THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Master 1
Wednesday, February 23
Cooling of a sphere Cooling of a cylinder
THE STURM-LIOUVILLE THEORY
Fourier
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
Jean-Baptiste Joseph FOURIER 1768 (Auxerre) - 1830 (Paris)
THE STURM-LIOUVILLE THEORY
Sturm
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
Jacques Charles Franc ¸ ois STURM 1803 (Gen`eve) - 1855 (Paris)
THE STURM-LIOUVILLE THEORY
Liouville
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
Joseph Liouville 1809 (Saint-Omer) - 1882 (Paris)
THE STURM-LIOUVILLE THEORY
FIRST PART :
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville
The birth of STURM-LIOUVILLE Theory.
Cooling of a sphere Cooling of a cylinder
THE STURM-LIOUVILLE THEORY
FIRST PART :
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville
The birth of STURM-LIOUVILLE Theory.
Cooling of a sphere Cooling of a cylinder
SECOND PART : The developement of STURM-LIOUVILLE Theory.
First part :
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
Cooling of a sphere.
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
We consider a homogeneous sphere center O radius R.
Let us suppose that the temperature θ(x, t) of this sphere depends only on r =| x | and t (time) so that heat equation : K ∇2 θ =
∂θ ∂t
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
Let us suppose that the temperature θ(x, t) of this sphere depends only on r =| x | and t (time) so that heat equation : K ∇2 θ =
∂θ ∂t
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere
becomes :
Cooling of a cylinder
∂θ ∂ 2 θ 2 ∂θ + = K −1 ∂r 2 r ∂t ∂t
(1)
Let us suppose that the temperature θ(x, t) of this sphere depends only on r =| x | and t (time) so that heat equation : K ∇2 θ =
∂θ ∂t
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere
becomes :
Cooling of a cylinder
∂θ ∂ 2 θ 2 ∂θ + = K −1 ∂r 2 r ∂t ∂t
(1)
Fourier considered the case which the heat loss at the boundary is proportionnal to the excess temperature, i.e : ∂θ = −hθ(R, t) ∂r r =R
(2)
∂2θ ∂r 2
+
2 ∂θ r ∂t
= K −1 ∂θ ∂t
(1)
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
∂2θ ∂r 2
+
2 ∂θ r ∂t
= K −1 ∂θ ∂t
(1)
Technique of separation of variables : we consider solutions of the form : θ(r , t) = ρ(r )τ (t)
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
∂2θ ∂r 2
+
2 ∂θ r ∂t
= K −1 ∂θ ∂t
(1)
Technique of separation of variables : we consider solutions of the form : θ(r , t) = ρ(r )τ (t) 2 ρ (r )τ (t) + ρ0 (r )τ (t) = K −1 ρ(r )τ 0 (t) r 00
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
∂2θ ∂r 2
+
2 ∂θ r ∂t
= K −1 ∂θ ∂t
(1)
Technique of separation of variables : we consider solutions of the form : θ(r , t) = ρ(r )τ (t) 2 ρ (r )τ (t) + ρ0 (r )τ (t) = K −1 ρ(r )τ 0 (t) r 00
(if τ (t) 6= 0)
∀r ∈]0, R] : 1 00 2 τ 0 (t) ρ (r ) + ρ0 (r ) = K −1 ρ(r ) r τ (t)
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
∂2θ ∂r 2
+
2 ∂θ r ∂t
= K −1 ∂θ ∂t
(1)
Technique of separation of variables : we consider solutions of the form : θ(r , t) = ρ(r )τ (t) 2 ρ (r )τ (t) + ρ0 (r )τ (t) = K −1 ρ(r )τ 0 (t) r 00
(if τ (t) 6= 0)
∀r ∈]0, R] : 1 00 2 τ 0 (t) ρ (r ) + ρ0 (r ) = K −1 ρ(r ) r τ (t)
Let be −λ2 a constant so that : 1 00 2 τ 0 (t) ρ (r ) + ρ0 (t) = −λ2 = K −1 ρ(r ) r τ (t)
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
THE STURM-LIOUVILLE THEORY
Thus : (
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
2 ρ00 (r ) + ρ0 (r ) + λ2 ρ(r ) = 0 r
(3)
K −1 τ 0 (t) + λ2 τ (t) = 0
(4)
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
THE STURM-LIOUVILLE THEORY
Thus : (
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
2 ρ00 (r ) + ρ0 (r ) + λ2 ρ(r ) = 0 r
(3)
K −1 τ 0 (t) + λ2 τ (t) = 0
(4)
Cooling of a sphere
Equation (4) is easily solved to give : τ (t) = A1 exp(−λ2 Kt) with A1 an arbitrary constant.
