Chapter 4 PDE
Excersise #16
#16
a)
Reduce the following BVP to Sturm-Liouville problem:
x2 u   2xu   u  0
u 1  0
u e  0
and find eigenvalues and eigenfunctions .
b) Use the obtained set of eigenfunctions for generalized Fourier series representation of the
function
f  x   xe x in the interval  1,e 
Sketch the graph for n  5 and n  20 .
Solution:
The 2nd order ODE includes parameter  .
We have to find the values of this parameter ( n )
for which ODE with boundary conditions has non-trivial solution un  x  .
The existence of such solution is provided by the Sturm-Liouville Theorem (4.5.3 p.268).
1) Reduce our BVP to SLP:
Rewrite equation in self-adjoint form with the help of the multiplication factor (4.5.4 Eq.19 p.271):
Identify coefficients


x 2 u   2x u    0    u  0
a0
a1
 a2


e a
a1
  x 
0
a0
dx
e x

x2
2x
2
dx
2
dx
e x
e 2 ln x eln x
x2




1
x2
x2
x2
x2
1
2
r  x   a0   x 2  0
p  x    1  0
q  x  0
 x 2 u     0    u  0


(self-adjoint form, Eq.5)

Lu    x 2 u    u
(operator form, Eq.13)
u 1  0
u e  0
(both conditions are of the Dirichlet type)
According to Sturm-Liouville Theorem, this SLP has
infinitely many positive eigenvalues n  0 for which
boundary value problem has non-trivial solution un  x  (eigenfunctions).
2) Find eigenvalues and eigenfunctions (solve BVP)
Chapter 4 PDE
Excersise #16
x2 u   2xu   u  0
u 1  0
u e  0
2nd order ODE, homogeneous, with variable coefficients, linear, Euler-Cauchy type
Auxiliary equation (see table “linear o.d.e.” Euler-Cauchy Equation):
1  x2 u   2 xu    u  0
a0
a1
a2
a0 m2   a1  a0  m  a2  0
m2   2  1 m    0
m2  m    0
m1,2 
case 1
1  1  4 
1
1
 

2
2
4
1
 0
4
 0
m1  m2
general solution u  c1 xm1  c2 xm2
apply boundary conditions:
x1
0  c1  c2
xe
0  c1em1  c2 em2
Rewrite as a system in matrix form:
1   c1  0 
 1
em em  c   0 

 2  
1
2
1 
 1
det  m1
 e m2  e m1  0 because m1  m2
m2 
e
e


Therefore, system has only the trivial solution
 c1  0 
c   0 
 2  
which yields the trivial solution of BVP.
Therefore, there is no eigenvalues in the interval
1
0
4
case 2
1
 0
4
m1  m2  
general solution u  c1 x

1
2
 c2 x

1
2
1
2
ln x
apply boundary conditions:
x1
0  c1  c2  0
1

2
 c1  0
xe
0  c2 e
 c2  0
It also yields the trivial solution.
1
4
Chapter 4 PDE
Excersise #16
case 3
1
 0
4
m1,2  
1
1
1
1

      1    i  
2
4
2
4

 
1
4




  1
1
1
general solution u  c1 cos    ln x   c2 sin    ln x   x 2




4
4





apply boundary conditions:
0  c1 cos0  c2 sin0  c1  0
x1
u  c2 x
xe

1
2


1
sin    ln x 


4


0  c2 e

1
2
1




1
1
sin    ln e   c2 e 2 sin    



4
4 



non-trivial solution only if

1
sin      0

4 



1
 n
4
n  1,2,...
1
 n 2 2
4
1
 n 2 2
4

un 
un
2
eigenvalues
sin  n ln x 
eigenfunctions
x
 sin  n ln x  
 
 dx
x

1

2
e
e

sin2  n ln x 
1

x
dx
1 e 2
sin  n ln x d  n ln x 
n 1
 n ln x sin  n ln x  cos  n ln x  



2
 2
 1
e
1

n
Chapter 4 PDE

1
n

1
2
Excersise #16

  n ln e sin  n ln e  cos  n ln e    n ln1 sin  n ln1 cos  n ln1  






2
2
2
2
 

 
 

b) Generalized Fourier series expansion:

f  x    cn un  x 
n 1
cn 
 f ,un  e
 2 f  x un  x  dx
 un ,un  1
Maple example for f  x   xe x :
> f:=x*exp(-x);
f := x e
> u[n]:=sin(n*Pi*ln(x))/sqrt(x);
un :=
> N2[n]:=int(u[n]^2,x=1..exp(1));
N2n :=
( x )
sin ( n  ln ( x ) )
x
1 cos( n  ) sin ( n  )n 
2
n
> c[n]:=int(f*u[n],x=1..exp(1));
e
( x )
cn := 
xe
sin ( n  ln ( x ) ) d x


1
> f5:=sum(c[n]*u[n]/N2[n],n=1..5):
> f20:=sum(c[n]*u[n]/N2[n],n=1..20):
> plot({f,f5,f20},x=1..exp(1),axes=boxed,color=black);