
= 0 (2.212)
We indicate 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
(2.213)
Nonnormalized eigenfunctions are
> phi[n](x):=sin(sqrt(lambda[n])*ln(x));
φ
n
(x) := sin
λ
n
ln(x)
(2.214)
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
√
λ
n
ln(x)
2
x
dx (2.215)
126 Chapter 2
Substitution of the eigenvalue equation simplifies the norm
> norm:=simplify(subs(sin(sqrt(lambda[n])*ln(b))=−sqrt(lambda[n])/
(h*b)*cos(sqrt(lambda[n])*ln(b)),norm));
norm :=
1
2
√
2
cos
√
λ
n
ln(b)
2
+ln(b)hb
hb
(2.216)
Orthonormal eigenfunctions
> phi[n](x):=phi[n](x)/norm;phi[m](x):=subs(n=m,phi[n](x)):
φ
n
(x) :=
sin
√
λ
n
ln(x)
√
2
cos
√
λ
n
ln(b)
2
+ln(b)hb
hb
(2.217)
Statement of orthonormality
> Int(phi[n](x)*phi[m](x)*w(x),x=1..b)=delta(n,m);
b
1
2 sin
√
λ
n
ln(x)
sin
√
λ
n
ln(x)
cos
√
λ
n
ln(b)
2
+ln(b)hb
hb
cos
√
λ
n
ln(b)
2
+ln(b)hb
hb
x
dx = δ(n, m) (2.218)
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)sin
√
λ
n
ln(x)
√
2
cos
√
λ
n
ln(b)
2
+ln(b)hb
hb
(2.219)
Fourier coefficients
> F(n):=Int(f(x)*phi[n](x)*w(x),x=1..b);F(n):=‘F(n)’:
F(n) :=
b
1
f(x)sin
√
λ
n
ln(x)
√
2
cos
√
λ
n
ln(b)
2
+ln(b)hb
hb
x
dx (2.220)
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) = x −1 over the
interval I ={x |1 <x<2} in terms of the preceding eigenfunctions for h = 1. We assign the
system values
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 127
> a:=1;b:=2;h:=1;f(x):=x−1;
a := 1
b := 2
h := 1
f(x) := x −1 (2.221)
SOLUTION: We evaluate the Fourier coefficients
> F(n):=eval(expand((Int(f(x)*phi[n](x)*w(x),x=a..b))));F(n):=simplify(value(%)):
F(n) :=
√
2
⎛
⎝
2
1
sin
λ
n
ln(x)
−
sin
√
λ
n
ln(x)
x
dx
⎞
⎠
1
2
cos
√
λ
n
ln(2)
2
+ln(2)
(2.222)
Substitution of the eigenvalue equation simplifies the preceding equation
> F(n):=(eval(subs(sin(sqrt(lambda[n])*ln(b))=−sqrt(lambda[n])/
(b*h)*cos(sqrt(lambda[n])*ln(b)),F(n))));
F(n) :=
1
2 cos
√
λ
n
ln(2)
2
+4ln(2)
√
λ
n
(
1 +λ
n
)
2
−2λ
n
cos
1
2
λ
n
ln(2)
2
+λ
n
+2 cos
1
2
λ
n
ln(2)
2
+ 4 sin
1
2
λ
n
ln(2)
λ
n
cos
1
2
λ
n
ln(2)
−2
√
2
(2.223)
> Series:=eval(Sum(F(n)*phi[n](x),n=1..infinity));
Series :=
∞
n=1
4
−2λ
n
cos
1
2
λ
n
ln(2)
2
+λ
n
+2 cos
1
2
λ
n
ln(2)
2
+ 4 sin
1
2
λ
n
ln(2)
λ
n
cos
1
2
λ
n
ln(2)
−2
sin
λ
n
ln(x)
2 cos
λ
n
ln(2)
2
+4ln(2)
λ
n
(
1 +λ
n
)
1
2
cos
λ
n
ln(2)
2
+ln(2)
(2.224)