Fourier-Sturm-Liouville
(5)
Cooling of a cylinder
ρ00 (r ) + 2r ρ0 (r ) + λ2 ρ(r ) = 0
(3)
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
To solve (3) we can use two extra conditions :
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
ρ00 (r ) + 2r ρ0 (r ) + λ2 ρ(r ) = 0
THE STURM-LIOUVILLE THEORY
(3)
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
To solve (3) we can use two extra conditions :
Fourier-Sturm-Liouville Cooling of a sphere
∂θ ∂r r =R
= −hθ(R, t)
(2) gives :
ρ0 (R) = −hρ(R)
Cooling of a cylinder
(3a)
ρ00 (r ) + 2r ρ0 (r ) + λ2 ρ(r ) = 0
THE STURM-LIOUVILLE THEORY
(3)
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
To solve (3) we can use two extra conditions :
Fourier-Sturm-Liouville Cooling of a sphere
∂θ ∂r r =R
= −hθ(R, t)
(2) gives :
ρ0 (R) = −hρ(R)
Cooling of a cylinder
(3a)
Physical reality demands that θ be bounded for all r ∈]0, R], so there exists an M such that : | ρ(r ) |≤ M
∀r ∈]0, R]
(3b)
ρ00 (r ) + 2r ρ0 (r ) + λ2 ρ(r ) = 0
THE STURM-LIOUVILLE THEORY
(3)
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
To solve (3) we can use two extra conditions :
Fourier-Sturm-Liouville Cooling of a sphere
∂θ ∂r r =R
= −hθ(R, t)
(2) gives :
ρ0 (R) = −hρ(R)
Cooling of a cylinder
(3a)
Physical reality demands that θ be bounded for all r ∈]0, R], so there exists an M such that : | ρ(r ) |≤ M
∀r ∈]0, R]
but (3) requires more thought...
(3b)
ρ00 (r ) + 2r ρ0 (r ) + λ2 ρ(r ) = 0
(3)
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
The key to solve (3) lies in the substitution : γ(r ) = r ρ(r ).
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
ρ00 (r ) + 2r ρ0 (r ) + λ2 ρ(r ) = 0
(3)
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
The key to solve (3) lies in the substitution : γ(r ) = r ρ(r ). Then γ 0 (r ) = r ρ0 (r ) + ρ(r )
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
ρ00 (r ) + 2r ρ0 (r ) + λ2 ρ(r ) = 0
(3)
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
The key to solve (3) lies in the substitution : γ(r ) = r ρ(r ). Then γ 0 (r ) = r ρ0 (r ) + ρ(r ) and γ 00 (r ) = r ρ00 (r ) + 2ρ0 (r )
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
ρ00 (r ) + 2r ρ0 (r ) + λ2 ρ(r ) = 0
THE STURM-LIOUVILLE THEORY
(3)
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
The key to solve (3) lies in the substitution : γ(r ) = r ρ(r ). Then γ 0 (r ) = r ρ0 (r ) + ρ(r )
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
and γ 00 (r ) = r ρ00 (r ) + 2ρ0 (r ) so that γ 00 (r ) + λ2 γ(r ) = 0
(6)
ρ00 (r ) + 2r ρ0 (r ) + λ2 ρ(r ) = 0
THE STURM-LIOUVILLE THEORY
(3)
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
The key to solve (3) lies in the substitution : γ(r ) = r ρ(r ). Then γ 0 (r ) = r ρ0 (r ) + ρ(r )
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
and γ 00 (r ) = r ρ00 (r ) + 2ρ0 (r ) so that γ 00 (r ) + λ2 γ(r ) = 0 Thus equation (6) have the general solution : γ(r ) = A2 e iλr + A3 e −iλr with A2 and A3 arbitrary constants.
(6)
ρ00 (r ) + 2r ρ0 (r ) + λ2 ρ(r ) = 0
THE STURM-LIOUVILLE THEORY
(3)
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
The key to solve (3) lies in the substitution : γ(r ) = r ρ(r ). Then γ 0 (r ) = r ρ0 (r ) + ρ(r )
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
and γ 00 (r ) = r ρ00 (r ) + 2ρ0 (r ) so that γ 00 (r ) + λ2 γ(r ) = 0
(6)
Thus equation (6) have the general solution : γ(r ) = A2 e iλr + A3 e −iλr with A2 and A3 arbitrary constants. i.e :
1 ρ(r ) = (A2 e iλr + A3 e −iλr ) r
!
