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