γ 00 (r ) + λ2 γ(r ) = 0
(6) =⇒
In addition for (6) we have :
γ(r ) = A2 e iλr + A3 e −iλr
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
γ 00 (r ) + λ2 γ(r ) = 0
(6) =⇒
γ(r ) = A2 e iλr + A3 e −iλr
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
In addition for (6) we have : | ρ(r ) |≤ M
∀r ∈]0, R]
THE STURM-LIOUVILLE THEORY
(3b) gives :
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
γ(r ) −→ 0 as r −→ 0+
γ 00 (r ) + λ2 γ(r ) = 0
(6) =⇒
γ(r ) = A2 e iλr + A3 e −iλr
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
In addition for (6) we have : | ρ(r ) |≤ M
∀r ∈]0, R]
THE STURM-LIOUVILLE THEORY
(3b) gives :
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
γ(r ) −→ 0 as r −→ 0+ Thus :
A2 + A3 = 0
so that
γ(r ) = A4 sin λr .
γ 00 (r ) + λ2 γ(r ) = 0
(6) =⇒
γ(r ) = A2 e iλr + A3 e −iλr
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
In addition for (6) we have : | ρ(r ) |≤ M
∀r ∈]0, R]
THE STURM-LIOUVILLE THEORY
(3b) gives :
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
γ(r ) −→ 0 as r −→ 0+ Thus :
A2 + A3 = 0
ρ0 (R) = −hρ(R)
so that
γ(r ) = A4 sin λr .
(3a) gives :
Rγ 0 (R) = (1 − Rh)γ(R)
γ 00 (r ) + λ2 γ(r ) = 0
(6) =⇒
γ(r ) = A2 e iλr + A3 e −iλr
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
In addition for (6) we have : | ρ(r ) |≤ M
∀r ∈]0, R]
THE STURM-LIOUVILLE THEORY
(3b) gives :
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
γ(r ) −→ 0 as r −→ 0+ Thus :
A2 + A3 = 0
ρ0 (R) = −hρ(R)
so that
γ(r ) = A4 sin λr .
(3a) gives :
Rγ 0 (R) = (1 − Rh)γ(R) Thus : Rλ cos λR = (1 − Rh) sin λR
γ 00 (r ) + λ2 γ(r ) = 0
(6) =⇒
γ(r ) = A2 e iλr + A3 e −iλr
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
In addition for (6) we have : | ρ(r ) |≤ M
∀r ∈]0, R]
THE STURM-LIOUVILLE THEORY
Fourier-Sturm-Liouville
(3b) gives :
Cooling of a sphere Cooling of a cylinder
γ(r ) −→ 0 as r −→ 0+ Thus :
A2 + A3 = 0
ρ0 (R) = −hρ(R)
so that
γ(r ) = A4 sin λr .
(3a) gives :
Rγ 0 (R) = (1 − Rh)γ(R) Thus : Rλ cos λR = (1 − Rh) sin λR That is why λ must be chosen to be a root of Rλ = 1 − Rh tan Rλ
(7)
∂2θ ∂r 2
Now we have :
+
2 ∂θ r ∂t
= K −1 ∂θ ∂t
(1)
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
∂2θ ∂r 2
+
2 ∂θ r ∂t
Now we have :
= K −1 ∂θ ∂t
(1)
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere
τ (t) = A1 exp(−λ2 Kt)
Cooling of a cylinder
∂2θ ∂r 2
+
2 ∂θ r ∂t
Now we have :
= K −1 ∂θ ∂t
(1)
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere
τ (t) = A1 exp(−λ2 Kt) ρ(r ) = A4 1r sin λr
Cooling of a cylinder
∂2θ ∂r 2
+
2 ∂θ r ∂t
= K −1 ∂θ ∂t
(1)
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
Now we have :
Fourier-Sturm-Liouville Cooling of a sphere
τ (t) = A1 exp(−λ2 Kt) ρ(r ) = A4 1r sin λr
Cooling of a cylinder
Direct substitution shows that, indeed, θ(r , t) =
1 exp(−λ2 Kt) sin λr r
is a solution of (1) whenever λ is a root of : Rλ = 1 − Rh tan Rλ
(7)
THE STURM-LIOUVILLE THEORY
Rλ tan Rλ
= 1 − Rh
(7)
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
Fourier now notes that (7) has an infinity of real roots : λ1 , λ2 ,. . ., and proposes that all solutions of the problem have the form : θ(r , t) =
∞ X an n=1
r
exp(−λ2n Kt) sin λn r
Thus if the initial heat distribution is given by : θ(r , 0) = f (r )
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
THE STURM-LIOUVILLE THEORY
Thus if the initial heat distribution is given by :
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
θ(r , 0) = f (r )
Fourier-Sturm-Liouville
we must be able to find a1 , a2 , . . .with rf (r ) =
∞ X n=1
Cooling of a sphere Cooling of a cylinder
an sin λn r
(8)
THE STURM-LIOUVILLE THEORY
Thus if the initial heat distribution is given by :
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
θ(r , 0) = f (r )
Fourier-Sturm-Liouville
we must be able to find a1 , a2 , . . .with rf (r ) =
∞ X
Cooling of a sphere Cooling of a cylinder
an sin λn r
(8)
n=1
In addition, Fourier remarks that if λm and λn are roots of (7) then : Rλn Rλm = 1 − Rh = tan Rλn tan Rλm
THE STURM-LIOUVILLE THEORY
Thus if the initial heat distribution is given by :
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
θ(r , 0) = f (r )
Fourier-Sturm-Liouville
we must be able to find a1 , a2 , . . .with rf (r ) =
∞ X
Cooling of a sphere Cooling of a cylinder
an sin λn r
(8)
n=1
In addition, Fourier remarks that if λm and λn are roots of (7) then : Rλn Rλm = 1 − Rh = tan Rλn tan Rλm and so λn sin λm R cos λn R − λm sin λn R cos λm R = 0
THE STURM-LIOUVILLE THEORY
Whence, if λm 6= λn Z
R
1 sin (λn − λm )r sin (λn + λm )r − 2 λn − λ m λn + λ m 0 1 sin λn R cos λm R − sin λm R cos λn R = 2 λn − λm sin λm R cos λn R + sin λn R cos λm R − λn + λ m λn sin λm R cos λn R − λm sin λn R cos λm R = λ2n − λ2m =0
sin λn r sin λm r dr = 0
R
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
THE STURM-LIOUVILLE THEORY
Whence, if λm 6= λn R
Z
R
1 sin (λn − λm )r sin (λn + λm )r − 2 λn − λ m λn + λ m 0 1 sin λn R cos λm R − sin λm R cos λn R = 2 λn − λm sin λm R cos λn R + sin λn R cos λm R − λn + λ m λn sin λm R cos λn R − λm sin λn R cos λm R = λ2n − λ2m =0
sin λn r sin λm r dr = 0
whilst Z Z R 1 R 2 (1 − cos 2λn r )dr = (sin λn r ) dr = 2 0 0
R sin 2λn R − 2 4λn
!
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
rf (r ) =
∞ X n=1
THE STURM-LIOUVILLE THEORY
an sin λn r
(8)
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
rf (r ) =
∞ X
THE STURM-LIOUVILLE THEORY
an sin λn r
(8)
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
n=1
Z
Fourier-Sturm-Liouville
R
Cooling of a sphere
sin λn r sin λm r dr = 0
Cooling of a cylinder
0
Z
R 2
(sin λn r ) dr = 0
R sin 2λn R − 2 4λn
!
rf (r ) =
∞ X
THE STURM-LIOUVILLE THEORY
an sin λn r
(8)
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
n=1
Z
Fourier-Sturm-Liouville
R
Cooling of a sphere
sin λn r sin λm r dr = 0
Cooling of a cylinder
0
Z
R 2
(sin λn r ) dr = 0
R sin 2λn R − 2 4λn
!
Thus multiplying both sides of (8) by sin λm r and integrating term by term we get Z
R
rf (r ) sin λm r dr = 0
∞ X n=1
Z
R
sin λn r sin λm r dr
an 0
rf (r ) =
∞ X
THE STURM-LIOUVILLE THEORY
an sin λn r
(8)
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
n=1
Z
Fourier-Sturm-Liouville
R
Cooling of a sphere
sin λn r sin λm r dr = 0
Cooling of a cylinder
0
Z
R 2
(sin λn r ) dr = 0
R sin 2λn R − 2 4λn
!
Thus multiplying both sides of (8) by sin λm r and integrating term by term we get Z
R
rf (r ) sin λm r dr = 0
∞ X n=1
= am
Z
R
sin λn r sin λm r dr
an 0
R sin 2λn R − 2 4λn
!
THE STURM-LIOUVILLE THEORY
so that (if our arguments are valid) , Z R
am = 2
rf (r ) sin λm r dr 0
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
sin 2λn R R− 2λn
!
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
THE STURM-LIOUVILLE THEORY
so that (if our arguments are valid) , Z R
am = 2
rf (r ) sin λm r dr 0
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
sin 2λn R R− 2λn
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
Critism : Fourier had not shown that all the roots of Rλ = 1 − Rh tan Rλ are real.
!
THE STURM-LIOUVILLE THEORY
so that (if our arguments are valid) , Z R
am = 2
rf (r ) sin λm r dr 0
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy
sin 2λn R R− 2λn
!
Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
Critism : Fourier had not shown that all the roots of Rλ = 1 − Rh tan Rλ are real. Indeed, when we notice that if λ is complex then 1 exp(−λ2 Kt) sin λr r will be oscillatory as t increases giving a physically implausible solution.
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
Cooling of a cylinder. We consider the specific problem of the axially symmetric flow of heat in a homogeneous cylinder of radius R.
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
Cooling of a cylinder. We consider the specific problem of the axially symmetric flow of heat in a homogeneous cylinder of radius R. Here the heat equation becomes :
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere
2
∂ θ 1 ∂θ ∂θ = K −1 + ∂r 2 r ∂r ∂t
Cooling of a cylinder
Cooling of a cylinder. We consider the specific problem of the axially symmetric flow of heat in a homogeneous cylinder of radius R. Here the heat equation becomes :
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere
2
∂ θ 1 ∂θ ∂θ = K −1 + ∂r 2 r ∂r ∂t and we have the boundary conditions ∂θ = −hθ(R, t) ∂r r =R
Cooling of a cylinder
Cooling of a cylinder. We consider the specific problem of the axially symmetric flow of heat in a homogeneous cylinder of radius R. Here the heat equation becomes :
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere
2
∂ θ 1 ∂θ ∂θ = K −1 + ∂r 2 r ∂r ∂t and we have the boundary conditions ∂θ = −hθ(R, t) ∂r r =R Separation of variables, we take θ(r , t) = ρ(r )τ (t)
Cooling of a cylinder
THE STURM-LIOUVILLE THEORY
Cooling of a cylinder. We consider the specific problem of the axially symmetric flow of heat in a homogeneous cylinder of radius R. Here the heat equation becomes :
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere
2
Cooling of a cylinder
∂ θ 1 ∂θ ∂θ = K −1 + ∂r 2 r ∂r ∂t and we have the boundary conditions ∂θ = −hθ(R, t) ∂r r =R Separation of variables, we take θ(r , t) = ρ(r )τ (t) and obtain r ρ00 (r ) + ρ0 (r ) + λ2 r ρ(r ) = 0
(9)
K −1 τ 0 (t) + λ2 τ (t) = 0
(10)
0
ρ (R) = −hρ(R) | ρ0 (r ) | , | ρ(r ) |≤ M
[0 < r ≤ R]
(11) (12)
In this case Fourier was able to show that there exists a sequence of real numbers λ0 < λ1 < . . . such that : if λ = λj , then (9), (11) and (12) can be simultaneously satisfied by a fonction uj .
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
In this case Fourier was able to show that there exists a sequence of real numbers λ0 < λ1 < . . . such that : if λ = λj , then (9), (11) and (12) can be simultaneously satisfied by a fonction uj . He showed further that : Z R rum (r )un (r )dr = 0 for m 6= n 0
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
In this case Fourier was able to show that there exists a sequence of real numbers λ0 < λ1 < . . . such that : if λ = λj , then (9), (11) and (12) can be simultaneously satisfied by a fonction uj . He showed further that : Z R rum (r )un (r )dr = 0 for m 6= n 0
suggesting the expansion f (r ) =
∞ X
an un (r )
n=0
with Z
R
Z rf (r )um (r )dr = am
0
0
R
rum (r )2 dr
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
In this case Fourier was able to show that there exists a sequence of real numbers λ0 < λ1 < . . . such that : if λ = λj , then (9), (11) and (12) can be simultaneously satisfied by a fonction uj . He showed further that : Z R rum (r )un (r )dr = 0 for m 6= n 0
suggesting the expansion f (r ) =
∞ X
an un (r )
n=0
with Z
R
Z rf (r )um (r )dr = am
0
R
rum (r )2 dr
0
(We can recognise the um as Bessel fonctions.)
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
Second part :
THE STURM-LIOUVILLE THEORY PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
The Sturm-Liouville Theory
THE STURM-LIOUVILLE THEORY
Second part :
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder
The Sturm-Liouville Theory
by...
THE STURM-LIOUVILLE THEORY
Mehdi
PERRIN Mathilde, COOBAR Mehdi, MOUGIN Davy Fourier-Sturm-Liouville Cooling of a sphere Cooling of a cylinder